Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

By Carsten Carstensen and Sören Bartels

Abstract

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

1. Introduction

Error control and efficient mesh-design in finite element simulations of computational engineering and scientific computing finite element simulations is frequently based on a posteriori error estimates. One of the more popular techniques is local or global averaging, e.g., in form of the ZZ-error indicator ReferenceZZ. Efficiency and reliability of this estimator were known only for very structured grids and for solutions of higher regularity and then we have even asymptotic exactness ReferenceV. Numerical experiments in ReferenceBaetal showed that averaging techniques were quite more reliable on irregular meshes than expected. For homogeneous Dirichlet conditions and conforming finite element methods, the reliability and efficiency of the ZZ-estimator is proven on unstructured, merely shape-regular grids ReferenceR2.

This work is devoted to give theoretical and numerical support for the robust reliability of all averaging techniques, robust with respect to violated (local) symmetry of meshes and superconvergence and robust with respect to other boundary conditions or other finite element methods.

For a more precise description of averaging techniques, let us discuss a discretisation of a conservation equation

StartLayout 1st Row with Label left-parenthesis 1.1 right-parenthesis EndLabel f plus d i v p equals 0 EndLayout

with a given right-hand side f element-of upper L squared left-parenthesis normal upper Omega right-parenthesis and a known approximation p Subscript h Baseline element-of upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript d to the unknown exact flux p element-of upper H left-parenthesis d i v semicolon normal upper Omega right-parenthesis in the bounded Lipschitz domain normal upper Omega subset-of double-struck upper R Superscript d with piecewise flat boundary. The test function finite element space should include the continuous piecewise affines script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis (with homogeneous Dirichlet boundary conditions) based on a regular (in the sense of ReferenceCi, cf. Section 2) triangulation script upper T of normal upper Omega . Suppose a Galerkin property for p minus p Subscript h with script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis , i.e.,

StartLayout 1st Row with Label left-parenthesis 1.2 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts p Subscript h Baseline dot nabla v Subscript h Baseline d x equals integral Underscript normal upper Omega Endscripts f v Subscript h Baseline d x for all v Subscript h Baseline element-of script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis period EndLayout

What can be said about the error double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis when we regard p as an unknown and p Subscript h as a known variable?

In averaging techniques, the error estimator is based on a smoother approximation, e.g., in script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d , the continuous script upper T -piecewise linears, to the (components of the) discrete solution p Subscript h . For instance,

StartLayout 1st Row with Label left-parenthesis 1.3 right-parenthesis EndLabel eta Subscript upper Z Baseline colon equals min Underscript q Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d Endscripts double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis EndLayout

may serve as a computable error estimator and the elementwise contributions as local error indicators in an adaptive mesh-refining algorithm.

The triangle inequality shows that eta Subscript upper Z is efficient with constant 1 up to higher order terms of the exact solution p , indeed,

StartLayout 1st Row with Label left-parenthesis 1.4 right-parenthesis EndLabel eta Subscript upper Z Baseline less-than-or-equal-to double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus min Underscript q Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts double-vertical-bar p minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline period EndLayout

The last term converges as upper O left-parenthesis h squared right-parenthesis (provided p is smooth enough and h denotes the maximal mesh–size in script upper T ) and so, generically, is of higher order than the error double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline equals upper O left-parenthesis h right-parenthesis in the lowest order finite element method. If the second term in the right-hand side of Equation1.4 fails to be of higher order, one can still prove efficiency of eta Subscript upper Z using equivalence of global and local averaging (cf. Theorem 3.2) and that local averaging is equivalent to weighted jumps across interelement boundaries. An efficiency estimate with higher order terms that depend on local smoothness of right-hand sides but with unknown constants then follows as in ReferenceV.

In practise, we may apply an averaging operator script upper A colon upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript d Baseline right-arrow script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d to p Subscript h and compute the upper bound double-vertical-bar p Subscript h Baseline minus script upper A p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis of eta Subscript upper Z . Then, efficiency depends strongly on the approximation properties of script upper A and deserves further investigation.

In this paper, the focus is on the reliability of eta Subscript upper Z , i.e., we investigate under which conditions an estimate

StartLayout 1st Row with Label left-parenthesis 1.5 right-parenthesis EndLabel double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to c 1 eta Subscript upper Z Baseline plus h period o period t period less-than-or-equal-to c 1 double-vertical-bar p Subscript h Baseline minus script upper A p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus h period o period t period EndLayout

holds, we study what the constant c 1 greater-than 0 depends on, what affects the higher-order contributions h period o period t period ”, how to modify the definition of eta Subscript upper Z in the presence of mixed boundary conditions, and how to modify the general setting presented for nonconforming and mixed lowest order finite element methods.

Recall from Equation1.5 that any averaging technique, described by script upper A , then is reliable up to higher order terms. We also prove equivalence to local modifications of eta Subscript upper Z where the minimisation is over smaller domains, e.g., patches of nodes or edges.

The outline of the paper is as follows. Preliminaries and notation are introduced in Section 2 where we state and prove stability and first order estimates for a certain approximation operator script upper J colon upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis right-arrow script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis essentially designed to yield further local orthogonality properties as in ReferenceCVReferenceC2. Basic estimates are provided in Section 3 for a local and global averaging technique and their equivalence. The subsequent Sections 4, 5, and 6 display the consequences to averaging techniques in a posteriori error control for first order conforming, nonconforming and mixed finite element schemes. Numerical evidence, reported in Section 7, supports the theoretical results for adaptively refined and evenly perturbed meshes. Although asymptotic exactness is not claimed in this paper, our numerical experiments illustrate that eta Subscript upper Z is a very good approximate to double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis even on perturbed grids.

The proofs are given for a simple elliptic model example with mixed boundary conditions for conforming, nonconforming, and mixed finite elements in two dimensions for notational simplicity. More interesting examples such as higher-order schemes, the application to the Stokes problem or the Navier-Lamé equations without incompressibility-locking will appear elsewhere ReferenceBCReferenceCF2ReferenceCF3ReferenceCF4ReferenceCF5.

2. Approximation in finite element spaces

The Lipschitz boundary normal upper Gamma equals partial-differential normal upper Omega of the bounded domain normal upper Omega is split into a closed Dirichlet part normal upper Gamma Subscript upper D with positive surface measure and a remaining, relatively open and possibly empty, Neumann part normal upper Gamma Subscript upper N Baseline colon equals normal upper Gamma minus normal upper Gamma Subscript upper D . Suppose script upper T be a regular triangulation of the domain normal upper Omega subset-of-or-equal-to double-struck upper R Superscript d , d equals 1 comma 2 comma 3 , in the sense of Ciarlet ReferenceBSReferenceCi (no hanging node, domain is matched exactly) with piecewise affine Lipschitz boundary normal upper Gamma equals partial-differential normal upper Omega equals normal upper Gamma Subscript upper D Baseline dot union normal upper Gamma Subscript upper N , i.e., script upper T consists of a finite number of closed subsets of normal upper Omega overbar , that cover normal upper Omega overbar equals union script upper T . Each element upper T element-of script upper T is either an interval upper T equals c o n v left-brace a comma b right-brace if d equals 1 , a triangle upper T equals c o n v left-brace a comma b comma c right-brace , or a parallelogram upper T equals c o n v left-brace a comma b comma c comma d right-brace if d equals 2 . The extremal points a comma b comma c are called vertices, the faces upper E subset-of-or-equal-to partial-differential upper T , e.g., upper E equals c o n v left-brace a comma b right-brace , are called edges. The set of all vertices and all edges appearing for some upper T in script upper T are denoted as script upper N and script upper E . Two distinct and intersecting upper T 1 and upper T 2 share either an entire edge or a vertex. Each edge upper E element-of script upper E on the boundary normal upper Gamma belongs either to normal upper Gamma Subscript upper D , written upper E element-of script upper E Subscript upper D , or to normal upper Gamma overbar Subscript upper N , written upper E element-of script upper E Subscript upper N . Therefore the set of edges is partitioned into script upper E Subscript normal upper Omega Baseline colon equals StartSet upper E element-of script upper E colon upper E not-a-subset-of normal upper Gamma EndSet , script upper E Subscript upper D , and script upper E Subscript upper N . We stress that union script upper E , the union of all egdes, denotes the skeleton of egdes in script upper T , i.e., the set of all points x that belong to some boundary x element-of partial-differential upper T of some element upper T element-of script upper T . Finally, script upper K colon equals script upper N minus normal upper Gamma Subscript upper D denotes the set of free nodes.

We do not explicitly distinguish between nodes and vertices when we consider conforming finite elements (and avoid these concepts for nonconforming or mixed finite element schemes).

