A projection-based error analysis of HDG methods

By Bernardo Cockburn, Jayadeep Gopalakrishnan, Francisco-Javier Sayas

Abstract

We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG method applied to a model second-order elliptic problem.

1. Introduction

This paper is dedicated to presenting a new technique for the error analysis of an emerging class of numerical methods called hybridizable discontinuous Galerkin (HDG) methods Reference 11. The main idea is to devise a projection that matches the form of the numerical traces of the HDG method. The analysis of the projection of the error then becomes simple, concise and independent of the particular choice of the stabilization parameters.

We do not aim for maximal generality, but rather to convey the advantages of our technique compared to other DG analyses. Hence we only consider one specific HDG method, namely, the LDG-hybridizable (LDG-H) method Reference 11 for the model problem

StartLayout 1st Row with Label left-parenthesis 1.1 a right-parenthesis EndLabel 1st Column c bold-italic q plus bold nabla u 2nd Column equals 0 3rd Column Blank 4th Column in normal upper Omega comma 2nd Row with Label left-parenthesis 1.1 b right-parenthesis EndLabel 1st Column nabla dot bold-italic q 2nd Column equals f 3rd Column Blank 4th Column in normal upper Omega comma 3rd Row with Label left-parenthesis 1.1 c right-parenthesis EndLabel 1st Column u 2nd Column equals g 3rd Column Blank 4th Column on partial-differential normal upper Omega comma EndLayout

where normal upper Omega is a Lipschitz polyhedral domain in double-struck upper R Superscript n ( n greater-than-or-equal-to 2 ). Here c colon normal upper Omega right-arrow from bar double-struck upper R Superscript n times n is a variable matrix-valued coefficient, which we assume to be symmetric and uniformly positive definite, f is in upper L squared left-parenthesis normal upper Omega right-parenthesis and g in upper H Superscript 1 slash 2 Baseline left-parenthesis partial-differential normal upper Omega right-parenthesis period

Let us put our result in historical perspective. To do that, we begin by describing the HDG method. Consider a partitioning of the domain normal upper Omega into elements upper K forming a mesh script upper T Subscript h satisfying the standard finite element conditions Reference 7; the faces of upper K are going to be denoted by upper F . This method yields a scalar approximation u Subscript h to u , a vector approximation bold-italic q Subscript h to bold-italic q , and a scalar approximation ModifyingAbove u With caret Subscript h to the trace of u on element boundaries, in spaces of the form

StartLayout 1st Row with Label left-parenthesis 1.2 a right-parenthesis EndLabel 1st Column bold-italic upper V Subscript h 2nd Column equals left-brace bold-italic v colon for all mesh elements upper K comma bold-italic v vertical-bar Subscript upper K Baseline element-of bold-italic upper V left-parenthesis upper K right-parenthesis right-brace comma 2nd Row with Label left-parenthesis 1.2 b right-parenthesis EndLabel 1st Column upper W Subscript h 2nd Column equals left-brace w colon for all mesh elements upper K comma w vertical-bar Subscript upper K Baseline element-of upper W left-parenthesis upper K right-parenthesis right-brace comma 3rd Row with Label left-parenthesis 1.2 c right-parenthesis EndLabel 1st Column upper M Subscript h 2nd Column equals left-brace mu colon for all mesh faces upper F comma mu vertical-bar Subscript upper F Baseline element-of upper M left-parenthesis upper F right-parenthesis right-brace comma EndLayout

respectively, where bold-italic upper V left-parenthesis upper K right-parenthesis , upper W left-parenthesis upper K right-parenthesis , and upper M left-parenthesis upper F right-parenthesis are finite dimensional spaces. The HDG approximations u Subscript h in upper W Subscript h , bold-italic q Subscript h in bold-italic upper V Subscript h , and the numerical trace ModifyingAbove u With caret Subscript h in upper M Subscript h , are determined by requiring that

StartLayout 1st Row with Label left-parenthesis 1.3 a right-parenthesis EndLabel 1st Column left-parenthesis c bold-italic q Subscript h Baseline comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Baseline minus left-parenthesis u Subscript h Baseline comma nabla dot bold-italic r right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle ModifyingAbove u With caret Subscript h Baseline comma bold-italic r dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals 0 comma 2nd Row with Label left-parenthesis 1.3 b right-parenthesis EndLabel 1st Column minus left-parenthesis bold-italic q Subscript h Baseline comma bold nabla w right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n comma w mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals left-parenthesis f comma w right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma 3rd Row with Label left-parenthesis 1.3 c right-parenthesis EndLabel 1st Column mathematical left-angle ModifyingAbove u With caret Subscript h Baseline comma mu mathematical right-angle Subscript partial-differential normal upper Omega 2nd Column equals mathematical left-angle g comma mu mathematical right-angle Subscript partial-differential normal upper Omega Baseline comma 4th Row with Label left-parenthesis 1.3 d right-parenthesis EndLabel 1st Column mathematical left-angle ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus partial-differential normal upper Omega 2nd Column equals 0 comma EndLayout

hold for all bold-italic r in bold-italic upper V Subscript h Baseline comma w element-of upper W Subscript h Baseline comma and mu element-of upper M Subscript h Baseline comma with a specific ModifyingAbove bold-italic q With caret Subscript h defined on partial-differential script upper T Subscript h Baseline equals StartSet partial-differential upper K colon upper K element-of script upper T Subscript h Baseline EndSet ; see Reference 11. Above and throughout, we use the notation

left-parenthesis v comma w right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts left-parenthesis v comma w right-parenthesis Subscript upper K Baseline and mathematical left-angle v comma w mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts mathematical left-angle v comma w mathematical right-angle Subscript partial-differential upper K Baseline comma

where we write left-parenthesis u comma v right-parenthesis Subscript upper D Baseline equals integral Underscript upper D Endscripts u v d x whenever upper D is a domain of double-struck upper R Superscript n , and mathematical left-angle u comma v mathematical right-angle Subscript upper D Baseline equals integral Underscript upper D Endscripts u v d x whenever upper D is a domain of double-struck upper R Superscript n minus 1 . For vector functions bold-italic v and bold-italic w , the notations are similarly defined with the integrand being the dot product bold-italic v dot bold-italic w . For HDG methods ModifyingAbove u With caret Subscript h is taken to be an unknown in upper M Subscript h , while the numerical trace ModifyingAbove bold-italic q With caret Subscript h is prescribed on partial-differential script upper T Subscript h in such a way that both bold-italic q Subscript h and u Subscript h can be eliminated from the above equations to give rise to a single equation for ModifyingAbove u With caret Subscript h ; see Reference 11. Thus the often made criticism that DG methods have too many unknowns does not apply to HDG methods. Additional advantages of HDG methods include the ability to postprocess to get higher order solutions as we shall see in Section 5.

In contrast, the DG methods of the last century for second-order elliptic problems use only the first two equations Equation 1.3aEquation 1.3b, together with specific prescriptions of both ModifyingAbove u With caret Subscript h and ModifyingAbove bold-italic q With caret Subscript h to define their approximations. For most such methods, the above mentioned elimination is not feasible. Moreover, their analysis is also very different from ours, as it does not require the use of any special projection. This can be seen in Reference 2, where most of the then known DG methods were analyzed in a single unifying framework. All of these methods were shown to converge with order k plus 1 in the scalar variable u and with order k in the flux bold-italic q . The order of convergence in u is optimal and that in bold-italic q sub-optimal since the methods use as local spaces bold-italic upper V left-parenthesis upper K right-parenthesis and upper W left-parenthesis upper K right-parenthesis the sets bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis and script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis , respectively, where bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis colon equals left-bracket script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis right-bracket Superscript n and script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis is the space of polynomials of total degree at most k .

Two particular DG methods of the local discontinuous Galerkin (LDG) type, fitting in the above-mentioned unifying framework, deserve special mention as they provide approximations to the flux which converge with better orders of convergence. Their analyses do use special projections. The first is defined in the one-dimensional case and provides an order of convergence for the approximate flux of k plus 1 ; see Reference 6. The second is an extension of that method to multiple dimensions. It uses Cartesian grids and takes as local spaces bold-italic upper V left-parenthesis upper K right-parenthesis and upper W left-parenthesis upper K right-parenthesis the sets bold-script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis and script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis , respectively, where bold-script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis colon equals left-bracket script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis right-bracket Superscript n and script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis is the space of polynomials of degree at most k in each variable. The order of convergence for the approximate flux can be proven to be k plus 1 slash 2 ; see Reference 13. In both cases, the special projections used in the analysis were carefully devised to capture the structure of the numerical fluxes ModifyingAbove u With caret Subscript h and ModifyingAbove bold-italic q With caret Subscript h .

Another example of a method requiring a special projection to carry out its analysis is the so-called minimal dissipation LDG method. This is a special LDG method whose penalization parameter is identically zero in all the interior faces; as a consequence, it cannot be analyzed as in Reference 2. However, by using a suitably defined projection, the lack of stabilization can be overcome and sharp error estimates can be obtained. The flux can be shown to converge with the sub-optimal but sharp order of k ; see Reference 8.

Now, let us consider the HDG methods for which

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel ModifyingAbove bold-italic q With caret Subscript h Baseline equals bold-italic q Subscript h Baseline plus tau left-parenthesis u Subscript h Baseline minus ModifyingAbove u With caret Subscript h Baseline right-parenthesis bold-italic n comma on partial-differential script upper T Subscript h Baseline comma EndLayout

for some nonnegative penalty function tau defined on partial-differential script upper T Subscript h which we assume to be constant on each face of the triangulation. As explained in Reference 11, these methods are called the LDG-hybridizable (LDG-H) methods because the above numerical trace is that of the LDG method applied separately on each mesh element upper K . The well-known hybridized versions of the Raviart-Thomas (RT) method Reference 18 and the Brezzi-Douglas-Marini (BDM) method Reference 5 can be considered to be HDG methods for which tau equals 0 ; their analyses are carried out by using the celebrated RT and BDM projections. The analysis of HDG methods for which tau is not identically equal to zero, has been carried out in Reference 12 also by using special projections. All of the above methods were proven to provide approximations to the flux converging with the optimal order of k plus 1 ; the first HDG method with this property was introduced in Reference 9.

