Approximation by quadrilateral finite elements

By Douglas N. Arnold, Daniele Boffi, Richard S. Falk

Abstract

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r plus 1 in upper L Superscript p and order r in upper W Subscript p Superscript 1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

1. Introduction

Finite element spaces are often constructed starting with a finite dimensional space ModifyingAbove upper V With caret of shape functions given on a reference element ModifyingAbove upper K With caret and a class sans-serif upper S of isomorphic mappings of the reference element. If upper F element-of sans-serif upper S , we obtain a space of functions upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis on the image element upper K equals upper F left-parenthesis ModifyingAbove upper K With caret right-parenthesis as the compositions of functions in ModifyingAbove upper V With caret with upper F Superscript negative 1 . Then, given a partition script upper T of a domain normal upper Omega into images of ModifyingAbove upper K With caret under mappings in sans-serif upper S , we obtain a finite element space as a subspaceFootnote1 The subspace is typically determined by some interelement continuity conditions. The imposition of such conditions through the association of local degrees of freedom is an important part of the construction of finite element spaces, but, not being directly relevant to the present work, will not be discussed. of the space upper V Superscript script upper T of all functions on normal upper Omega which restrict to an element of upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis on each upper K element-of script upper T .

For example, if the reference element ModifyingAbove upper K With caret is the unit triangle, the reference space ModifyingAbove upper V With caret is the space script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis of polynomials of degree at most r on ModifyingAbove upper K With caret , and the mapping class sans-serif upper S is the space sans-serif upper A times sans-serif f times sans-serif f left-parenthesis ModifyingAbove upper K With caret right-parenthesis of affine isomorphisms of ModifyingAbove upper K With caret into double-struck upper R squared , then upper V Superscript script upper T is the familiar space of all piecewise polynomials of degree at most r on an arbitrary triangular mesh script upper T . When sans-serif upper S equals sans-serif upper A times sans-serif f times sans-serif f left-parenthesis ModifyingAbove upper K With caret right-parenthesis , as in this case, we speak of affine finite elements.

If the reference element ModifyingAbove upper K With caret is the unit square, then it is often useful to take sans-serif upper S equal to a larger space than sans-serif upper A times sans-serif f times sans-serif f left-parenthesis ModifyingAbove upper K With caret right-parenthesis , namely the space sans-serif upper B times sans-serif i times sans-serif l left-parenthesis ModifyingAbove upper K With caret right-parenthesis of all bilinear isomorphisms of ModifyingAbove upper K With caret into double-struck upper R squared . Indeed, if we allow only affine images of the unit square, then we obtain only parallelograms, and we are quite limited as to the domains that we can mesh (e.g., it is not possible to mesh a triangle with parallelograms). On the other hand, with bilinear images of the square we obtain arbitrary convex quadrilaterals, which can be used to mesh arbitrary polygons.

The above framework is also well suited to studying the approximation properties of finite element spaces (e.g., see Reference2 and Reference1). A fundamental result holds in the case of affine finite elements: sans-serif upper S equals sans-serif upper A times sans-serif f times sans-serif f left-parenthesis ModifyingAbove upper K With caret right-parenthesis . Under the assumption that the reference space ModifyingAbove upper V With caret superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , the following result is well known: if script upper T 1 , script upper T 2 , …is any shape-regular sequence of triangulations of a domain normal upper Omega and u is any smooth function on normal upper Omega , then the upper L Superscript p error in the best approximation of u by functions in upper V Superscript script upper T Super Subscript n is upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis and the piecewise upper W Subscript p Superscript 1 error is upper O left-parenthesis h Superscript r Baseline right-parenthesis , where h equals h left-parenthesis script upper T Subscript n Baseline right-parenthesis is the maximum element diameter. (Here, and throughout the paper, p can take any value between 1 and normal infinity , inclusive.) It is also true, even if less well-known, that the condition that ModifyingAbove upper V With caret superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis is necessary if these estimates are to hold. In fact, the problem of characterizing what is needed for optimal order approximation arises naturally in the study of the finite element method and has been investigated for some time (see Reference9).