For upper T element-of script upper T , let upper P Subscript upper T Superscript k Baseline colon equals script upper P Subscript k Baseline left-parenthesis upper T right-parenthesis if upper T is a triangle or upper P Subscript upper T Superscript k Baseline colon equals script upper Q Subscript k Baseline left-parenthesis upper T right-parenthesis if upper T is a parallelogram. Here, script upper P Subscript k Baseline left-parenthesis upper K right-parenthesis , resp. script upper Q Subscript k Baseline left-parenthesis upper K right-parenthesis , denotes the set of algebraic polynomials in d variables on upper K of total, resp. partial, degree less-than-or-equal-to k . The space script upper L Superscript k Baseline left-parenthesis script upper T right-parenthesis of (possibly discontinuous) script upper T -piecewise polynomials of degree less-than-or-equal-to k is the set of all upper U element-of upper L Superscript normal infinity Baseline left-parenthesis normal upper Omega right-parenthesis with upper U vertical-bar Subscript upper T Baseline element-of upper P Subscript upper T Superscript k for all upper T in script upper T . Set

script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis colon equals script upper L Superscript k Baseline left-parenthesis script upper T right-parenthesis intersection upper C left-parenthesis normal upper Omega right-parenthesis and script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis colon equals StartSet u Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis colon u Subscript h Baseline vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals 0 EndSet period

Let left-parenthesis phi Subscript z Baseline vertical-bar z element-of script upper N right-parenthesis denote the nodal basis of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis , i.e., phi Subscript z Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis satisfies phi Subscript z Baseline left-parenthesis x right-parenthesis equals 0 if x element-of script upper N minus StartSet z EndSet and phi Subscript z Baseline left-parenthesis z right-parenthesis equals 1 . Note that left-parenthesis phi Subscript z Baseline vertical-bar z element-of script upper N right-parenthesis is a partition of unity and the open patches

StartLayout 1st Row with Label left-parenthesis 2.1 right-parenthesis EndLabel omega Subscript z Baseline colon equals StartSet x element-of normal upper Omega colon 0 less-than phi Subscript z Baseline left-parenthesis x right-parenthesis EndSet EndLayout

form an open cover left-parenthesis omega Subscript z Baseline colon z element-of script upper N right-parenthesis of normal upper Omega with finite overlap.

In order to define a weak interpolation operator script upper J colon upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis right-arrow script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis , we modify left-parenthesis phi Subscript z Baseline vertical-bar z element-of script upper K right-parenthesis to a partition of unity left-parenthesis psi Subscript z Baseline vertical-bar z element-of script upper K right-parenthesis . For each fixed node z element-of script upper N minus script upper K , we choose a node zeta left-parenthesis z right-parenthesis element-of script upper K and let zeta left-parenthesis z right-parenthesis colon equals z if z element-of script upper K . In this way, we define a partition of script upper N into c a r d left-parenthesis script upper K right-parenthesis classes upper I left-parenthesis z right-parenthesis colon equals StartSet z overTilde element-of script upper N colon zeta left-parenthesis z overTilde right-parenthesis equals z EndSet , z element-of script upper K . For each z element-of script upper K set

StartLayout 1st Row with Label left-parenthesis 2.2 right-parenthesis EndLabel psi Subscript z Baseline colon equals sigma-summation Underscript zeta element-of upper I left-parenthesis z right-parenthesis Endscripts phi Subscript zeta EndLayout

and notice that left-parenthesis psi Subscript z Baseline vertical-bar z element-of script upper K right-parenthesis is a partition of unity. It is required that

StartLayout 1st Row with Label left-parenthesis 2.3 right-parenthesis EndLabel normal upper Omega Subscript z Baseline colon equals StartSet x element-of normal upper Omega colon 0 less-than psi Subscript z Baseline left-parenthesis x right-parenthesis EndSet EndLayout

is connected and that psi Subscript z Baseline not-equals phi Subscript z implies that normal upper Gamma Subscript upper D Baseline intersection partial-differential normal upper Omega Subscript z has a positive surface measure.

For g element-of upper L Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis and z element-of script upper K let g Subscript z Baseline element-of double-struck upper R be

StartLayout 1st Row with Label left-parenthesis 2.4 right-parenthesis EndLabel g Subscript z Baseline colon equals StartFraction integral Underscript normal upper Omega Subscript z Baseline Endscripts g psi Subscript z Baseline d x Over integral Underscript normal upper Omega Subscript z Baseline Endscripts phi Subscript z Baseline d x EndFraction comma EndLayout

and then define

StartLayout 1st Row with Label left-parenthesis 2.5 right-parenthesis EndLabel script upper J g colon equals sigma-summation Underscript z element-of script upper K Endscripts g Subscript z Baseline phi Subscript z Baseline element-of script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis period EndLayout

The local mesh-sizes are denoted by h Subscript script upper T and h Subscript script upper E , where h Subscript script upper T Baseline element-of script upper L Superscript 0 Baseline left-parenthesis script upper T right-parenthesis denotes the element-size, h Subscript script upper T Baseline vertical-bar Subscript upper T Baseline colon equals h Subscript upper T Baseline colon equals d i a m left-parenthesis upper T right-parenthesis for upper T element-of script upper T , and the edge-size h Subscript script upper E Baseline element-of upper L Superscript normal infinity Baseline left-parenthesis union script upper E right-parenthesis is defined on the union or skeleton union script upper E of all edges upper E in script upper E by h Subscript script upper E Baseline vertical-bar Subscript upper E Baseline colon equals h Subscript upper E Baseline colon equals d i a m left-parenthesis upper E right-parenthesis . The patch-size h Subscript z Baseline colon equals d i a m left-parenthesis normal upper Omega Subscript z Baseline right-parenthesis is defined for each node z element-of script upper K separately.

Theorem 2.1

There exist left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constants c 2 comma c 3 comma c 4 comma c 5 greater-than 0 such that for all g element-of upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis and f element-of upper L squared left-parenthesis normal upper Omega right-parenthesis there holds

StartLayout 1st Row with Label left-parenthesis 2.6 right-parenthesis EndLabel 1st Column double-vertical-bar nabla script upper J g minus nabla g double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis 2nd Column less-than-or-equal-to c 2 double-vertical-bar nabla g double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline comma 2nd Row with Label left-parenthesis 2.7 right-parenthesis EndLabel 1st Column integral Underscript normal upper Omega Endscripts f left-parenthesis g minus script upper J g right-parenthesis d x 2nd Column less-than-or-equal-to c 3 double-vertical-bar nabla g double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f minus f Subscript z Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega Sub Subscript z Subscript right-parenthesis Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline comma 3rd Row with Label left-parenthesis 2.8 right-parenthesis EndLabel 1st Column double-vertical-bar h Subscript script upper T Superscript negative 1 Baseline left-parenthesis g minus script upper J g right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis 2nd Column less-than-or-equal-to c 4 double-vertical-bar nabla g double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline comma 4th Row with Label left-parenthesis 2.9 right-parenthesis EndLabel 1st Column double-vertical-bar h Subscript script upper E Superscript negative 1 slash 2 Baseline left-parenthesis g minus script upper J g right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis 2nd Column less-than-or-equal-to c 5 double-vertical-bar nabla g double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline period EndLayout

The constants c 2 comma c 3 comma c 4 comma c 5 only depend on normal upper Omega , normal upper Gamma Subscript upper D , normal upper Gamma Subscript upper N and the shape of the elements and patches (not on their sizes).

Remark 2.1

The assertion of the theorem holds verbatim for three space dimensions where script upper T consists of tetrahedra or parallelepipeds with the same proof.

Proof.

In this proof and at similar occasions, less-than-or-equivalent-to abbreviates an inequality less-than-or-equal-to up to a constant left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent factor. Also, double-vertical-bar dot double-vertical-bar Subscript p comma upper K abbreviates double-vertical-bar dot double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis and we neglect upper K if normal upper Omega is meant, i.e., double-vertical-bar dot double-vertical-bar Subscript 2 Baseline colon equals double-vertical-bar dot double-vertical-bar Subscript 2 comma normal upper Omega . Hence, e.g., Equation2.6 could be phrased as double-vertical-bar nabla script upper J g minus nabla g double-vertical-bar Subscript 2 Baseline less-than-or-equivalent-to double-vertical-bar nabla g double-vertical-bar Subscript 2 .