In this paper, we provide a new approach to the error analysis of HDG methods. It is an alternative to the techniques in Reference 12 for general HDG methods, and those in Reference 9 for the particular method treated therein. We recover all the results of Reference 9Reference 12 and obtain new superconvergence results for the projection of the scalar variable with our new approach. The novelty of our analysis is the use of new projections bold-italic upper Pi Subscript upper V Baseline comma upper Pi Subscript upper W Baseline which are fitted to the structure of the numerical traces of the scheme in the sense that

upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis equals bold-italic upper Pi Subscript upper V Baseline bold-italic q dot bold-italic n plus tau left-parenthesis upper Pi Subscript upper W Baseline u minus upper P Subscript upper M Baseline u right-parenthesis on partial-differential script upper T Subscript h Baseline comma

where upper P Subscript upper M is the upper L squared -projection into upper M Subscript h ; cf. the expression for ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n from Equation 1.4.

Projections bold-italic upper Pi Subscript upper V Baseline comma upper Pi Subscript upper W Baseline tailored to the numerical traces have actually been widely used. Indeed, beginning with the simplest example, notice that the projections used to analyze both the RT and BDM methods capture the structure of its numerical fluxes by satisfying

StartLayout 1st Row 1st Column upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis 2nd Column equals bold-italic upper Pi Subscript upper V Baseline bold-italic q dot bold-italic n on partial-differential script upper T Subscript h Baseline comma EndLayout

This should be compared with the definition of their numerical trace Reference 11

StartLayout 1st Row 1st Column ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n 2nd Column equals bold-italic q Subscript h Baseline dot bold-italic n on partial-differential script upper T Subscript h Baseline period EndLayout

Besides the projections used to carry out the analysis of the DG methods in Reference 8, Reference 6 and Reference 13, we have the projections used in Reference 9 to analyze an HDG type method called single-face hybridizable (SFH) method. The name arises due to the fact that for every simplex upper K , the penalty function tau is nonzero just on a single face, say  upper F Subscript upper K . The projections bold-italic upper Pi Subscript upper V and upper Pi Subscript upper W used there are such that for each simplex upper K element-of script upper T Subscript h ,

StartLayout 1st Row 1st Column upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis equals 2nd Column bold-italic upper Pi Subscript upper V Baseline bold-italic q dot bold-italic n 3rd Column Blank 4th Column on partial-differential upper K minus upper F Subscript upper K Baseline comma 2nd Row 1st Column upper P Subscript upper M Baseline u equals 2nd Column upper Pi Subscript upper W Baseline u 3rd Column Blank 4th Column on upper F Subscript upper K Baseline period EndLayout

This is in accordance with the numerical traces of the SFH method given by

StartLayout 1st Row 1st Column ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n equals 2nd Column bold-italic q Subscript h Baseline dot bold-italic n 3rd Column Blank 4th Column on partial-differential upper K minus upper F Subscript upper K Baseline comma 2nd Row 1st Column ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n equals 2nd Column bold-italic q Subscript h Baseline dot bold-italic n plus tau left-parenthesis u Subscript h Baseline minus ModifyingAbove u With caret Subscript h Baseline right-parenthesis 3rd Column Blank 4th Column on upper F Subscript upper K Baseline comma EndLayout

The SFH method is also an HDG method of the LDG-H type, so our projection can also be used to analyze the SFH method. It is interesting to point out that the component bold-italic upper Pi Subscript upper V Baseline bold-italic q of the SFH projection introduced in Reference 9 and ours do coincide when tau is as above, whereas the component upper Pi Subscript upper W Baseline u does not, except when nabla dot bold-italic q is a polynomial of degree k minus 1 on each simplex upper K element-of script upper T Subscript h . The final estimates for the SFH method are the same with both approaches.

The remainder of this paper is organized as follows. In Section 2, we define the new projection upper Pi Subscript h and state its key properties. In Section 3, we use an energy argument to obtain an optimal error estimate for the approximate flux bold-italic q Subscript h and its numerical trace ModifyingAbove bold-italic q With caret Subscript h . In Section 4, we use a duality argument to obtain a superconvergence estimate of the projection of the error in u Subscript h and its numerical trace ModifyingAbove u With caret Subscript h . In Section 5 we discuss previously known element-by-element postprocessing of the flux and the scalar variable. They will result in further approximations bold-italic q Subscript h Superscript star and u Subscript h Superscript star with interesting properties. Finally, we conclude in Section 6 by summarizing our results and relating them to the previous work Reference 12. Proof of the properties of the projection have been gathered in the Appendix.

2. The projection

The main ingredient of our error analysis is a new projection upper Pi Subscript h into the product space bold-italic upper V Subscript h Baseline times upper W Subscript h . In this section, we introduce it and establish its properties. From now on, we are going to use the following local spaces:

bold-italic upper V left-parenthesis upper K right-parenthesis equals bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis comma upper W left-parenthesis upper K right-parenthesis equals script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis and upper M left-parenthesis upper F right-parenthesis equals script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis period

Also, we consider the stabilization function tau colon partial-differential script upper T Subscript h Baseline right-arrow double-struck upper R to be constant on each face.

The projected function is denoted by upper Pi Subscript h Baseline left-parenthesis bold-italic q comma u right-parenthesis or by left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q comma upper Pi Subscript upper W Baseline u right-parenthesis where bold-italic upper Pi Subscript upper V Baseline bold-italic q and upper Pi Subscript upper W Baseline u are the components of the projection in bold-italic upper V Subscript h and upper W Subscript h , respectively. The values of the projection on any simplex upper K are fixed by requiring that the components satisfy the equations

StartLayout 1st Row with Label left-parenthesis 2.1 a right-parenthesis EndLabel 1st Column left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold-italic v right-parenthesis Subscript upper K 2nd Column equals left-parenthesis bold-italic q comma bold-italic v right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column for all bold-italic v element-of bold-script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row with Label left-parenthesis 2.1 b right-parenthesis EndLabel 1st Column left-parenthesis upper Pi Subscript upper W Baseline u comma w right-parenthesis Subscript upper K 2nd Column equals left-parenthesis u comma w right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column for all w element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 3rd Row with Label left-parenthesis 2.1 c right-parenthesis EndLabel 1st Column mathematical left-angle bold-italic upper Pi Subscript upper V Baseline bold-italic q dot bold-italic n plus tau upper Pi Subscript upper W Baseline u comma mu mathematical right-angle Subscript upper F 2nd Column equals mathematical left-angle bold-italic q dot bold-italic n plus tau u comma mu mathematical right-angle Subscript upper F Baseline 3rd Column Blank 4th Column for all mu element-of script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis comma EndLayout

for all faces upper F of the simplex upper K . If k equals 0 , then Equation 2.1a and Equation 2.1b are vacuous and upper Pi Subscript h is defined solely by Equation 2.1c. Note that although we denoted the first component of the projection by bold-italic upper Pi Subscript upper V Baseline bold-italic q , it depends not just on bold-italic q , but rather on both bold-italic q and u , as we see from 2.1. The same is true for upper Pi Subscript upper W Baseline u . Hence the notation left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q comma upper Pi Subscript upper W Baseline u right-parenthesis for upper Pi Subscript h Baseline left-parenthesis bold-italic q comma u right-parenthesis is somewhat misleading, but its convenience outweighs this disadvantage.

The domain of upper Pi Subscript h is a subspace of upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript n times upper L squared left-parenthesis normal upper Omega right-parenthesis on which the right-hand sides of 2.1 are well defined. Indeed, all functions bold-italic q and u that are regular enough for their traces bold-italic q dot bold-italic n and u to be in upper L squared left-parenthesis partial-differential upper K right-parenthesis are in the domain of upper Pi Subscript h , for example, left-parenthesis bold-italic q comma u right-parenthesis element-of bold-italic upper H Superscript 1 Baseline left-parenthesis script upper T Subscript h Baseline right-parenthesis times upper H Superscript 1 Baseline left-parenthesis script upper T Subscript h Baseline right-parenthesis , where

upper H Superscript 1 Baseline left-parenthesis script upper T Subscript h Baseline right-parenthesis equals product Underscript upper K element-of script upper T Subscript h Baseline Endscripts upper H Superscript 1 Baseline left-parenthesis upper K right-parenthesis comma bold-italic upper H Superscript 1 Baseline left-parenthesis script upper T Subscript h Baseline right-parenthesis equals upper H Superscript 1 Baseline left-parenthesis script upper T Subscript h Baseline right-parenthesis Superscript n Baseline period

That the left-hand sides of 2.1 uniquely determine bold-italic upper Pi Subscript upper V Baseline bold-italic q and upper Pi Subscript upper W Baseline u is proved as part of the next theorem.

To state the theorem, we need to introduce additional notation and conventions. We use double-vertical-bar dot double-vertical-bar Subscript upper D to denote the upper L squared left-parenthesis upper D right-parenthesis -norm for any upper D . More generally, we denote the norm and seminorm on any Sobolev space upper X by double-vertical-bar dot double-vertical-bar Subscript upper X and StartAbsoluteValue dot EndAbsoluteValue Subscript upper X , respectively. Also, as is usual in finite element analysis, we restrict elements to have shape regularity, i.e., all mesh elements upper K under consideration satisfy h Subscript upper K Baseline slash rho Subscript upper K Baseline less-than-or-equal-to gamma , where h Subscript upper K Baseline equals d i a m left-parenthesis upper K right-parenthesis , rho Subscript upper K is the diameter of the largest ball contained in upper K , and gamma is a fixed constant. We use  upper C , with or without subscripts, to denote a generic constant, independent of the elements and functions involved in our inequalities (but it may depend on  gamma ). The value of  upper C at different occurrences may differ. While we absorb dependencies on c into upper C , the dependencies on tau will always be explicitly mentioned.

Our first result states that the projection upper Pi Subscript h is well defined and has reasonable approximation properties. See Appendix A for a detailed proof.

Theorem 2.1.

Suppose k greater-than-or-equal-to 0 , tau vertical-bar Subscript partial-differential upper K Baseline is nonnegative and tau Subscript upper K Superscript max Baseline colon equals max tau vertical-bar Subscript partial-differential upper K Baseline greater-than 0 . Then the system 2.1 is uniquely solvable for bold-italic upper Pi Subscript upper V Baseline bold-italic q Subscript h and  upper Pi Subscript upper W Baseline u . Furthermore, there is a constant upper C Subscript upper Pi independent of upper K and tau such that

StartLayout 1st Row 1st Column double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript upper K Baseline less-than-or-equal-to 2nd Column upper C h Subscript upper K Superscript script l Super Subscript bold-italic q Superscript plus 1 Baseline StartAbsoluteValue bold-italic q EndAbsoluteValue Subscript bold-italic upper H Sub Superscript script l Sub Super Subscript bold-italic q Sub Superscript plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline plus upper C h Subscript upper K Superscript script l Super Subscript u Superscript plus 1 Baseline tau Subscript upper K Superscript asterisk Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript u Sub Superscript plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline comma 2nd Row 1st Column double-vertical-bar upper Pi Subscript upper W Baseline u minus u double-vertical-bar Subscript upper K Baseline less-than-or-equal-to 2nd Column upper C h Subscript upper K Superscript script l Super Subscript u Superscript plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript u Sub Superscript plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline plus upper C StartFraction h Subscript upper K Superscript script l Super Subscript bold-italic q Superscript plus 1 Baseline Over tau Subscript upper K Superscript max Baseline EndFraction StartAbsoluteValue nabla dot bold-italic q EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript bold-italic q Sub Superscript Subscript left-parenthesis upper K right-parenthesis Baseline comma EndLayout

for script l Subscript u Baseline comma script l Subscript bold-italic q Baseline in left-bracket 0 comma k right-bracket . Here tau Subscript upper K Superscript asterisk Baseline colon equals max tau vertical-bar Subscript partial-differential upper K minus upper F Sub Superscript asterisk Baseline , where upper F Superscript asterisk is a face of upper K at which tau vertical-bar Subscript partial-differential upper K Baseline is maximum.

Note that if tau is of unit order, both approximation errors converge with the optimal order of k plus 1 , when the functions bold-italic q and u are smooth enough. If tau identical-to 1 , or if tau is bounded above and below uniformly by fixed constants, then a standard Bramble-Hilbert argument proves the order  k plus 1 approximation estimates of the theorem. However, to track the dependence of the constants on tau , and to compare with previous works, we need to perform a more careful analysis. This is done in Appendix A in elaborate detail. To make an immediate comparison, note that for the tau used in the SFH method Reference 9, tau Subscript upper K Superscript asterisk Baseline equals 0 and the approximation properties of bold-italic upper Pi Subscript upper V Baseline bold-italic q are independent of the stabilization function  tau .

We conclude this section with a property of the projection that we use critically in the error analysis of the method.

Proposition 2.1 (A weak commutativity property).

For any w in upper W Subscript h and any left-parenthesis bold-italic upper Phi comma upper Psi right-parenthesis in the domain of upper Pi Subscript h , we have

StartLayout 1st Row 1st Column left-parenthesis w comma nabla dot bold-italic upper Phi right-parenthesis Subscript upper K Baseline equals 2nd Column left-parenthesis w comma nabla dot bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript upper K Baseline plus mathematical left-angle w comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential upper K Baseline period EndLayout

Proof.

For any w element-of upper W Subscript h ,

StartLayout 1st Row 1st Column left-parenthesis w comma nabla dot bold-italic upper Phi right-parenthesis Subscript upper K 2nd Column equals minus left-parenthesis bold nabla w comma bold-italic upper Phi right-parenthesis Subscript upper K Baseline plus mathematical left-angle w comma bold-italic upper Phi dot bold-italic n mathematical right-angle Subscript partial-differential upper K Baseline 2nd Row 1st Column Blank 2nd Column equals minus left-parenthesis bold nabla w comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript upper K Baseline plus mathematical left-angle w comma bold-italic upper Phi dot bold-italic n mathematical right-angle Subscript partial-differential upper K Baseline 3rd Column Blank 4th Column by 2.1 a comma 3rd Row 1st Column Blank 2nd Column equals minus left-parenthesis bold nabla w comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript upper K Baseline plus mathematical left-angle w comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi dot bold-italic n plus tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential upper K Baseline 3rd Column Blank 4th Column by 2.1 c comma 4th Row 1st Column Blank 2nd Column equals left-parenthesis w comma nabla dot bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript upper K Baseline plus mathematical left-angle w comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential upper K Baseline period 3rd Column Blank 4th Column Blank 5th Column Blank EndLayout

3. Flux error estimates by an energy argument

The purpose of this section is to give error estimates under minimal regularity assumptions on the solution. Since the projection  upper Pi Subscript h is designed to fit the structure of the numerical trace, the equations satisfied by the projection of the errors have a form amenable to simple analysis, as we see next. We will need the upper L squared -orthogonal projection onto upper M Subscript h , which we denote by upper P Subscript upper M . Note that because tau is piecewise constant,

mathematical left-angle tau left-parenthesis upper P Subscript upper M Baseline u minus u right-parenthesis comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline equals 0 for all mu element-of upper M Subscript h Baseline comma

a fact that we will use repeatedly without explicit mention. The projection of the errors satisfy the following.

Lemma 3.1.

Let bold-italic epsilon Subscript h Superscript q Baseline equals bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q Subscript h , epsilon Subscript h Superscript u Baseline equals upper Pi Subscript upper W Baseline u minus u Subscript h , and epsilon Subscript h Superscript ModifyingAbove u With caret Baseline equals upper P Subscript upper M Baseline u minus ModifyingAbove u With caret Subscript h Baseline period Then

StartLayout 1st Row with Label left-parenthesis 3.1 a right-parenthesis EndLabel 1st Column left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Baseline minus left-parenthesis epsilon Subscript h Superscript u Baseline comma nabla dot bold-italic r right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma bold-italic r dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma 2nd Row with Label left-parenthesis 3.1 b right-parenthesis EndLabel 1st Column minus left-parenthesis bold-italic epsilon Subscript h Superscript q Baseline comma bold nabla w right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n comma w mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals 0 comma 3rd Row with Label left-parenthesis 3.1 c right-parenthesis EndLabel 1st Column mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma mu mathematical right-angle Subscript partial-differential normal upper Omega 2nd Column equals 0 comma 4th Row with Label left-parenthesis 3.1 d right-parenthesis EndLabel 1st Column mathematical left-angle ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus partial-differential normal upper Omega 2nd Column equals 0 comma EndLayout

for all bold-italic r element-of bold-italic upper V Subscript h Baseline comma w element-of upper W Subscript h Baseline comma and mu element-of upper M Subscript h , where

StartLayout 1st Row with Label left-parenthesis 3.1 e right-parenthesis EndLabel 1st Column ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n colon equals bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n plus tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis equals upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis 2nd Column minus ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n on partial-differential script upper T Subscript h Baseline minus partial-differential normal upper Omega period EndLayout

Proof.

Let us begin by noting that the exact solution bold-italic q and u satisfies

StartLayout 1st Row 1st Column left-parenthesis c bold-italic q comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Baseline minus left-parenthesis u comma nabla dot bold-italic r right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle u comma bold-italic r dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals 0 comma 2nd Row 1st Column minus left-parenthesis bold-italic q comma bold nabla w right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle bold-italic q dot bold-italic n comma w mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals left-parenthesis f comma w right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma EndLayout

for all bold-italic r element-of bold-italic upper V Subscript h and w element-of upper W Subscript h . By the definition of upper Pi Subscript h and upper P Subscript upper M , the above implies

StartLayout 1st Row 1st Column left-parenthesis c bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Baseline minus left-parenthesis upper Pi Subscript upper W Baseline u comma nabla dot bold-italic r right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle upper P Subscript upper M Baseline u comma bold-italic r dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic r right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma 2nd Row 1st Column minus left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold nabla w right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle bold-italic upper Pi Subscript upper V Baseline bold-italic q dot bold-italic n minus tau left-parenthesis u minus upper Pi Subscript upper W Baseline u right-parenthesis comma w mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Column equals left-parenthesis f comma w right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma EndLayout

for all bold-italic r element-of bold-italic upper V Subscript h and w element-of upper W Subscript h . Subtracting Equation 1.3a and Equation 1.3b from the above two equations, respectively, we obtain Equation 3.1a and Equation 3.1b. The equation Equation 3.1c follows directly from the boundary condition Equation 1.3c. To prove Equation 3.1d we proceed as follows:

StartLayout 1st Row 1st Column mathematical left-angle mu comma ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus partial-differential normal upper Omega 2nd Column equals mathematical left-angle left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis dot bold-italic n plus tau left-parenthesis upper Pi Subscript upper W Baseline u minus u Subscript h Baseline minus upper P Subscript upper M Baseline u plus ModifyingAbove u With caret Subscript h Baseline right-parenthesis comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus partial-differential normal upper Omega Baseline 2nd Row 1st Column Blank 2nd Column equals mathematical left-angle left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis dot bold-italic n plus tau left-parenthesis u minus u Subscript h Baseline minus u plus ModifyingAbove u With caret Subscript h Baseline right-parenthesis comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus partial-differential normal upper Omega Baseline 3rd Row 1st Column Blank 2nd Column equals mathematical left-angle bold-italic q dot bold-italic n comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript minus partial-differential normal upper Omega Baseline minus mathematical left-angle ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n comma mu mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript minus partial-differential normal upper Omega Baseline comma EndLayout

where we have used the definition of upper Pi Subscript h . Note that this proves the identity Equation 3.1e because ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n element-of upper M Subscript h . Since bold-italic q is in upper H left-parenthesis normal d normal i normal v comma normal upper Omega right-parenthesis and since ModifyingAbove bold-italic q With caret Subscript h satisfies Equation 1.3d, both terms above are zero. This completes the proof.