The key estimate for the stability and the approximation property of script upper J will be

StartLayout 1st Row with Label left-parenthesis 2.10 right-parenthesis EndLabel double-vertical-bar g Subscript z Baseline phi Subscript z Baseline minus g psi Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline left-parenthesis z element-of script upper K right-parenthesis period EndLayout

For the proof of Equation2.10, let g overbar Subscript z denote the integral mean of g on normal upper Omega Subscript z . Then, using the definition Equation2.4 for the coefficients g Subscript z , Cauchy’s and Young’s inequality, we infer, with c 6 colon equals double-vertical-bar 1 double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript slash double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript ,

StartLayout 1st Row with Label left-parenthesis 2.11 right-parenthesis EndLabel StartLayout 1st Row 1st Column double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 2nd Column equals integral Underscript normal upper Omega Subscript z Baseline Endscripts phi Subscript z Baseline left-parenthesis g overbar Subscript z Baseline minus g right-parenthesis left-parenthesis g overbar Subscript z Baseline minus g Subscript z Baseline right-parenthesis d x plus integral Underscript normal upper Omega Subscript z Baseline Endscripts left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis g left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to one-fourth double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline plus double-vertical-bar g minus g overbar Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline plus StartFraction 1 Over 4 c 6 squared EndFraction double-vertical-bar g Subscript z Baseline minus g overbar Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline 3rd Row 1st Column Blank 2nd Column plus c 6 squared double-vertical-bar left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline period EndLayout EndLayout

Absorbing StartFraction 1 Over 4 c 6 squared EndFraction double-vertical-bar g Subscript z Baseline minus g overbar Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline less-than-or-equal-to one-fourth double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 , we deduce

StartLayout 1st Row with Label left-parenthesis 2.12 right-parenthesis EndLabel double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline less-than-or-equivalent-to double-vertical-bar g minus g overbar Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline plus double-vertical-bar left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline period EndLayout

A Poincaré inequality yields

double-vertical-bar g minus g overbar Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline period

Note that left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis g is nonzero only if normal upper Gamma Subscript upper D Baseline intersection left-parenthesis partial-differential normal upper Omega Subscript z Baseline right-parenthesis has positive surface measure. Since g vanishes there, Friedrichs’ inequality shows

double-vertical-bar g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline period

Therefore, (Equation2.12) results in

StartLayout 1st Row with Label left-parenthesis 2.13 right-parenthesis EndLabel double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline period EndLayout

To prove (Equation2.10), we use the triangle inequality, (Equation2.13), and again Cauchy’s and Friedrichs’ inequality to verify

StartLayout 1st Row with Label left-parenthesis 2.14 right-parenthesis EndLabel double-vertical-bar g Subscript z Baseline phi Subscript z Baseline minus g psi Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equal-to double-vertical-bar left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis phi Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline plus double-vertical-bar left-parenthesis g minus g overbar Subscript z Baseline right-parenthesis phi Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline 2nd Row plus double-vertical-bar left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline period EndLayout

To prove (Equation2.7), we use that left-parenthesis psi Subscript z Baseline vertical-bar z element-of script upper K right-parenthesis is a partition of unity and obtain with (Equation2.10), (Equation2.4) for any f Subscript z Baseline element-of double-struck upper R that

StartLayout 1st Row with Label left-parenthesis 2.15 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts f left-parenthesis g minus script upper J g right-parenthesis d x equals sigma-summation Underscript z element-of script upper K Endscripts integral Underscript normal upper Omega Subscript z Baseline Endscripts f left-parenthesis g psi Subscript z Baseline minus g Subscript z Baseline phi Subscript z Baseline right-parenthesis d x equals sigma-summation Underscript z element-of script upper K Endscripts integral Underscript normal upper Omega Subscript z Baseline Endscripts left-parenthesis f minus f Subscript z Baseline right-parenthesis left-parenthesis g psi Subscript z Baseline minus g Subscript z Baseline phi Subscript z Baseline right-parenthesis d x 2nd Row less-than-or-equivalent-to sigma-summation Underscript z element-of script upper K Endscripts double-vertical-bar f minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline h Subscript z Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline double-vertical-bar f minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Baseline period EndLayout

In the last step we used that left-parenthesis phi Subscript z Baseline vertical-bar z element-of script upper K right-parenthesis has a finite overlap that depends on the shape of the elements only. This concludes the proof of (Equation2.7).

The remaining part of the proof uses standard arguments and is therefore sketched for brevity. To prove (Equation2.8) we let f colon equals h Subscript script upper T Superscript negative 2 Baseline left-parenthesis g minus script upper J g right-parenthesis and f Subscript z Baseline equals 0 , z element-of script upper K , in (Equation2.7). To verify (Equation2.6) we use sigma-summation Underscript z element-of script upper K Endscripts psi Subscript z Baseline equals 1 and sigma-summation Underscript z element-of script upper K Endscripts nabla psi Subscript z Baseline equals 0 and repeat the triangle inequality several times for

StartLayout 1st Row with Label left-parenthesis 2.16 right-parenthesis EndLabel double-vertical-bar nabla g minus nabla script upper J g double-vertical-bar Subscript 2 Superscript 2 Baseline less-than-or-equivalent-to sigma-summation Underscript z element-of script upper K Endscripts double-vertical-bar nabla left-parenthesis psi Subscript z Baseline g minus phi Subscript z Baseline g Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 Superscript 2 Baseline period EndLayout

With Friedrichs’ and Poincaré’s inequality we infer

StartLayout 1st Row with Label left-parenthesis 2.17 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis g psi Subscript z Baseline minus g Subscript z Baseline phi Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equal-to double-vertical-bar left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline plus double-vertical-bar nabla left-parenthesis phi Subscript z Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline 2nd Row plus double-vertical-bar nabla left-parenthesis phi Subscript z Baseline left-parenthesis g overbar Subscript z Baseline minus g right-parenthesis right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline plus double-vertical-bar g nabla left-parenthesis psi Subscript z Baseline minus phi Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline less-than-or-equivalent-to double-vertical-bar nabla g double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline plus double-vertical-bar nabla left-parenthesis phi Subscript z Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Baseline period EndLayout

It remains to estimate double-vertical-bar nabla left-parenthesis phi Subscript z Baseline left-parenthesis g Subscript z Baseline minus g overbar Subscript z Baseline right-parenthesis right-parenthesis double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript with Equation2.10, Friedrichs’ inequality, and the above arguments. A trace inequality ReferenceBSReferenceClReferenceCF1 of the form

StartLayout 1st Row with Label left-parenthesis 2.18 right-parenthesis EndLabel double-vertical-bar w double-vertical-bar Subscript 2 comma upper E Baseline less-than-or-equivalent-to h Subscript upper E Superscript negative 1 slash 2 Baseline double-vertical-bar w double-vertical-bar Subscript 2 comma upper T Baseline plus h Subscript upper E Superscript 1 slash 2 Baseline double-vertical-bar nabla w double-vertical-bar Subscript 2 comma upper T EndLayout

for upper E element-of script upper E Subscript upper N and upper T element-of script upper T with upper E subset-of partial-differential upper T intersection normal upper Gamma overbar Subscript upper N together with Equation2.6 and Equation2.8 implies Equation2.9.

3. Basic estimates

In this section we first derive with the approximation operator script upper J a global error estimate for a posteriori error control by averaging processes in an abstract setting. We then show the equivalence of local and global averaging techniques. The estimates of this section are then specified, and thereby proved to be substantial, in the subsequent sections to conforming, nonconforming, and mixed finite element methods.

Theorem 3.1

Suppose p comma q element-of upper H left-parenthesis d i v semicolon normal upper Omega right-parenthesis and p Subscript h Baseline element-of script upper L Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d with p dot n comma q dot n element-of upper L squared left-parenthesis normal upper Gamma Subscript upper N Baseline right-parenthesis and

StartLayout 1st Row with Label left-parenthesis 3.1 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla w Subscript h Baseline d x equals 0 for all w Subscript h Baseline element-of script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis period EndLayout

Then there holds

StartLayout 1st Row with Label left-parenthesis 3.2 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column sup Underscript StartLayout 1st Row w element-of upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis 2nd Row double-vertical-bar nabla w double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline equals 1 EndLayout Endscripts integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla w d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to c 2 double-vertical-bar p Subscript h Baseline minus q double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar d i v left-parenthesis p minus q right-parenthesis minus f Subscript z Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega Sub Subscript z Subscript right-parenthesis Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline 3rd Row 1st Column Blank 2nd Column plus c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis p minus q right-parenthesis dot n double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline period EndLayout EndLayout

Proof.