Lemma 3.2.

We have

left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline plus mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline equals left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline period

Proof.

Taking bold-italic r colon equals bold-italic epsilon Subscript h Superscript q in Equation 3.1a, w equals epsilon Subscript h Superscript u in Equation 3.1b, mu equals minus ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n in Equation 3.1c and mu equals minus epsilon Subscript h Superscript ModifyingAbove u With caret in Equation 3.1d, and adding the resulting four equations, we get

left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline plus normal upper Theta Subscript h Baseline equals left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline comma

where

StartLayout 1st Row 1st Column normal upper Theta Subscript h Baseline equals 2nd Column minus left-parenthesis epsilon Subscript h Superscript u Baseline comma nabla dot bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus left-parenthesis bold-italic epsilon Subscript h Superscript q Baseline comma bold nabla epsilon Subscript h Superscript u Baseline right-parenthesis Subscript script upper T Sub Subscript h 2nd Row 1st Column Blank 2nd Column plus mathematical left-angle ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n comma epsilon Subscript h Superscript u Baseline mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline minus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline period EndLayout

But, by integration by parts, we get

StartLayout 1st Row 1st Column normal upper Theta Subscript h Baseline equals 2nd Column minus mathematical left-angle epsilon Subscript h Superscript u Baseline comma bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n mathematical right-angle Subscript script upper T Sub Subscript h plus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Baseline plus mathematical left-angle ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n comma epsilon Subscript h Superscript u Baseline mathematical right-angle Subscript partial-differential script upper T Sub Subscript h minus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 2nd Row 1st Column equals 2nd Column minus mathematical left-angle bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n comma epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline mathematical right-angle Subscript script upper T Sub Subscript h plus mathematical left-angle ModifyingAbove bold-italic epsilon With caret Subscript h Baseline dot bold-italic n comma epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Row 1st Column equals 2nd Column mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline comma EndLayout

by the definition of ModifyingAbove bold-italic epsilon With caret Subscript h , namely Equation 3.1e. This completes the proof.

While Lemma 3.2 identifies an “energy” norm, Lemma 3.1 gives “consistency” relations. These are enough to immediately prove an error estimate for the flux. To state it, we need the norm double-vertical-bar dot double-vertical-bar Subscript h defined by double-vertical-bar mu double-vertical-bar Subscript h Superscript 2 Baseline equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts h Subscript upper K Baseline double-vertical-bar mu double-vertical-bar Subscript partial-differential upper K Superscript 2 for any function mu element-of upper L squared left-parenthesis partial-differential script upper T Subscript h Baseline right-parenthesis colon equals product Underscript upper K element-of script upper T Subscript h Endscripts upper L squared left-parenthesis partial-differential upper K right-parenthesis , and the c -weighted upper L squared left-parenthesis normal upper Omega right-parenthesis -norm double-vertical-bar bold-italic q double-vertical-bar Subscript c Baseline colon equals left-parenthesis c bold-italic q comma bold-italic q right-parenthesis Subscript script upper T Sub Subscript h Subscript Superscript 1 slash 2 Baseline period

Theorem 3.1 (Flux error estimates).

Let bold-italic q Subscript h Baseline comma u Subscript h Baseline comma and ModifyingAbove u With caret Subscript h solve the HDG equations Equation 1.3 and let the exact solution bold-italic q comma u be in the domain of upper Pi Subscript h . Then, for k greater-than-or-equal-to 0 ,

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel 1st Column double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q Subscript h Baseline double-vertical-bar Subscript c 2nd Column less-than-or-equal-to double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript c Baseline comma 2nd Row with Label left-parenthesis 3.3 right-parenthesis EndLabel 1st Column double-vertical-bar upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis minus ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n double-vertical-bar Subscript h 2nd Column less-than-or-equal-to upper C Subscript 1 comma tau Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript c Baseline comma EndLayout

where upper C Subscript 1 comma tau Baseline equals upper C max left-brace right-brace comma colon 1 comma left-parenthesis right-parenthesis times times hK tau Kmax slash slash 12 colon element-of element-of KTh .

Proof.

By the Cauchy-Schwarz inequality applied to the identity of Lemma 3.2, we get

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic epsilon Subscript h Superscript q Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline plus mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline less-than-or-equal-to double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript c Baseline double-vertical-bar bold-italic epsilon Subscript h Superscript q Baseline double-vertical-bar Subscript c Baseline comma EndLayout

and Equation 3.2 follows since tau greater-than-or-equal-to 0 . For Equation 3.3, we start by using the identity Equation 3.1e to get

StartLayout 1st Row 1st Column double-vertical-bar upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis minus ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n double-vertical-bar Subscript h 2nd Column less-than-or-equal-to double-vertical-bar bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n double-vertical-bar Subscript h Baseline plus double-vertical-bar tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis double-vertical-bar Subscript h Baseline 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C double-vertical-bar bold-italic epsilon Subscript h Superscript q Baseline double-vertical-bar Subscript c Baseline plus max Underscript upper K element-of script upper T Subscript h Baseline Endscripts left-parenthesis h Subscript upper K Baseline tau Subscript upper K Superscript max Baseline right-parenthesis Superscript 1 slash 2 Baseline mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Superscript 1 slash 2 EndLayout

where we used an inverse inequality and the fact that c Superscript negative 1 is uniformly bounded. Then Equation 3.3 follows from Equation 3.4.

4. Superconvergence of the scalar variableby a duality argument

In the previous section, we established an error estimate for bold-italic q Subscript h that only requires that the solution be as regular as required for the application of the projection. On domains permitting higher regularity estimates, we can perform an analogue of the Aubin-Nitsche duality argument Reference 3Reference 17 to get higher rates of convergence. In particular, such arguments will give us error estimates for u Subscript h and ModifyingAbove u With caret Subscript h .

We thus begin by introducing the dual problem for any given upper Theta in upper L squared left-parenthesis normal upper Omega right-parenthesis :

StartLayout 1st Row with Label left-parenthesis 4.1 a right-parenthesis EndLabel 1st Column c bold-italic upper Phi minus bold nabla upper Psi 2nd Column equals 0 3rd Column Blank 4th Column on normal upper Omega comma 2nd Row with Label left-parenthesis 4.1 b right-parenthesis EndLabel 1st Column nabla dot bold-italic upper Phi 2nd Column equals upper Theta 3rd Column Blank 4th Column on normal upper Omega comma 3rd Row with Label left-parenthesis 4.1 c right-parenthesis EndLabel 1st Column upper Psi 2nd Column equals 0 3rd Column Blank 4th Column on partial-differential normal upper Omega period EndLayout

We assume that this boundary value problem admits the regularity estimate

StartLayout 1st Row with Label left-parenthesis 4.2 right-parenthesis EndLabel double-vertical-bar bold-italic upper Phi double-vertical-bar Subscript upper H Sub Superscript 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline plus double-vertical-bar upper Psi double-vertical-bar Subscript upper H squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C Subscript normal r normal e normal g Baseline double-vertical-bar upper Theta double-vertical-bar Subscript normal upper Omega EndLayout

for all upper Theta in upper L squared left-parenthesis normal upper Omega right-parenthesis . This is well known to hold in several cases, e.g., if c identical-to 1 and normal upper Omega is a convex polygon Reference 16. Recall that we have been tacitly assuming that left-parenthesis bold-italic q comma u right-parenthesis is in the domain of upper Pi Subscript h . By Equation 4.2, left-parenthesis bold-italic upper Phi comma upper Psi right-parenthesis is also regular enough to apply upper Pi Subscript h , so we have the following lemma.

Lemma 4.1 (The duality argument).

For any upper Psi Subscript h Baseline element-of upper W Subscript h , we have

left-parenthesis epsilon Subscript h Superscript u Baseline comma upper Theta right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline equals left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline plus left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold nabla upper Psi minus bold nabla upper Psi Subscript h Baseline right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline period

Consequently,

double-vertical-bar epsilon Subscript h Superscript u Baseline double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline less-than-or-equal-to upper C upper H left-parenthesis upper Theta right-parenthesis double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline comma

where

upper H left-parenthesis upper Theta right-parenthesis colon equals sup Underscript upper Theta element-of upper L squared left-parenthesis normal upper Omega right-parenthesis minus 0 Endscripts StartFraction double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline Over double-vertical-bar upper Theta double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline EndFraction plus sup Underscript upper Theta element-of upper L squared left-parenthesis normal upper Omega right-parenthesis minus 0 Endscripts inf Underscript upper Psi Subscript h Baseline element-of upper W Subscript h Baseline Endscripts StartFraction double-vertical-bar bold nabla upper Psi minus bold nabla upper Psi Subscript h Baseline double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline Over double-vertical-bar upper Theta double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline EndFraction period

Proof.