According to Equation3.1, Equation2.6, Cauchy’s inequality, and an integration by parts we have, for each w element-of upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis with double-vertical-bar nabla w double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline equals 1 , that

StartLayout 1st Row with Label left-parenthesis 3.3 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla w d x equals integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla left-parenthesis w minus script upper J w right-parenthesis d x 2nd Row equals integral Underscript normal upper Omega Endscripts left-parenthesis p minus q right-parenthesis dot nabla left-parenthesis w minus script upper J w right-parenthesis d x plus integral Underscript normal upper Omega Endscripts left-parenthesis q minus p Subscript h Baseline right-parenthesis dot nabla left-parenthesis w minus script upper J w right-parenthesis d x 3rd Row less-than-or-equal-to integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts left-parenthesis w minus script upper J w right-parenthesis left-parenthesis p minus q right-parenthesis dot n d x minus integral Underscript normal upper Omega Endscripts left-parenthesis w minus script upper J w right-parenthesis d i v left-parenthesis p minus q right-parenthesis d x plus c 2 double-vertical-bar p Subscript h Baseline minus q double-vertical-bar Subscript 2 comma normal upper Omega EndLayout

since w and script upper J w vanish on partial-differential normal upper Omega minus normal upper Gamma Subscript upper N . Owing to Equation2.7 and Equation2.9 in Theorem 2.1, we conclude Equation3.2 from Equation3.3 and Cauchy’s inequality.

The second result justifies local averaging. For each edge upper E element-of script upper E Subscript normal upper Omega , let omega Subscript upper E Baseline colon equals i n t left-parenthesis upper T 1 union upper T 2 right-parenthesis and script upper T Subscript upper E Baseline colon equals StartSet upper T 1 comma upper T 2 EndSet for the two distinct elements upper T 1 comma upper T 2 element-of script upper T with upper E equals upper T 1 intersection upper T 2 and for each edge upper E element-of script upper E Subscript upper N , let omega Subscript upper E Baseline colon equals i n t left-parenthesis upper T right-parenthesis and script upper T Subscript upper E Baseline colon equals StartSet upper T EndSet for the element upper T element-of script upper T with upper E equals upper T intersection normal upper Gamma overbar Subscript upper N . Let script upper L Superscript k Baseline left-parenthesis script upper E Subscript upper N Baseline right-parenthesis denote the (possibly discontinuous) script upper E Subscript upper N -piecewise polynomials of degree less-than-or-equal-to k on normal upper Gamma Subscript upper N and let script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis colon equals script upper L Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis intersection upper C left-parenthesis omega Subscript upper E Baseline right-parenthesis .

Theorem 3.2

There exists an left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constant c 7 greater-than 0 which depends on the shape of the elements in script upper T and on the polynomial degree k greater-than-or-equal-to 1 , c 8 equals max Underscript upper T element-of script upper T Endscripts c a r d left-brace upper E element-of script upper E Subscript normal upper Omega Baseline union script upper E Subscript upper N Baseline colon upper E subset-of-or-equal-to partial-differential upper T right-brace , such that, for all left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis element-of script upper L Superscript k minus 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d times script upper L Superscript k Baseline left-parenthesis script upper E Subscript upper N Baseline right-parenthesis , we have

StartLayout 1st Row with Label left-parenthesis 3.4 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column c 7 min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Superscript 2 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to sigma-summation Underscript upper E element-of script upper E Subscript normal upper Omega Baseline union script upper E Subscript upper N Baseline Endscripts min Underscript q Subscript upper E Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript upper E Baseline double-vertical-bar Subscript upper L squared left-parenthesis omega Sub Subscript upper E Subscript right-parenthesis Superscript 2 Baseline plus h Subscript upper E Baseline double-vertical-bar g Subscript h Baseline minus q Subscript upper E Baseline dot n double-vertical-bar Subscript upper L squared left-parenthesis upper E intersection normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Superscript 2 Baseline right-parenthesis 3rd Row 1st Column Blank 2nd Column less-than-or-equal-to c 8 min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Superscript 2 Baseline right-parenthesis period EndLayout EndLayout

Proof.

The upper estimate follows from q Subscript upper E Baseline colon equals q Subscript h Baseline vertical-bar Subscript omega Sub Subscript upper E Subscript Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d for all q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d and a rearrangement of the sums over edges and elements. To verify the lower estimate in Equation3.4 we consider a subspace ModifyingAbove script upper S With tilde Superscript k Baseline left-parenthesis script upper T right-parenthesis of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis ,

ModifyingAbove script upper S With tilde Superscript k Baseline left-parenthesis script upper T right-parenthesis colon equals StartSet sigma-summation Underscript z element-of script upper N Endscripts q Subscript z Baseline phi Subscript z Baseline colon q Subscript z Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis EndSet subset-of-or-equal-to script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis comma

where script upper T Subscript z Baseline equals StartSet upper T element-of script upper T colon upper T subset-of-or-equal-to omega overbar Subscript z Baseline EndSet denotes the restriction of the triangulation script upper T to omega Subscript z . Since StartSet left-parenthesis q Subscript h Baseline comma q Subscript h Baseline dot n vertical-bar Subscript normal upper Gamma Sub Subscript upper N Subscript Baseline right-parenthesis colon q Subscript h Baseline element-of script upper S overTilde Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline EndSet is a closed convex subset of upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript d times upper L squared left-parenthesis normal upper Gamma Subscript upper N Baseline right-parenthesis , the best-approximation problem

StartLayout 1st Row with Label left-parenthesis 3.5 right-parenthesis EndLabel min Underscript q Subscript h Baseline element-of script upper S overTilde Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Superscript 2 Baseline right-parenthesis EndLayout

defines an orthogonal relation, namely, for all q Subscript z Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis Superscript d ,

StartLayout 1st Row with Label left-parenthesis 3.6 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts left-parenthesis p Subscript h Baseline minus q Subscript h Baseline overTilde right-parenthesis dot q Subscript z Baseline phi Subscript z Baseline d x plus integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts h Subscript script upper E Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript h Baseline dot n right-parenthesis q Subscript z Baseline dot n phi Subscript z Baseline d s equals 0 comma EndLayout

where q overTilde Subscript h Baseline equals sigma-summation Underscript z element-of script upper N Endscripts q overTilde Subscript z Baseline phi Subscript z Baseline element-of script upper S overTilde Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d , q overTilde Subscript z Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis Superscript d , denotes the minimiser in Equation3.5. From sigma-summation Underscript z element-of script upper N Endscripts phi Subscript z Baseline equals 1 , (Equation3.6), and Cauchy’s inequality we deduce, for arbitrary q Subscript z Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis Superscript d ,

StartLayout 1st Row with Label left-parenthesis 3.7 right-parenthesis EndLabel StartLayout 1st Row 1st Column double-vertical-bar p Subscript h Baseline 2nd Column minus q overTilde Subscript h Baseline double-vertical-bar plus double-vertical-bar Subscript 2 Superscript 2 Baseline h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline Superscript 2 Baseline 2nd Row 1st Column equals 2nd Column sigma-summation Underscript z element-of script upper N Endscripts left-parenthesis integral Underscript normal upper Omega Endscripts left-parenthesis p Subscript h Baseline minus q overTilde Subscript h Baseline right-parenthesis phi Subscript z Baseline left-parenthesis p Subscript h Baseline minus q overTilde Subscript z Baseline right-parenthesis d x plus integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts h Subscript script upper E Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript h Baseline dot n right-parenthesis phi Subscript z Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript z Baseline dot n right-parenthesis d s right-parenthesis 3rd Row 1st Column equals 2nd Column sigma-summation Underscript z element-of script upper N Endscripts left-parenthesis integral Underscript normal upper Omega Endscripts left-parenthesis p Subscript h Baseline minus q overTilde Subscript h Baseline right-parenthesis phi Subscript z Baseline left-parenthesis p Subscript h Baseline minus q Subscript z Baseline right-parenthesis d x plus integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts h Subscript script upper E Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript h Baseline dot n right-parenthesis phi Subscript z Baseline left-parenthesis g Subscript h Baseline minus q Subscript z Baseline dot n right-parenthesis d s right-parenthesis 4th Row 1st Column less-than-or-equal-to 2nd Column left-parenthesis double-vertical-bar p Subscript h Baseline minus q overTilde Subscript h Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q overTilde Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline right-parenthesis 5th Row 1st Column Blank 2nd Column times left-parenthesis sigma-summation Underscript z element-of script upper N Endscripts left-parenthesis double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis p Subscript h Baseline minus q Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma omega Sub Subscript z Subscript Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript z Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Superscript 2 Baseline right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline period EndLayout EndLayout