We have

StartLayout 1st Row 1st Column left-parenthesis epsilon Subscript h Superscript u Baseline comma upper Theta right-parenthesis Subscript script upper T Sub Subscript h Baseline equals 2nd Column left-parenthesis epsilon Subscript h Superscript u Baseline comma nabla dot bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h 3rd Column Blank 4th Column by 4.1 b comma 2nd Row 1st Column equals 2nd Column left-parenthesis epsilon Subscript h Superscript u Baseline comma nabla dot bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis plus mathematical left-angle epsilon Subscript h Superscript u Baseline comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by Prop period 2.1 comma 3rd Row 1st Column equals 2nd Column left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 4th Row 1st Column Blank 2nd Column minus left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h plus mathematical left-angle epsilon Subscript h Superscript u Baseline comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by 3.1 a comma 5th Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi right-parenthesis dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 6th Row 1st Column Blank 2nd Column plus mathematical left-angle epsilon Subscript h Superscript u Baseline comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline comma EndLayout

by the continuity of bold-italic upper Phi dot bold-italic n and the fact that epsilon Subscript h Superscript ModifyingAbove u With caret Baseline equals 0 on partial-differential normal upper Omega by Equation 3.1c. Then

StartLayout 1st Row 1st Column left-parenthesis epsilon Subscript h Superscript u Baseline comma upper Theta right-parenthesis Subscript script upper T Sub Subscript h Baseline equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma tau left-parenthesis upper Pi Subscript upper W Baseline upper Psi minus upper Psi right-parenthesis mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by 2.1 c comma 2nd Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma upper Pi Subscript upper W Baseline upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Row 1st Column Blank 2nd Column minus mathematical left-angle epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline comma tau upper P Subscript upper M Baseline upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 4th Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma upper Pi Subscript upper W Baseline upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 5th Row 1st Column Blank 2nd Column minus mathematical left-angle bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n comma upper P Subscript upper M Baseline upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by 3.1 d comma 6th Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus mathematical left-angle tau left-parenthesis epsilon Subscript h Superscript u Baseline minus epsilon Subscript h Superscript ModifyingAbove u With caret Baseline right-parenthesis comma upper Pi Subscript upper W Baseline upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 7th Row 1st Column Blank 2nd Column minus mathematical left-angle bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n comma upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript Baseline period EndLayout

Moreover,

StartLayout 1st Row 1st Column left-parenthesis epsilon Subscript h Superscript u Baseline comma upper Theta right-parenthesis Subscript script upper T Sub Subscript h Baseline equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus left-parenthesis nabla dot bold-italic epsilon Subscript h Superscript q Baseline comma upper Pi Subscript upper W Baseline upper Psi right-parenthesis Subscript script upper T Sub Subscript h minus mathematical left-angle bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n comma upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by 3.1 b comma 2nd Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus left-parenthesis nabla dot bold-italic epsilon Subscript h Superscript q Baseline comma upper Psi right-parenthesis Subscript script upper T Sub Subscript h minus mathematical left-angle bold-italic epsilon Subscript h Superscript q Baseline dot bold-italic n comma upper Psi mathematical right-angle Subscript partial-differential script upper T Sub Subscript h 3rd Column Blank 4th Column by 2.1 b comma 3rd Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline minus left-parenthesis bold-italic epsilon Subscript h Superscript q Baseline comma bold nabla upper Psi right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline period EndLayout

To bring this into the needed form, we continue:

StartLayout 1st Row 1st Column left-parenthesis epsilon Subscript h Superscript u Baseline comma upper Theta right-parenthesis Subscript script upper T Sub Subscript h Baseline equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h 2nd Row 1st Column Blank 2nd Column minus left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q Subscript h Baseline comma bold nabla upper Psi right-parenthesis Subscript script upper T Sub Subscript h 3rd Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold nabla upper Psi right-parenthesis Subscript script upper T Sub Subscript h 3rd Column Blank 4th Column by 4.1 a comma 4th Row 1st Column equals 2nd Column left-parenthesis c left-parenthesis bold-italic q minus bold-italic q Subscript h Baseline right-parenthesis comma bold-italic upper Pi Subscript upper V Baseline bold-italic upper Phi minus bold-italic upper Phi right-parenthesis Subscript script upper T Sub Subscript h Baseline plus left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q comma bold nabla upper Psi minus bold nabla upper Psi Subscript h Baseline right-parenthesis Subscript script upper T Sub Subscript h 3rd Column Blank 4th Column by 2.1 a period EndLayout

Finally, the inequality of the lemma follows by applying the Cauchy-Schwarz inequality to the identity and using the first estimate of Theorem 3.1.

Theorem 4.1.

Suppose the regularity assumption Equation 4.2 holds. Then

StartLayout 1st Row 1st Column double-vertical-bar upper Pi Subscript upper W Baseline u minus u Subscript h Baseline double-vertical-bar Subscript script upper T Sub Subscript h 2nd Column less-than-or-equal-to upper C Subscript 2 comma tau Baseline h Superscript min left-brace right-brace comma k comma 1 Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Baseline 3rd Column Blank 4th Column for k greater-than-or-equal-to 0 comma 2nd Row 1st Column double-vertical-bar upper P Subscript upper M Baseline u minus ModifyingAbove u With caret Subscript h Baseline double-vertical-bar Subscript h 2nd Column less-than-or-equal-to upper C Subscript 2 comma tau Baseline h double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Baseline 3rd Column Blank 4th Column for k greater-than-or-equal-to 1 comma EndLayout

where h equals max left-brace right-brace colon hK colon element-of element-of KTh and upper C Subscript 2 comma tau Baseline equals upper C max left-brace right-brace comma colon 1 comma times times hK tau upper K asterisk colon element-of element-of KTh .

Proof.

Let us prove the first inequality. By the first estimate of Theorem 2.1 with script l Subscript bold-italic q set to 0 and script l Subscript u Baseline equals min left-brace right-brace comma k comma 1 , we get that

StartLayout 1st Row 1st Column upper H left-parenthesis upper Theta right-parenthesis 2nd Column less-than-or-equal-to upper C h Superscript min left-brace right-brace comma k comma 1 Baseline upper C Subscript 2 comma tau Baseline sup Underscript upper Theta element-of upper L squared left-parenthesis normal upper Omega right-parenthesis minus 0 Endscripts StartFraction double-vertical-bar bold-italic upper Phi double-vertical-bar Subscript upper H Sub Superscript 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline plus double-vertical-bar upper Psi double-vertical-bar Subscript upper H Sub Superscript script l Sub Super Subscript u Sub Superscript plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline Over double-vertical-bar upper Theta double-vertical-bar Subscript normal upper Omega Baseline EndFraction 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C h Superscript min left-brace right-brace comma k comma 1 Baseline upper C Subscript 2 comma tau Baseline upper C Subscript normal r normal e normal g Baseline period EndLayout

by the regularity estimate Equation 4.2, and the first estimate follows.

The second estimate follows from the first from the same local argument used in Reference 5 to obtain a similar estimate for the BDM method. Indeed, when k greater-than-or-equal-to 1 , we can select a function bold-italic r element-of bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis such that bold-italic r dot bold-italic n equals epsilon Subscript h Superscript ModifyingAbove u With caret on partial-differential upper K and double-vertical-bar bold-italic r double-vertical-bar Subscript upper K Baseline less-than-or-equal-to upper C h Subscript upper K Superscript 1 slash 2 Baseline double-vertical-bar epsilon Subscript h Superscript ModifyingAbove u With caret Baseline double-vertical-bar Subscript partial-differential upper K Baseline period Using h Subscript upper K Baseline bold-italic r as the test function in Equation 3.1a, and applying an inverse inequality, we find that

StartLayout 1st Row 1st Column h Subscript upper K Baseline double-vertical-bar epsilon Subscript h Superscript ModifyingAbove u With caret Baseline double-vertical-bar Subscript partial-differential upper K Superscript 2 2nd Column equals h Subscript upper K Baseline left-parenthesis c left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q right-parenthesis comma bold-italic r right-parenthesis Subscript upper K Baseline plus h Subscript upper K Baseline left-parenthesis epsilon Subscript h Superscript u Baseline comma nabla dot bold-italic r right-parenthesis Subscript upper K Baseline minus h Subscript upper K Baseline left-parenthesis c bold-italic epsilon Subscript h Superscript q Baseline comma bold-italic r right-parenthesis Subscript upper K Baseline 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to upper C h Subscript upper K Baseline double-vertical-bar bold-italic r double-vertical-bar Subscript upper K Baseline left-parenthesis double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript upper K Baseline plus double-vertical-bar bold-italic epsilon Subscript h Superscript q Baseline double-vertical-bar Subscript upper K Baseline right-parenthesis plus upper C double-vertical-bar bold-italic r double-vertical-bar Subscript upper K Baseline double-vertical-bar epsilon Subscript h Superscript u Baseline double-vertical-bar Subscript upper K Baseline period EndLayout

Applying the first estimate of the theorem for epsilon Subscript h Superscript u and the estimate in Theorem 3.1 for bold-italic epsilon Subscript h Superscript q , we get the second inequality of the theorem. This completes the proof.

5. Enhanced accuracy by postprocessing

In this section we describe a few techniques to postprocess the approximate solution and flux.

5.1. Flux postprocessing

We can obtain a postprocessed flux bold-italic q Subscript h Superscript star with better conservation properties. Although bold-italic q Subscript h Superscript star converges at the same order as bold-italic q Subscript h , it is in upper H left-parenthesis normal d normal i normal v comma normal upper Omega right-parenthesis and its divergence converges at one higher order than bold-italic q Subscript h .