For each z element-of script upper N , we consider the semi-norms on a finite dimensional subspace of upper L squared left-parenthesis omega Subscript z Baseline right-parenthesis Superscript d times upper L squared left-parenthesis left-parenthesis partial-differential omega Subscript z Baseline right-parenthesis intersection normal upper Gamma Subscript upper N Baseline right-parenthesis

StartLayout 1st Row 1st Column StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 1 2nd Column colon equals min Underscript q Subscript z Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline left-parenthesis p Subscript h Baseline minus q Subscript z Baseline right-parenthesis double-vertical-bar Subscript 2 comma omega Sub Subscript z Subscript Baseline plus double-vertical-bar phi Subscript z Superscript 1 slash 2 Baseline h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript z Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline right-parenthesis comma 2nd Row 1st Column StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 2 2nd Column colon equals left-parenthesis sigma-summation Underscript StartLayout 1st Row upper E element-of script upper E 2nd Row z element-of upper E EndLayout Endscripts min Underscript q Subscript upper E Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript upper E Baseline double-vertical-bar Subscript 2 comma omega Sub Subscript upper E Subscript Superscript 2 Baseline plus h Subscript upper E Baseline double-vertical-bar g Subscript h Baseline minus q Subscript upper E Baseline dot n double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript intersection upper E Superscript 2 Baseline right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

Then, Equation3.7 and ModifyingAbove script upper S With tilde Superscript k Baseline left-parenthesis script upper T right-parenthesis subset-of-or-equal-to script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis yield

StartLayout 1st Row with Label left-parenthesis 3.8 right-parenthesis EndLabel min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Superscript 2 Baseline right-parenthesis less-than-or-equivalent-to sigma-summation Underscript z element-of script upper N Endscripts StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 1 Superscript 2 Baseline period EndLayout

We claim StartAbsoluteValue double-vertical-bar dot double-vertical-bar EndAbsoluteValue Subscript z comma 1 Baseline less-than-or-equivalent-to StartAbsoluteValue double-vertical-bar dot double-vertical-bar EndAbsoluteValue Subscript z comma 2 . For a proof, suppose StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 2 Baseline equals 0 . Then, for each upper E that is an inner edge of omega Subscript z , we have p Subscript h Baseline equals q Subscript upper E on the open set omega Subscript upper E for some q Subscript upper E Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d . Since p Subscript h Baseline element-of script upper L Superscript k minus 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d , we find that p Subscript h Baseline vertical-bar Subscript omega Sub Subscript upper E Subscript Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis . The set of all such omega Subscript upper E is a cover of omega Subscript z and there is a sequence upper E 1 comma period period period comma upper E Subscript m Baseline of inner edges such that omega Subscript upper E Sub Subscript j Baseline intersection omega Subscript upper E Sub Subscript j plus 1 Baseline not-equals normal empty-set , so that we deduce p Subscript h Baseline vertical-bar Subscript omega Sub Subscript z Subscript Baseline element-of script upper S Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis . Moreover, g Subscript h Baseline equals p Subscript h Baseline dot n on each edge upper E subset-of-or-equal-to normal upper Gamma overbar Subscript upper N with z element-of upper E , while for edges upper E subset-of-or-equal-to partial-differential omega Subscript z intersection normal upper Gamma overbar Subscript upper N with z not-an-element-of upper E we have phi Subscript z Baseline vertical-bar Subscript upper E Baseline equals 0 . Altogether, we deduce StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 1 Baseline equals 0 . A compactness and scaling argument then shows our claim

StartLayout 1st Row with Label left-parenthesis 3.9 right-parenthesis EndLabel StartAbsoluteValue double-vertical-bar dot double-vertical-bar EndAbsoluteValue Subscript z comma 1 Baseline less-than-or-equivalent-to StartAbsoluteValue double-vertical-bar dot double-vertical-bar EndAbsoluteValue Subscript z comma 2 Baseline on script upper L Superscript k minus 1 Baseline left-parenthesis script upper T Subscript z Baseline right-parenthesis Superscript d Baseline times script upper L Superscript k Baseline left-parenthesis StartSet upper E element-of script upper E colon upper E subset-of-or-equal-to partial-differential omega Subscript z Baseline EndSet right-parenthesis period EndLayout

Utilizing Equation3.9 in Equation3.8, we conclude

StartLayout 1st Row with Label left-parenthesis 3.10 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Superscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Superscript 2 Baseline right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equivalent-to sigma-summation Underscript z element-of script upper N Endscripts StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 1 Superscript 2 Baseline less-than-or-equivalent-to sigma-summation Underscript z element-of script upper N Endscripts StartAbsoluteValue double-vertical-bar left-parenthesis p Subscript h Baseline comma g Subscript h Baseline right-parenthesis double-vertical-bar EndAbsoluteValue Subscript z comma 2 Superscript 2 Baseline 3rd Row 1st Column Blank 2nd Column less-than-or-equivalent-to sigma-summation Underscript StartLayout 1st Row upper E element-of script upper E EndLayout Endscripts min Underscript q Subscript upper E Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar p Subscript h Baseline minus q Subscript upper E Baseline double-vertical-bar Subscript 2 comma omega Sub Subscript upper E Subscript Superscript 2 Baseline plus h Subscript upper E Baseline double-vertical-bar g Subscript h Baseline minus q Subscript upper E Baseline dot n double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript intersection upper E Superscript 2 Baseline right-parenthesis period EndLayout EndLayout

Remark 3.1

The assertions of Theorems 3.1 and 3.2 hold verbatim for three space dimensions where script upper T consists of tetrahedra or parallelepipeds with the same proofs.

4. Applications to conforming finite element schemes

Given right-hand sides f element-of upper L squared left-parenthesis normal upper Omega right-parenthesis , g element-of upper L squared left-parenthesis normal upper Gamma Subscript upper N Baseline right-parenthesis , and u Subscript upper D Baseline element-of upper H Superscript 1 Baseline left-parenthesis normal upper Gamma Subscript upper D Baseline right-parenthesis , let u element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis denote the unique weak solution to

StartLayout 1st Row with Label left-parenthesis 4.1 right-parenthesis EndLabel 1st Column minus normal upper Delta u 2nd Column equals f 3rd Column Blank 4th Column in normal upper Omega comma 2nd Row with Label left-parenthesis 4.2 right-parenthesis EndLabel 1st Column u 2nd Column equals u Subscript upper D Baseline 3rd Column Blank 4th Column on normal upper Gamma Subscript upper D Baseline comma 3rd Row with Label left-parenthesis 4.3 right-parenthesis EndLabel 1st Column partial-differential u slash partial-differential n 2nd Column equals g 3rd Column Blank 4th Column on normal upper Gamma Subscript upper N Baseline period EndLayout

Suppose a finite element scheme, based on a regular triangulation script upper T , provided a discrete flux p Subscript h Baseline colon equals nabla u Subscript h to the exact flux p colon equals nabla u element-of upper H left-parenthesis d i v semicolon normal upper Omega right-parenthesis such that u Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis , u Subscript h Baseline left-parenthesis z right-parenthesis equals u Subscript upper D Baseline left-parenthesis z right-parenthesis for all z element-of script upper N intersection normal upper Gamma Subscript upper D and

StartLayout 1st Row with Label left-parenthesis 4.4 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts nabla u Subscript h Baseline dot nabla w Subscript h Baseline d x equals integral Underscript normal upper Omega Endscripts f w Subscript h Baseline d x plus integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts g w Subscript h Baseline d s for all w Subscript h Baseline element-of script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis period EndLayout

Theorem 4.1

There exists an left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constant c 9 greater-than 0 (that depends on k and the shape of the elements and patches) such that

StartLayout 1st Row with Label left-parenthesis 4.5 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis u minus u Subscript h Baseline right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts left-parenthesis c 9 double-vertical-bar nabla u Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus 2 c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline right-parenthesis 2nd Row plus inf Underscript v vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals u Subscript upper D Baseline Endscripts double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus 2 c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f minus f Subscript z Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega Sub Subscript z Subscript right-parenthesis Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

In the infimum, v vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals u Subscript upper D Baseline stands for all v element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis with v equals u Subscript upper D on normal upper Gamma Subscript upper D .

Proof.