To define bold-italic q Subscript h Superscript star , we use a slight modification of the Raviart-Thomas projection Reference 18, as used in the framework of Darcy flows Reference 4, or for the Navier-Stokes equations Reference 14, or for the postprocessing of superconvergent DG methods for second-order elliptic problems Reference 12. We follow Reference 12. Thus, on each simplex upper K element-of script upper T Subscript h , we take bold-italic q Subscript h Superscript star Baseline colon equals bold-italic q Subscript h plus bold-italic eta Subscript h where bold-italic eta Subscript h is the only element of bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis plus bold-italic x script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis satisfying

StartLayout 1st Row 1st Column left-parenthesis bold-italic eta Subscript h Baseline comma bold-italic v right-parenthesis Subscript upper K 2nd Column equals 0 3rd Column Blank 4th Column for all bold-italic v element-of bold-script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row 1st Column mathematical left-angle bold-italic eta Subscript h Baseline dot bold-italic n comma mu mathematical right-angle Subscript upper F 2nd Column equals mathematical left-angle left-parenthesis ModifyingAbove bold-italic q With caret Subscript h Baseline minus bold-italic q Subscript h Baseline right-parenthesis dot bold-italic n comma mu mathematical right-angle Subscript upper F Baseline 3rd Column Blank 4th Column for all mu element-of script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis and all faces upper F of upper K period EndLayout

If upper L squared left-parenthesis upper K right-parenthesis -orthogonal hierarchical basis functions are used, computation of bold-italic eta Subscript h reduces to solving, on each element, a linear system of size n plus 1 times the dimension of the space of homogeneous polynomials of degree k . It is instructive to compare the above definition to that of the Raviart-Thomas projection, namely on each simplex upper K element-of script upper T Subscript h , we set bold-italic upper Pi Subscript k Superscript normal upper R normal upper T Baseline bold-italic q as the only element of bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis plus bold-italic x script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis satisfying

StartLayout 1st Row 1st Column left-parenthesis bold-italic upper Pi Subscript k Superscript normal upper R normal upper T Baseline bold-italic q minus bold-italic q comma bold-italic v right-parenthesis Subscript upper K 2nd Column equals 0 3rd Column Blank 4th Column for all bold-italic v element-of bold-script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row 1st Column mathematical left-angle left-parenthesis bold-italic upper Pi Subscript k Superscript normal upper R normal upper T Baseline bold-italic q minus bold-italic q right-parenthesis dot bold-italic n comma mu mathematical right-angle Subscript upper F 2nd Column equals 0 comma 3rd Column Blank 4th Column for all mu element-of script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis and all faces upper F of upper K period EndLayout

This comparison yields the following theorem. For a proof, see Reference 12.

Theorem 5.1.

For any k greater-than-or-equal-to 0 , we have that bold-italic q Subscript h Superscript star Baseline element-of upper H left-parenthesis div comma normal upper Omega right-parenthesis . Moreover,

StartLayout 1st Row 1st Column double-vertical-bar bold-italic q minus bold-italic q Subscript h Superscript star Baseline double-vertical-bar Subscript normal upper Omega 2nd Column less-than-or-equal-to upper C double-vertical-bar bold-italic q minus bold-italic upper Pi Subscript k Superscript normal upper R normal upper T Baseline bold-italic q double-vertical-bar Subscript normal upper Omega Baseline plus upper C Subscript 1 comma tau Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline comma 2nd Row 1st Column double-vertical-bar nabla dot left-parenthesis bold-italic q minus bold-italic q Subscript h Superscript star Baseline right-parenthesis double-vertical-bar Subscript normal upper Omega 2nd Column equals inf Underscript f Subscript h Baseline element-of upper W Subscript h Baseline Endscripts double-vertical-bar f minus f Subscript h Baseline double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline period EndLayout

5.2. Postprocessing the approximate scalar variable

There are a few well-known ways to postprocess to obtain a new approximation  u Subscript h Superscript star of enhanced accuracy.

As the first postprocessing method, we define u Subscript h comma 1 Superscript star in script upper P Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis satisfying

StartLayout 1st Row with Label left-parenthesis 5.1 a right-parenthesis EndLabel 1st Column left-parenthesis c Superscript negative 1 Baseline bold nabla u Subscript h comma 1 Superscript star Baseline comma bold nabla w right-parenthesis Subscript upper K 2nd Column equals left-parenthesis f comma w right-parenthesis Subscript upper K Baseline minus mathematical left-angle bold-italic q Subscript h Baseline dot bold-italic n comma w mathematical right-angle Subscript partial-differential upper K Baseline comma 3rd Column Blank 4th Column for all w element-of script upper P Subscript k plus 1 Superscript 0 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row with Label left-parenthesis 5.1 b right-parenthesis EndLabel 1st Column m Subscript upper K Baseline left-parenthesis u Subscript h Superscript star Baseline right-parenthesis 2nd Column equals m Subscript upper K Baseline left-parenthesis u Subscript h Baseline right-parenthesis comma EndLayout

where m Subscript upper K Baseline left-parenthesis v right-parenthesis equals m e a s left-parenthesis upper K right-parenthesis Superscript negative 1 Baseline left-parenthesis v comma 1 right-parenthesis Subscript upper K and script upper P Subscript k plus 1 Superscript 0 Baseline left-parenthesis upper K right-parenthesis equals StartSet p element-of script upper P Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis colon m Subscript upper K Baseline left-parenthesis p right-parenthesis equals 0 EndSet period Clearly, u Subscript h comma 1 Superscript star satisfies a discrete Neumann problem on each element with the computed approximate solution as data. This method was introduced in Reference 15Reference 19Reference 20 in the framework of mixed methods.

A variation of this postprocessing was proposed in Reference 9Reference 12 for DG methods. It is obtained simply by substituting ModifyingAbove bold-italic q With caret Subscript h in place of bold-italic q Subscript h on the right-hand side of Equation 5.1a. This yields our second postprocessing alternative. The solution so obtained is denoted by  u Subscript h comma 2 Superscript star .

As a third alternative, we define u Subscript h comma 3 Superscript star obtained by following Reference 19Reference 20. Let script upper W Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis denote the upper L squared left-parenthesis upper K right-parenthesis -orthogonal complement of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis in script upper P Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis . The solution u Subscript h comma 3 Superscript star is of the form u Subscript h Baseline plus eta Subscript h where eta Subscript h is the unique function in script upper W Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis satisfying

StartLayout 1st Row with Label left-parenthesis 5.2 right-parenthesis EndLabel 1st Column left-parenthesis bold nabla eta Subscript h Baseline comma bold nabla w right-parenthesis Subscript upper K Baseline equals 2nd Column minus left-parenthesis bold nabla u Subscript h Baseline plus c bold-italic q Subscript h Baseline comma bold nabla w right-parenthesis Subscript upper K 3rd Column Blank 4th Column for all w element-of script upper W Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis period EndLayout

As in the case of the flux postprocessing, if we are using an upper L squared left-parenthesis upper K right-parenthesis -orthogonal hierarchical basis to find eta Subscript h , we need only invert a symmetric, positive definite matrix whose order is the dimension of script upper W Subscript k plus 1 Baseline left-parenthesis upper K right-parenthesis . Note also that to evaluate the right-hand side of Equation 5.2, we need only use n minus 1 -dimensional quadratures, as

minus left-parenthesis bold nabla u Subscript h Baseline plus c bold-italic q Subscript h Baseline comma bold nabla w right-parenthesis Subscript script upper T Sub Subscript h Subscript Baseline equals mathematical left-angle ModifyingAbove u With caret Subscript h Baseline minus u Subscript h Baseline comma bold nabla w dot bold-italic n mathematical right-angle Subscript partial-differential script upper T Sub Subscript h Subscript minus partial-differential normal upper Omega Baseline plus mathematical left-angle g minus u Subscript h Baseline comma bold nabla w dot bold-italic n mathematical right-angle Subscript partial-differential normal upper Omega Baseline period

This follows by setting bold-italic r equals bold nabla w in Equation 1.3a, the first equation of the HDG method.

All the above postprocessed solutions converge at a higher rate than  u Subscript h (whenever k greater-than-or-equal-to 1 ) as stated in the next theorem. It can be proved using the superconvergence estimate of Theorem 4.1 in a standard way (see Reference 20) so we omit it.

Theorem 5.2.

Under the same assumption as Theorem 4.1, the result u Subscript h comma i Superscript star of any of the three above mentioned postprocessings satisfy

double-vertical-bar u minus u Subscript h comma i Superscript star Baseline double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline less-than-or-equal-to left-parenthesis upper C Subscript 2 comma tau Baseline plus delta Subscript i Baseline 2 Baseline upper C Subscript 1 comma tau Baseline right-parenthesis h Superscript min left-brace right-brace comma k comma 1 Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline plus upper C h Superscript script l plus 2 Baseline double-vertical-bar u double-vertical-bar Subscript upper H Sub Superscript script l plus 2 Subscript left-parenthesis script upper T Sub Subscript h Subscript right-parenthesis Baseline comma

for any k greater-than-or-equal-to 0 and i equals 1 comma 2 comma 3 comma and any script l element-of left-bracket 0 comma k right-bracket .

6. Concluding remarks

We have presented a technique for error analysis of HDG methods that is remarkable for its brevity, especially in comparison with previous DG analyses Reference 2Reference 9Reference 12. We achieved this through the use of a new projection  upper Pi Subscript h . While brevity and elegance is traditionally achieved in the analysis of mixed methods, like RT and BDM methods, via the use of projections with commutativity properties, in the case of DG methods, commutativity seems not to be of paramount importance. Rather, what seems important is a projection tailored to fit the structure of the DG method, such as our upper Pi Subscript h .

Because upper Pi Subscript h is adapted to the structure of our numerical traces, we found it easy to estimate the projection of the errors. To summarize, we proved,

StartLayout 1st Row 1st Column double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q Subscript h Baseline double-vertical-bar Subscript c 2nd Column less-than-or-equal-to double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript c Baseline comma 2nd Row 1st Column double-vertical-bar upper P Subscript upper M Baseline left-parenthesis bold-italic q dot bold-italic n right-parenthesis minus ModifyingAbove bold-italic q With caret Subscript h Baseline dot bold-italic n double-vertical-bar Subscript h 2nd Column less-than-or-equal-to upper C Subscript 1 comma tau Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript c Baseline comma 3rd Row 1st Column double-vertical-bar upper Pi Subscript upper W Baseline u minus u Subscript h Baseline double-vertical-bar Subscript script upper T Sub Subscript h 2nd Column less-than-or-equal-to upper C Subscript 2 comma tau Baseline h Superscript min left-brace right-brace comma k comma 1 Baseline double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline comma 4th Row 1st Column double-vertical-bar upper P Subscript upper M Baseline u minus ModifyingAbove u With caret Subscript h Baseline double-vertical-bar Subscript h 2nd Column less-than-or-equal-to upper C Subscript 2 comma tau Baseline h double-vertical-bar bold-italic upper Pi Subscript upper V Baseline bold-italic q minus bold-italic q double-vertical-bar Subscript script upper T Sub Subscript h Subscript Baseline comma EndLayout

where the first three estimates hold for all k greater-than-or-equal-to 0 and the last for k greater-than-or-equal-to 1 . Thus, by the approximation properties of the projection upper Pi Subscript h (Theorem 2.1), if the penalty function tau is such that tau Subscript upper K Superscript normal m normal a normal x is of order one on each upper K element-of script upper T Subscript h , we obtain the optimal order of convergence of k plus 1 for the approximate flux and its numerical trace. Of course, by the triangle inequality, the above estimates imply that the error of these variables converges to zero at the optimal order. If k greater-than-or-equal-to 1 , the projection of the errors for u and its trace superconverge at order k plus 2 . This can be exploited to get locally postprocessed solutions of enhanced accuracy.

To end, note that the above estimates imply that

double-vertical-bar left-parenthesis ModifyingAbove bold-italic q With caret Subscript h Baseline minus bold-italic q Subscript h Baseline right-parenthesis dot bold-italic n double-vertical-bar Subscript h Baseline equals upper O left-parenthesis h Superscript k plus 1 Baseline right-parenthesis period

According to the main result in Reference 12, such an inequality implies optimal order of convergence for the numerical flux bold-italic q Subscript h and its postprocessing bold-italic q Subscript h Superscript star for k greater-than-or-equal-to 0 . It also implies, for k greater-than-or-equal-to 1 , the superconvergence of the orthogonal projection of the error in  u into script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis , and furthermore the superconvergence of ModifyingAbove u With caret Subscript h . This is in perfect agreement with our results.

The extension of our approach to other equations of practical interest appearing in, for example, fluid flow and solid mechanics, can prove to be useful not only to analyze already existing HDG methods but also to devise new ones. This constitutes the objective of ongoing research.

Appendix A. Proof of Theorem 2.1

We begin by observing that once we prove the approximation estimates of the theorem, then the unisolvency of the equations defining the projection follows as a corollary. This is because the number of equations and unknowns in 2.1, namely

dimension left-parenthesis bold-script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis right-parenthesis plus dimension left-parenthesis script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis right-parenthesis plus left-parenthesis n plus 1 right-parenthesis dimension left-parenthesis script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis right-parenthesis

and

dimension left-parenthesis bold-script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis right-parenthesis plus dimension left-parenthesis script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis right-parenthesis comma

respectively, coincide, i.e. 2.1 is a square linear system. Hence setting u equals 0 and bold-italic q equals 0 in the approximation estimates we find that the projection must vanish.

In view of this, in the remainder of this section, we develop estimates for any bold-italic upper Pi Subscript upper V Baseline bold-italic q and upper Pi Subscript upper W Baseline u satisfying 2.1 without assuming uniqueness a priori (although it will follow a posteriori). But first, we begin with two auxiliary lemmas.

A.1. Two estimates involving orthogonal polynomials

Let

script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis colon equals StartSet w element-of script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis colon left-parenthesis w comma zeta right-parenthesis Subscript upper K Baseline equals 0 for-all zeta element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis EndSet period

Lemma A.1.

Let upper F be any face of a simplex upper K . The trace map

gamma Subscript upper F Baseline colon script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis long right-arrow from bar script upper P Subscript k Baseline left-parenthesis upper F right-parenthesis defined by gamma Subscript upper F Baseline left-parenthesis p right-parenthesis equals p vertical-bar Subscript upper F Baseline

is a bijection. Moreover,

StartLayout 1st Row 1st Column double-vertical-bar p double-vertical-bar Subscript upper K 2nd Column less-than-or-equal-to upper C h Subscript upper K Superscript 1 slash 2 Baseline double-vertical-bar p double-vertical-bar Subscript upper F Baseline 3rd Column Blank 4th Column for all p element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis period EndLayout

Proof.

We first prove that gamma Subscript upper F is injective. Suppose gamma Subscript upper F Baseline left-parenthesis p right-parenthesis equals 0 for some p element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis . Then we can write p equals lamda Subscript upper F Baseline q , where lamda Subscript upper F denotes the barycentric coordinate function of upper K that vanishes on upper F and q is some function in script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis . But since left-parenthesis p comma w right-parenthesis Subscript upper K Baseline equals 0 for all w element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis , we then have left-parenthesis lamda Subscript upper F Baseline q comma q right-parenthesis Subscript upper K Baseline equals 0 comma so q equals 0 and hence p equals 0 . The surjectivity of gamma Subscript upper F now follows by counting dimensions and using the injectivity.

Finally, the estimate of the lemma follows from the injectivity and a standard scaling argument.

Lemma A.2.

Let eta be a nonnegative function on partial-differential upper K , constant on each face of upper K , and such that eta Superscript max Baseline colon equals max eta greater-than 0 . Let p element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis satisfy the equation

StartLayout 1st Row 1st Column mathematical left-angle eta p comma w mathematical right-angle Subscript partial-differential upper K Baseline equals 2nd Column b left-parenthesis w right-parenthesis 3rd Column Blank 4th Column for all w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis comma EndLayout

where b colon script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis right-arrow double-struck upper R is linear. Then

double-vertical-bar p double-vertical-bar Subscript upper K Baseline less-than-or-equal-to upper C StartFraction h Subscript upper K Baseline Over eta Superscript max Baseline EndFraction double-vertical-bar b double-vertical-bar comma

where double-vertical-bar b double-vertical-bar colon equals sup Underscript w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis minus 0 Endscripts b left-parenthesis w right-parenthesis slash double-vertical-bar w double-vertical-bar Subscript upper K .

Proof.

Let upper F be a face of upper K at which eta equals eta Superscript max . Then, by Lemma A.1,

StartLayout 1st Row 1st Column double-vertical-bar p double-vertical-bar Subscript upper K Superscript 2 2nd Column less-than-or-equal-to upper C h Subscript upper K Baseline double-vertical-bar p double-vertical-bar Subscript upper F Superscript 2 Baseline equals upper C StartFraction h Subscript upper K Baseline Over eta Superscript max Baseline EndFraction mathematical left-angle eta p comma p mathematical right-angle Subscript upper F Baseline less-than-or-equal-to upper C StartFraction h Subscript upper K Baseline Over eta Superscript max Baseline EndFraction mathematical left-angle eta p comma p mathematical right-angle Subscript partial-differential upper K Baseline comma 3rd Column Blank 4th Column since eta greater-than-or-equal-to 0 comma 2nd Row 1st Column Blank 2nd Column equals upper C StartFraction h Subscript upper K Baseline Over eta Superscript max Baseline EndFraction b left-parenthesis p right-parenthesis less-than-or-equal-to upper C StartFraction h Subscript upper K Baseline Over eta Superscript max Baseline EndFraction double-vertical-bar b double-vertical-bar double-vertical-bar p double-vertical-bar Subscript upper K Baseline comma EndLayout

and the wanted estimate follows.

A.2. Decoupling the projection component  upper Pi Subscript upper W Baseline u

Now we characterize the second (scalar) component of the projection upper Pi Subscript h Baseline left-parenthesis bold-italic q comma u right-parenthesis identical-to left-parenthesis bold-italic upper Pi Subscript upper V Baseline bold-italic q comma upper Pi Subscript upper W Baseline u right-parenthesis , namely upper Pi Subscript upper W Baseline u , independently of the first, and prove its approximation properties.

Proposition A.1.

On each element upper K element-of script upper T Subscript h , the component upper Pi Subscript upper W Baseline u satisfies

StartLayout 1st Row with Label left-parenthesis upper A .1 a right-parenthesis EndLabel 1st Column left-parenthesis upper Pi Subscript upper W Baseline u comma v right-parenthesis Subscript upper K 2nd Column equals left-parenthesis u comma v right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column for all v element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row with Label left-parenthesis upper A .1 b right-parenthesis EndLabel 1st Column mathematical left-angle tau upper Pi Subscript upper W Baseline u comma w mathematical right-angle Subscript partial-differential upper K 2nd Column equals left-parenthesis nabla dot bold-italic q comma w right-parenthesis Subscript upper K Baseline plus mathematical left-angle tau u comma w mathematical right-angle Subscript partial-differential upper K Baseline 3rd Column Blank 4th Column for all w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis period EndLayout

Proof.

The first equation Equation A.1a is the same as an equation defining the projection Equation 2.1b. For the second, note that Equation 2.1c implies

StartLayout 1st Row 1st Column mathematical left-angle tau upper Pi Subscript upper W Baseline u comma w mathematical right-angle Subscript partial-differential upper K 2nd Column equals mathematical left-angle left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q right-parenthesis dot bold-italic n plus tau u comma w mathematical right-angle Subscript partial-differential upper K Baseline 3rd Column Blank 4th Column for all w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis period EndLayout

Simplifying the right-hand side using

StartLayout 1st Row 1st Column mathematical left-angle left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q right-parenthesis dot bold-italic n comma w mathematical right-angle Subscript partial-differential upper K 2nd Column equals left-parenthesis nabla dot left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q right-parenthesis comma w right-parenthesis Subscript upper K Baseline plus left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q comma nabla w right-parenthesis Subscript upper K Baseline 2nd Row 1st Column Blank 2nd Column equals left-parenthesis nabla dot left-parenthesis bold-italic q minus bold-italic upper Pi Subscript upper V Baseline bold-italic q right-parenthesis comma w right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column by 2.1 a comma 3rd Row 1st Column Blank 2nd Column equals left-parenthesis nabla dot bold-italic q comma w right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column as w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis comma EndLayout

we finish the proof.