Abbreviate e colon equals u minus u Subscript h and let q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d . Assume that v element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis satisfies v equals u Subscript upper D on normal upper Gamma Subscript upper D and double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript 2 Baseline less-than-or-equal-to double-vertical-bar nabla e double-vertical-bar Subscript 2 . Recall p equals nabla u and p Subscript h Baseline equals nabla u Subscript h . Then Equation4.1-Equation4.4 imply Equation3.1. Hence, we may choose q equals q Subscript h and w equals u minus v in Theorem 3.1 to obtain with Cauchy’s inequality for the second term that

StartLayout 1st Row with Label left-parenthesis 4.6 right-parenthesis EndLabel StartLayout 1st Row 1st Column double-vertical-bar nabla e double-vertical-bar Subscript 2 Superscript 2 2nd Column equals integral Underscript normal upper Omega Endscripts nabla e dot nabla w d x plus integral Underscript normal upper Omega Endscripts nabla e dot nabla left-parenthesis v minus u Subscript h Baseline right-parenthesis d x 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to double-vertical-bar nabla w double-vertical-bar Subscript 2 Baseline left-parenthesis c 2 double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Baseline plus c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline 3rd Row 1st Column Blank 2nd Column plus c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f plus d i v q Subscript h Baseline minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline right-parenthesis 4th Row 1st Column Blank 2nd Column plus double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript 2 Baseline double-vertical-bar nabla e double-vertical-bar Subscript 2 Baseline period EndLayout EndLayout

Since double-vertical-bar nabla w double-vertical-bar Subscript 2 Baseline less-than-or-equal-to double-vertical-bar nabla e double-vertical-bar Subscript 2 Baseline plus double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript 2 Baseline less-than-or-equal-to 2 double-vertical-bar nabla e double-vertical-bar Subscript 2 , we can divide Equation4.6 by double-vertical-bar nabla e double-vertical-bar Subscript 2 to verify

StartLayout 1st Row with Label left-parenthesis 4.7 right-parenthesis EndLabel double-vertical-bar nabla e double-vertical-bar Subscript 2 Baseline less-than-or-equal-to 2 c 2 double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Baseline plus 2 c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline plus double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript 2 Baseline 2nd Row plus 2 c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f plus d i v q Subscript h Baseline minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline period EndLayout

Let d i v Subscript script upper T denote the script upper T -piecewise action of the d i v -operator. The triangle inequality in the last summand in Equation4.7 and h Subscript z Baseline less-than-or-equivalent-to h Subscript upper T for z element-of upper T intersection script upper N and upper T element-of script upper T and a summation over elements show

StartLayout 1st Row with Label left-parenthesis 4.8 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f plus d i v q Subscript h Baseline minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 2nd Row 1st Column Blank 2nd Column less-than-or-equivalent-to double-vertical-bar h Subscript script upper T Baseline d i v Subscript script upper T Baseline left-parenthesis p Subscript h Baseline minus q Subscript h Baseline right-parenthesis double-vertical-bar Subscript 2 Superscript 2 Baseline plus sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f plus d i v Subscript script upper T Baseline p Subscript h Baseline minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline period EndLayout EndLayout

Note that d i v Subscript script upper T Baseline p Subscript h Baseline equals normal upper Delta Subscript script upper T Baseline u Subscript h Baseline equals 0 for our choices of u Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis . A script upper T -elementwise inverse estimate shows double-vertical-bar h Subscript script upper T Baseline d i v Subscript script upper T Baseline left-parenthesis p Subscript h Baseline minus q Subscript h Baseline right-parenthesis double-vertical-bar Subscript 2 Baseline less-than-or-equivalent-to double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 (with a constant that depends on the shape of the finite elements only). Utilising this in Equation4.7Equation4.8, we deduce Equation4.5.

Remark 4.1

In the proof of Theorem 4.1 we used the assumption that u Subscript h is of lowest order, i.e., nabla u Subscript h element-of script upper L Superscript 0 Baseline left-parenthesis script upper T right-parenthesis Superscript d , for the purpose of estimating double-vertical-bar h Subscript script upper T Baseline d i v q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis by double-vertical-bar q Subscript h Baseline minus nabla u Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis . We refer to ReferenceBC for related error estimates for higher order methods.

The subsequent lemma shows that inf Underscript v vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals u Subscript upper D Baseline Endscripts double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis is a higher order term.

Lemma 4.1

Suppose that u Subscript h Baseline left-parenthesis z right-parenthesis equals u Subscript upper D Baseline left-parenthesis z right-parenthesis for all z element-of script upper N intersection normal upper Gamma Subscript upper D . Then there exists an h Subscript script upper E -independent constant c 10 greater-than 0 (that depends on the shapes of the elements only) such that

StartLayout 1st Row with Label left-parenthesis 4.9 right-parenthesis EndLabel inf Underscript v vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals u Subscript upper D Baseline Endscripts double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to c 10 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline partial-differential left-parenthesis u Subscript h Baseline minus u Subscript upper D Baseline right-parenthesis slash partial-differential s double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper D Subscript right-parenthesis Baseline period EndLayout

If u Subscript upper D Baseline element-of upper H squared left-parenthesis script upper E Subscript upper D Baseline right-parenthesis colon equals StartSet v element-of upper L squared left-parenthesis normal upper Gamma Subscript upper D Baseline right-parenthesis colon for-all upper E element-of script upper E Subscript upper D Baseline comma v vertical-bar Subscript upper E Baseline element-of upper H squared left-parenthesis upper E right-parenthesis EndSet , we have

StartLayout 1st Row with Label left-parenthesis 4.10 right-parenthesis EndLabel inf Underscript v vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals u Subscript upper D Baseline Endscripts double-vertical-bar nabla left-parenthesis u Subscript h Baseline minus v right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to c 10 double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Superscript 2 Baseline u Subscript upper D Baseline slash partial-differential s squared double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper D Subscript right-parenthesis Baseline period EndLayout

Proof.

Let upper E element-of script upper E Subscript upper D belong to some upper T element-of script upper T and denote gamma colon equals normal upper Gamma Subscript upper D intersection partial-differential upper T . We determine w element-of upper H Superscript 1 Baseline left-parenthesis upper T right-parenthesis by extending the boundary values w vertical-bar Subscript gamma Baseline equals u Subscript h Baseline minus u Subscript upper D Baseline and w vertical-bar Subscript partial-differential upper T minus gamma Baseline equals 0 . Note that w is continuous on partial-differential upper T since u Subscript h interpolates u Subscript upper D Baseline equals v at each node on normal upper Gamma Subscript upper D . An harmonic extension of w vertical-bar Subscript partial-differential upper T Baseline to w element-of upper H Superscript 1 Baseline left-parenthesis script upper T right-parenthesis yields

StartLayout 1st Row with Label left-parenthesis 4.11 right-parenthesis EndLabel double-vertical-bar nabla w double-vertical-bar Subscript 2 comma upper T Baseline less-than-or-equivalent-to double-vertical-bar w double-vertical-bar Subscript upper H Sub Superscript 1 slash 2 Subscript left-parenthesis partial-differential upper T right-parenthesis Baseline less-than-or-equivalent-to double-vertical-bar w double-vertical-bar Subscript 2 comma partial-differential upper T Superscript 1 slash 2 Baseline double-vertical-bar partial-differential w slash partial-differential s double-vertical-bar Subscript 2 comma partial-differential upper T Superscript 1 slash 2 Baseline comma EndLayout

where we applied an interpolation estimate. A one-dimensional integration argument shows double-vertical-bar w double-vertical-bar Subscript 2 comma partial-differential upper T Baseline less-than-or-equal-to h Subscript upper T Baseline double-vertical-bar partial-differential w slash partial-differential s double-vertical-bar Subscript 2 comma partial-differential upper T . Consequently,

StartLayout 1st Row with Label left-parenthesis 4.12 right-parenthesis EndLabel double-vertical-bar nabla w double-vertical-bar Subscript 2 comma upper T Baseline less-than-or-equivalent-to h Subscript upper T Superscript 1 slash 2 Baseline double-vertical-bar partial-differential w slash partial-differential s double-vertical-bar Subscript 2 comma partial-differential upper T Baseline equals h Subscript upper T Superscript 1 slash 2 Baseline double-vertical-bar partial-differential left-parenthesis u Subscript h Baseline minus u Subscript upper D Baseline right-parenthesis slash partial-differential s double-vertical-bar Subscript 2 comma gamma Baseline period EndLayout

A scaling argument guarantees that the constant in Equation4.12 is h Subscript upper T -independent. Defining v by u Subscript h Baseline minus w on elements on normal upper Gamma Subscript upper D and by zero on other elements then shows the lemma. The second estimate follows from double-vertical-bar w double-vertical-bar Subscript 2 comma partial-differential upper T Baseline less-than-or-equal-to h Subscript upper T Superscript 2 Baseline double-vertical-bar partial-differential squared w slash partial-differential s squared double-vertical-bar Subscript 2 comma partial-differential upper T .

Lemma 4.2

Suppose g element-of upper H Superscript 1 Baseline left-parenthesis script upper E Subscript upper N Baseline right-parenthesis and, for each node z element-of script upper N intersection normal upper Gamma overbar Subscript upper N where the outer unit normal n on normal upper Gamma Subscript upper N is continuous (hence constant in a neighbourhood of z as normal upper Gamma Subscript upper N is a polygon), let g be continuous. Then, the set

StartLayout 1st Row with Label left-parenthesis 4.13 right-parenthesis EndLabel script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T comma g right-parenthesis colon equals StartSet q Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline colon for-all upper E element-of script upper E Subscript upper N Baseline for-all z element-of upper E intersection script upper N comma q Subscript h Baseline left-parenthesis z right-parenthesis dot n Subscript upper E Baseline equals g left-parenthesis z right-parenthesis EndSet EndLayout

is nonvoid and, for each q Subscript h Baseline element-of script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T comma g right-parenthesis ,

StartLayout 1st Row with Label left-parenthesis 4.14 right-parenthesis EndLabel double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline less-than-or-equal-to double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Baseline g slash partial-differential s double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline period EndLayout

Proof.

Elementary estimates on each edge on normal upper Gamma Subscript upper N verify Equation4.14; the proof of script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T comma g right-parenthesis not-equals normal empty-set follows from an explicit construction in Example 4.1.

Example 4.1

We define an operator script upper A colon upper L squared left-parenthesis normal upper Omega right-parenthesis squared right-arrow script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T comma g right-parenthesis by

StartLayout 1st Row with Label left-parenthesis 4.15 right-parenthesis EndLabel script upper A p colon equals sigma-summation Underscript z element-of script upper N Endscripts p Subscript z Baseline phi Subscript z Baseline comma EndLayout

where p Subscript z Baseline colon equals minus integral Underscript omega Subscript z Endscripts p d x colon equals StartFraction 1 Over StartAbsoluteValue omega Subscript z Baseline EndAbsoluteValue EndFraction integral Underscript omega Subscript z Endscripts p d x element-of double-struck upper R squared for z element-of script upper N minus normal upper Gamma overbar Subscript upper N while we incorporate script upper A p left-parenthesis z right-parenthesis dot n Subscript upper E Baseline equals g left-parenthesis z right-parenthesis for z element-of script upper N intersection normal upper Gamma overbar Subscript upper N . In case z equals upper E 1 intersection upper E 2 for two distinct edges upper E 1 comma upper E 2 element-of script upper E Subscript upper N Baseline with distinct outer unit normals n Subscript upper E 1 , n Subscript upper E 2 on upper E 1 , upper E 2 at a corner z we choose p Subscript z Baseline element-of double-struck upper R squared to be the unique solution of the 2 times 2 linear system

StartLayout 1st Row with Label left-parenthesis 4.16 a right-parenthesis EndLabel n Subscript upper E 1 Baseline dot p Subscript z Baseline equals g StartAbsoluteValue left-parenthesis z right-parenthesis and n Subscript upper E 2 Baseline dot p Subscript z Baseline equals g EndAbsoluteValue Subscript upper E 1 Baseline Subscript upper E 2 Baseline left-parenthesis z right-parenthesis period EndLayout

In the remaining cases z element-of upper E 1 intersection normal upper Gamma Subscript upper D for upper E 1 element-of script upper E Subscript upper N or z equals upper E 1 intersection upper E 2 with two parallel edges upper E 1 comma upper E 2 element-of script upper E Subscript upper N Baseline with the unit tangent vector t Subscript upper E 1 let p Subscript z Baseline element-of double-struck upper R squared solve

StartLayout 1st Row with Label left-parenthesis 4.16 b right-parenthesis EndLabel n Subscript upper E 1 Baseline dot p Subscript z Baseline equals g vertical-bar Subscript upper E 1 Baseline left-parenthesis z right-parenthesis and t Subscript upper E 1 Baseline dot p Subscript z Baseline equals minus integral Underscript omega Subscript z Baseline Endscripts t Subscript upper E 1 Baseline dot p d x period EndLayout

The following corollary is Equation1.5 with a constant c 1 equals c 9 as in Theorem 4.1 and with specified higher order terms from Lemma 4.1 and 4.2 and a Poincaré inequality.

Corollary 4.1

Under the conditions of Theorem 4.1 and Lemmas 4.1 and 4.2 we have for f element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis that

StartLayout 1st Row with Label left-parenthesis 4.17 right-parenthesis EndLabel double-vertical-bar nabla left-parenthesis u minus u Subscript h Baseline right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to c 9 min Underscript q Subscript h Baseline element-of script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T comma g right-parenthesis Endscripts double-vertical-bar nabla u Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline 2nd Row plus c 11 left-parenthesis double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Superscript 2 Baseline u Subscript upper D Baseline slash partial-differential s squared double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper D Subscript right-parenthesis Baseline plus double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Baseline g slash partial-differential s double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline plus double-vertical-bar h Subscript script upper T Superscript 2 Baseline nabla f double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline right-parenthesis period EndLayout

The left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constant c 11 greater-than 0 depends on the shape of the elements and patches only.

Remark 4.2

Let us emphasise that the derivatives along normal upper Gamma are required only script upper E -piecewisely while f needs to be patch-wise (not only elementwise) in upper H Superscript 1 and so f element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis . For a nonsmooth right-hand side f , double-vertical-bar h Subscript script upper T Superscript 2 Baseline nabla f double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis may be replaced by a patch-wise upper L squared –best approximation error in the approximation through constants of f (cf. Equation2.7).

The global averaging process might be too expensive or its approximation may be inefficient and hence a local averaging process of interest. Recall that omega Subscript upper E is the (interior of the) union of all elements in script upper T that share the edge upper E element-of script upper E .

Corollary 4.2

Under the conditions of Theorem 4.1 and Lemmas 4.1 and 4.2 we have for f element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis that

StartLayout 1st Row with Label left-parenthesis 4.18 right-parenthesis EndLabel StartLayout 1st Row 1st Column double-vertical-bar nabla left-parenthesis u minus u Subscript h Baseline right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis 2nd Column less-than-or-equal-to c 12 left-parenthesis sigma-summation Underscript upper E element-of script upper E Endscripts min Underscript q Subscript upper E Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T Subscript upper E Baseline right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar nabla u Subscript h Baseline minus q Subscript upper E Baseline double-vertical-bar Subscript upper L squared left-parenthesis omega Sub Subscript upper E Subscript right-parenthesis Superscript 2 Baseline 2nd Row 1st Column Blank 2nd Column plus h Subscript upper E Baseline double-vertical-bar g Subscript h Baseline minus q Subscript upper E Baseline dot n double-vertical-bar Subscript upper L squared left-parenthesis upper E intersection normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Superscript 2 Baseline right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline 3rd Row 1st Column Blank 2nd Column plus c 11 left-parenthesis double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Superscript 2 Baseline u Subscript upper D Baseline slash partial-differential s squared double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper D Subscript right-parenthesis Baseline 4th Row 1st Column Blank 2nd Column plus double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Baseline g slash partial-differential s double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline plus double-vertical-bar h Subscript script upper T Superscript 2 Baseline nabla f double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline right-parenthesis period EndLayout EndLayout

The left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constant c 12 equals max left-brace c 9 comma 2 c 5 right-brace slash c 7 depends on the shape of the elements and patches only.

Proof.

Theorem 4.1, Lemma 4.1, an approximation g Subscript h of g as in Lemma 4.2, and a Poincaré inequality show

StartLayout 1st Row with Label left-parenthesis 4.19 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column double-vertical-bar nabla left-parenthesis u minus u Subscript h Baseline right-parenthesis double-vertical-bar Subscript 2 2nd Row 1st Column Blank 2nd Column less-than-or-equivalent-to min Underscript q Subscript h Baseline element-of script upper S Superscript 1 Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts left-parenthesis double-vertical-bar nabla u Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Baseline plus double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g Subscript h Baseline minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline right-parenthesis plus double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Baseline g slash partial-differential s double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline 3rd Row 1st Column Blank 2nd Column plus double-vertical-bar h Subscript script upper E Superscript 3 slash 2 Baseline partial-differential Subscript script upper E Superscript 2 Baseline u Subscript upper D Baseline slash partial-differential s squared double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper D Subscript Baseline plus double-vertical-bar h Subscript script upper T Superscript 2 Baseline nabla f double-vertical-bar Subscript 2 Baseline period EndLayout EndLayout

This and the first inequality of Theorem 3.2 imply the assertion.

Remark 4.3

The results of this section hold also in three dimensions where script upper T consists of tetrahedra or parallelepipeds. The proofs of some details as Lemma 4.1 or Lemma 4.2 require much more technical preparations and so are omitted in this overview.