Proposition A.1 permits comparison with the SFH method. Suppose tau is selected as in the SFH method and suppose nabla dot bold-italic q element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis . Then the system A.1 becomes

StartLayout 1st Row 1st Column left-parenthesis upper Pi Subscript upper W Baseline u comma v right-parenthesis Subscript upper K 2nd Column equals left-parenthesis u comma v right-parenthesis Subscript upper K Baseline 3rd Column Blank 4th Column for all v element-of script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row 1st Column mathematical left-angle tau upper Pi Subscript upper W Baseline u comma mu mathematical right-angle Subscript upper F Sub Subscript upper K 2nd Column equals mathematical left-angle tau u comma mu mathematical right-angle Subscript upper F Sub Subscript upper K Baseline 3rd Column Blank 4th Column for all mu element-of script upper P Subscript k Baseline left-parenthesis upper F Subscript upper K Baseline right-parenthesis period EndLayout

Note that to obtain the last equation, we used the surjectivity of gamma Subscript upper F (Lemma A.1). Thus, in this case, upper Pi Subscript upper W coincides with the projection used in the analysis of the SFH method Reference 9 (denoted there by  double-struck upper P ).

We are now ready to obtain the estimate of upper Pi Subscript upper W Baseline u minus u in Theorem 2.1.

Proposition A.2.

Suppose the assumptions on tau in Theorem 2.1 hold. Then,

StartLayout 1st Row 1st Column double-vertical-bar upper Pi Subscript upper W Baseline u minus u double-vertical-bar Subscript upper K Baseline less-than-or-equal-to 2nd Column upper C h Subscript upper K Superscript script l Super Subscript u Superscript plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript u Sub Superscript plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline plus upper C StartFraction h Subscript upper K Superscript script l Super Subscript bold-italic q Superscript plus 1 Baseline Over tau Subscript upper K Superscript max Baseline EndFraction StartAbsoluteValue nabla dot bold-italic q EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript bold-italic q Sub Superscript Subscript left-parenthesis upper K right-parenthesis Baseline comma EndLayout

for script l Subscript u Baseline comma script l Subscript bold-italic q Baseline in left-bracket 0 comma k right-bracket (and consequently, upper Pi Subscript upper W Baseline u is uniquely determined by A.1).

Proof.

To prove the result, we set delta Superscript u Baseline colon equals upper Pi Subscript upper W Baseline u minus u Subscript k , where u Subscript k is the upper L squared left-parenthesis upper K right-parenthesis -orthogonal projection of u into script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis , and note that

double-vertical-bar upper Pi Subscript upper W Baseline u minus u double-vertical-bar Subscript upper K Baseline less-than-or-equal-to double-vertical-bar u minus u Subscript k Baseline double-vertical-bar Subscript upper K Baseline plus double-vertical-bar delta Superscript u Baseline double-vertical-bar Subscript upper K Baseline period

The first term can be readily estimated by using the standard approximation properties of the upper L squared -projection. Let us estimate the second term.

Equation Equation A.1a shows that delta Superscript u belongs to script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis , and Equation A.1b implies

StartLayout 1st Row 1st Column mathematical left-angle tau delta Superscript u Baseline comma w mathematical right-angle Subscript partial-differential upper K Baseline equals 2nd Column b Subscript bold-italic q Baseline left-parenthesis w right-parenthesis plus b Subscript u Baseline left-parenthesis w right-parenthesis for all w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis comma EndLayout

where b Subscript bold-italic q Baseline left-parenthesis w right-parenthesis colon equals left-parenthesis nabla dot bold-italic q comma w right-parenthesis Subscript upper K and b Subscript u Baseline left-parenthesis w right-parenthesis colon equals mathematical left-angle tau left-parenthesis u minus u Subscript k Baseline right-parenthesis comma w mathematical right-angle Subscript partial-differential upper K . By Lemma A.2 with eta colon equals tau , p colon equals delta Superscript u and b equals b Subscript bold-italic q Baseline plus b Subscript u , this implies that

double-vertical-bar delta Superscript u Baseline double-vertical-bar Subscript upper K Baseline less-than-or-equal-to upper C StartFraction h Subscript upper K Baseline Over tau Subscript upper K Superscript max Baseline EndFraction left-parenthesis double-vertical-bar b Subscript bold-italic q Baseline double-vertical-bar plus double-vertical-bar b Subscript u Baseline double-vertical-bar right-parenthesis period

Let us estimate double-vertical-bar b Subscript bold-italic q Baseline double-vertical-bar . Since w element-of script upper P Subscript k Superscript up-tack Baseline left-parenthesis upper K right-parenthesis , we have

StartLayout 1st Row 1st Column b Subscript bold-italic q Baseline left-parenthesis w right-parenthesis equals 2nd Column left-parenthesis nabla dot bold-italic q minus left-parenthesis nabla dot bold-italic q right-parenthesis Subscript k minus 1 Baseline comma w right-parenthesis Subscript upper K Baseline comma EndLayout

where left-parenthesis nabla dot bold-italic q right-parenthesis Subscript k minus 1 is the upper L squared left-parenthesis upper K right-parenthesis -projection of nabla dot bold-italic q into script upper P Subscript k minus 1 Baseline left-parenthesis upper K right-parenthesis when k greater-than-or-equal-to 1 . If k equals 0 , we set left-parenthesis nabla dot bold-italic q right-parenthesis Subscript negative 1 Baseline identical-to 0 . Hence,

StartLayout 1st Row 1st Column double-vertical-bar b Subscript bold-italic q Baseline double-vertical-bar less-than-or-equal-to 2nd Column double-vertical-bar nabla dot bold-italic q minus left-parenthesis nabla dot bold-italic q right-parenthesis Subscript k minus 1 Baseline double-vertical-bar Subscript upper K Baseline less-than-or-equal-to upper C h Subscript upper K Superscript script l Super Subscript bold-italic q Superscript Baseline StartAbsoluteValue nabla dot bold-italic q EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript bold-italic q Sub Superscript Subscript left-parenthesis upper K right-parenthesis Baseline comma EndLayout

for script l Subscript bold-italic q in left-bracket 0 comma k right-bracket , by the approximation properties of the upper L squared -projection.

Finally, let us estimate double-vertical-bar b Subscript u Baseline double-vertical-bar . By a scaling argument,

StartLayout 1st Row 1st Column b Subscript u Baseline left-parenthesis w right-parenthesis less-than-or-equal-to 2nd Column tau Subscript upper K Superscript max Baseline double-vertical-bar u minus u Subscript k Baseline double-vertical-bar Subscript partial-differential upper K Baseline double-vertical-bar w double-vertical-bar Subscript partial-differential upper K Baseline less-than-or-equal-to upper C h Subscript upper K Superscript negative 1 slash 2 Baseline tau Subscript upper K Superscript max Baseline double-vertical-bar u minus u Subscript k Baseline double-vertical-bar Subscript partial-differential upper K Baseline double-vertical-bar w double-vertical-bar Subscript upper K Baseline comma EndLayout

A trace inequality and the approximation properties of the upper L squared -projection imply

StartLayout 1st Row 1st Column double-vertical-bar b Subscript u Baseline double-vertical-bar less-than-or-equal-to 2nd Column upper C tau Subscript upper K Superscript max Baseline h Subscript upper K Superscript negative 1 Baseline left-parenthesis double-vertical-bar u minus u Subscript k Baseline double-vertical-bar Subscript upper K Baseline plus h Subscript upper K Baseline StartAbsoluteValue u minus u Subscript k Baseline EndAbsoluteValue Subscript upper H Sub Superscript 1 Subscript left-parenthesis upper K right-parenthesis Baseline right-parenthesis less-than-or-equal-to upper C tau Subscript upper K Superscript max Baseline h Subscript upper K Superscript script l Super Subscript u Superscript Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper H Sub Superscript script l Sub Super Subscript u Sub Superscript plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline comma EndLayout

for any script l Subscript u in left-bracket 0 comma k right-bracket . This completes the proof.

A.3. Properties of the flux component  upper P Subscript upper V Baseline bold-italic q

To study the flux component, we recall another projection bold-italic upper B Subscript upper V introduced and studied in Reference 8 and later used in Reference 9Reference 12. Let upper F Superscript asterisk be a face of upper K at which tau vertical-bar Subscript partial-differential upper K Baseline is a maximum. For any function bold-italic q in the domain of bold-italic upper Pi Subscript upper V , the restriction of bold-italic upper B Subscript upper V Baseline bold-italic q to upper K is defined to be the unique element of bold-script upper P Superscript k Baseline left-parenthesis upper K right-parenthesis satisfying

StartLayout 1st Row with Label left-parenthesis upper A .2 a right-parenthesis EndLabel 1st Column left-parenthesis bold-italic upper B Subscript upper V Baseline bold-italic q comma bold-italic v right-parenthesis Subscript upper K Baseline equals 2nd Column left-parenthesis bold-italic q comma bold-italic v right-parenthesis Subscript upper K 3rd Column Blank 4th Column for-all bold-italic v element-of bold-script upper P Superscript k minus 1 Baseline left-parenthesis upper K right-parenthesis comma 2nd Row with Label left-parenthesis upper A .2 b right-parenthesis EndLabel 1st Column mathematical left-angle bold-italic upper B Subscript upper V Baseline bold-italic q dot bold-italic n comma mu mathematical right-angle Subscript upper F Baseline equals 2nd Column mathematical left-angle bold-italic q dot bold-italic n comma mu mathematical right-angle Subscript upper F 3rd Column Blank 4th Column for-all mu element-of script upper P Superscript k Baseline left-parenthesis upper F right-parenthesis comma EndLayout