Remark 4.4

It is shown in ReferenceCVReferenceC2 that the edge-contributions (jump differences in the normal fluxes components across edges) dominate in standard residual a posteriori error estimates ReferenceBaRReferenceBReferenceBSReferenceCF1ReferenceEEHJReferenceV. Arguing as in ReferenceR1ReferenceR2ReferenceDMR, one can hence derive alternative proofs of Equation4.18 and then of Equation4.17.

Remark 4.5

In an upper L Superscript normal infinity -estimate of ReferenceHSWW it is suggested to average over a domain of size upper O left-parenthesis h log left-parenthesis 1 slash h right-parenthesis right-parenthesis instead of merely over patches or the entire domain to obtain asymptotic exact results.

5. Applications to nonconforming finite element schemes

In the Laplace problem with mixed boundary conditions Equation4.1Equation4.3, we suppose that the discrete flux p Subscript h Baseline colon equals nabla Subscript script upper T Baseline u Subscript h element-of script upper L Superscript 0 Baseline left-parenthesis script upper T right-parenthesis Superscript d , where nabla Subscript script upper T denotes the script upper T -piecewise application of the gradient, satisfies

StartLayout 1st Row with Label left-parenthesis 5.1 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts nabla Subscript script upper T Baseline u Subscript h Baseline dot nabla w Subscript h Baseline d x equals integral Underscript normal upper Omega Endscripts f w Subscript h Baseline d x plus integral Underscript normal upper Gamma Subscript upper N Baseline Endscripts g w Subscript h Baseline d s for all w Subscript h Baseline element-of script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis period EndLayout

The usual conformity conditions read for all upper E element-of script upper E Subscript normal upper Omega union script upper E Subscript upper D ,

StartLayout 1st Row with Label left-parenthesis 5.2 right-parenthesis EndLabel integral Underscript upper E Endscripts left-bracket u Subscript h Baseline right-bracket d s equals 0 comma EndLayout

where left-bracket u Subscript h Baseline right-bracket vertical-bar Subscript upper E Baseline denotes the jump of u Subscript h across upper E element-of script upper E Subscript normal upper Omega and denotes u Subscript upper D Baseline minus u Subscript h on normal upper Gamma Subscript upper D . Those conditions are satisfied by construction for Crouzeix–Raviart finite elements of lowest order.

Remark 5.1

It is stressed that script upper S Subscript upper D Superscript 1 Baseline left-parenthesis script upper T right-parenthesis is a conforming test function space which is included in the nonconforming finite element spaces for triangles or tetrahedra. For parallelograms, Equation5.1 means that the polynomial degrees are at least of second order to include the conforming term x 1 x 2 . This technical detail could actually be dropped since the contribution from an enhanced finite element space leads to a higher order term ReferenceKS. We restrict our analysis to triangles or tetrahedra for simplicity.

Theorem 5.1

Suppose that normal upper Gamma Subscript upper N is connected and that normal upper Gamma Subscript upper D belongs to only one connectivity component of partial-differential normal upper Omega . Then, there exists an left-parenthesis h Subscript script upper T Baseline comma h Subscript script upper E Baseline right-parenthesis -independent constant c 13 greater-than 0 (that depends on k greater-than-or-equal-to 1 and the shape of the elements and patches) such that

StartLayout 1st Row with Label left-parenthesis 5.3 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column double-vertical-bar nabla Subscript script upper T Baseline left-parenthesis u minus u Subscript h Baseline right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis 2nd Row 1st Column Blank 2nd Column less-than-or-equal-to min Underscript q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d Baseline Endscripts left-parenthesis c 13 double-vertical-bar nabla Subscript script upper T Baseline u Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Baseline plus c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper N Subscript right-parenthesis Baseline 3rd Row 1st Column Blank 2nd Column plus c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis q Subscript h Baseline dot t minus partial-differential u Subscript upper D Baseline slash partial-differential s right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis normal upper Gamma Sub Subscript upper D Subscript right-parenthesis Baseline right-parenthesis 4th Row 1st Column Blank 2nd Column plus c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f minus f Subscript z Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega Sub Subscript z Subscript right-parenthesis Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline period EndLayout EndLayout

Here, t element-of script upper L Superscript 0 Baseline left-parenthesis script upper E Subscript upper D Baseline right-parenthesis Superscript d denotes the unit tangent vector on normal upper Gamma Subscript upper D .

Remark 5.2

The following lemma is based on the Helmholtz decomposition of a vector field. The decomposition is available in three dimensions as well (cf., e.g., [GR]) but the notation is more involved so we restrict the discussion to the two-dimensional setting for brevity.

Lemma 5.1

For all p minus p Subscript h element-of upper L squared left-parenthesis normal upper Omega right-parenthesis squared , there exist alpha comma beta element-of upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis that satisfy boundary conditions alpha vertical-bar Subscript normal upper Gamma Sub Subscript upper D Subscript Baseline equals 0 and beta vertical-bar Subscript normal upper Gamma Sub Subscript upper N Subscript Baseline is constant such that

StartLayout 1st Row with Label left-parenthesis 5.4 right-parenthesis EndLabel p minus p Subscript h Baseline equals nabla alpha plus upper C u r l beta and double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript 2 Baseline equals double-vertical-bar nabla alpha double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript 2 Baseline plus double-vertical-bar nabla beta double-vertical-bar Subscript upper L squared left-parenthesis normal upper Omega right-parenthesis Superscript 2 Baseline period EndLayout

Proof.

The lemma follows from the Helmholtz decomposition where alpha element-of upper H Subscript upper D Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis solves normal upper Delta alpha equals d i v left-parenthesis p minus p Subscript h Baseline right-parenthesis and normal upper Delta beta equals c u r l left-parenthesis p minus p Subscript h Baseline right-parenthesis with proper boundary conditions (cf., e.g., ReferenceGR).

Proof of Theorem 5.1.

For p equals nabla u and p Subscript h Baseline equals nabla Subscript script upper T Baseline u Subscript h , Lemma 5.1 yields

StartLayout 1st Row with Label left-parenthesis 5.5 right-parenthesis EndLabel double-vertical-bar p minus p Subscript h Baseline double-vertical-bar Subscript 2 Superscript 2 Baseline equals integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla alpha d x plus integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot upper C u r l beta d x period EndLayout

Since normal upper Gamma Subscript upper N is connected, we may and will assume without loss of generality that beta equals 0 on normal upper Gamma Subscript upper N . According to Equation4.1Equation4.3 and Equation5.1, we infer Equation3.1 and hence may choose q equals q Subscript h Baseline element-of script upper S Superscript k Baseline left-parenthesis script upper T right-parenthesis Superscript d and, in case alpha not-identical-to 0 , w equals alpha slash double-vertical-bar alpha double-vertical-bar Subscript 2 comma normal upper Omega in Theorem 3.1 to obtain

StartLayout 1st Row with Label left-parenthesis 5.6 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot nabla alpha d x less-than-or-equal-to double-vertical-bar nabla alpha double-vertical-bar Subscript 2 Baseline left-parenthesis c 2 double-vertical-bar p Subscript h Baseline minus q Subscript h Baseline double-vertical-bar Subscript 2 Baseline plus c 5 double-vertical-bar h Subscript script upper E Superscript 1 slash 2 Baseline left-parenthesis g minus q Subscript h Baseline dot n right-parenthesis double-vertical-bar Subscript 2 comma normal upper Gamma Sub Subscript upper N Subscript Baseline 2nd Row plus c 3 left-parenthesis sigma-summation Underscript z element-of script upper K Endscripts h Subscript z Superscript 2 Baseline min Underscript f Subscript z Baseline element-of double-struck upper R Endscripts double-vertical-bar f plus d i v q Subscript h Baseline minus f Subscript z Baseline double-vertical-bar Subscript 2 comma normal upper Omega Sub Subscript z Subscript Superscript 2 Baseline right-parenthesis Superscript 1 slash 2 Baseline right-parenthesis period EndLayout

The estimate of the last term in Equation5.5 will follow from Theorem 3.1 as well once we establish an analogy to Equation3.1, namely

StartLayout 1st Row with Label left-parenthesis 5.7 right-parenthesis EndLabel integral Underscript normal upper Omega Endscripts left-parenthesis p minus p Subscript h Baseline right-parenthesis dot upper C u r l w Subscript h Baseline d x equals 0 for all w Subscript h Baseline element-of script upper S Subscript upper N Superscript 1 Baseline left-parenthesis script upper T right-parenthesis comma EndLayout