The above result does not restrict the choice of reference element ModifyingAbove upper K With caret , so it applies to rectangular and parallelogram meshes by taking ModifyingAbove upper K With caret to be the unit square. But it does not apply to general quadrilateral meshes, since to obtain them we must choose sans-serif upper S equals sans-serif upper B times sans-serif i times sans-serif l left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and the result only applies to affine finite elements. In this case there is a standard result analogous to the positive result in the previous paragraph (Reference2, Reference1, Reference4, Section I.A.2). Namely, if ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , then for any shape-regular sequence of quadrilateral partitions of a domain normal upper Omega and any smooth function u on normal upper Omega , we again obtain that the error in the best approximation of u by functions in upper V Superscript script upper T Super Subscript n is upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis in upper L Superscript p and upper O left-parenthesis h Superscript r Baseline right-parenthesis in (piecewise) upper W Subscript p Superscript 1 . It turns out, as we shall show in this paper, that the hypothesis that ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis is strictly necessary for these estimates to hold. In particular, if ModifyingAbove upper V With caret superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis but ModifyingAbove upper V With caret neither-a-superset-of-nor-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , then the rate of approximation achieved on general shape-regular quadrilateral meshes will be strictly lower than is obtained using meshes of rectangles or parallelograms.

More precisely, we shall exhibit in Section 3 a domain normal upper Omega and a sequence, script upper T 1 , script upper T 2 , …of quadrilateral meshes of it, and prove that whenever upper V left-parenthesis ModifyingAbove upper K With caret right-parenthesis neither-a-superset-of-nor-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , then there is a function u on normal upper Omega such that

inf Underscript v element-of upper V Superscript script upper T Super Subscript n Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline not-equals o left-parenthesis h Superscript r Baseline right-parenthesis

(and so, a fortiori, is not-equals upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis ). A similar result holds for upper W Subscript p Superscript 1 approximation. The counterexample is far from pathological. Indeed, the domain normal upper Omega is as simple as possible, namely a square; the mesh sequence script upper T Subscript n is simple and as shape-regular as possible in that all elements at all mesh levels are similar to a single fixed trapezoid; and the function u is as smooth as possible, namely a polynomial of degree r .Footnote2 However, as discussed at the end of Section 3 and illustrated numerically in Section 4, for a sequence of meshes which is asymptotically parallelogram (as defined at the end of Section 3), the hypothesis upper V left-parenthesis ModifyingAbove upper K With caret right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis is sufficient for optimal order approximation.

The use of a reference space which contains script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis but not script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis is not unusual, but the degradation of convergence order that this implies on general quadrilateral meshes in comparison to rectangular (or parallelogram) meshes is not widely appreciated.

We finish this introduction by considering some examples. Henceforth we shall always use ModifyingAbove upper K With caret to denote the unit square. First, consider finite elements with the simple polynomial spaces as shape functions: ModifyingAbove upper V With caret equals script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . These of course yield upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis approximation in upper L Superscript p for rectangular meshes. However, since script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis superset-of-or-equal-to script upper Q Subscript left floor r slash 2 right floor Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis but script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis neither-a-superset-of-nor-equal-to script upper Q Subscript left floor r slash 2 right floor plus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , on general quadrilateral meshes they only afford upper O left-parenthesis h Superscript left floor r slash 2 right floor plus 1 Baseline right-parenthesis approximation.

A similar situation holds for serendipity finite element spaces, which have been popular in engineering computation for thirty years. These spaces are constructed using as reference shape functions the space script upper S Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis which is the span of script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis together with the two monomials ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret and ModifyingAbove y With caret ModifyingAbove x With caret Superscript r . (The purpose of the additional two functions is to allow local degrees of freedom which can be used to ensure interelement continuity.) For r equals 1 , script upper S 1 left-parenthesis ModifyingAbove upper K With caret right-parenthesis equals script upper Q 1 left-parenthesis ModifyingAbove upper K With caret right-parenthesis , but for r greater-than 1 the situation is similar to that for script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , namely script upper S Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis superset-of-or-equal-to script upper Q Subscript left floor r slash 2 right floor Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis but script upper S Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis neither-a-superset-of-nor-equal-to script upper Q Subscript left floor r slash 2 right floor plus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . So, again, the asymptotic accuracy achieved for general quadrilateral meshes is only about half that achieved for rectangular meshes: upper O left-parenthesis h Superscript left floor r slash 2 right floor plus 1 Baseline right-parenthesis in upper L Superscript p and upper O left-parenthesis h Superscript left floor r slash 2 right floor Baseline right-parenthesis in upper W Subscript p Superscript 1 . In Section 4 we illustrate this with a numerical example. The fact that the eight-node serendipity space script upper S 2 does not perform as well as the nine-node space script upper Q 2 on distorted meshes has been noted previously by several authors, often as a result of numerical experiments. See, for example, Reference11, Section 8.7, Reference6, Reference5, Reference10.

While the serendipity elements are commonly used for solving second order differential equations, the pure polynomial spaces script upper P Subscript r can only be used on quadrilaterals when interelement continuity is not required. This is the case in several mixed methods. For example, a popular element choice to solve the stationary Stokes equations is bilinearly mapped piecewise continuous script upper Q 2 elements for the two components of velocity, and discontinuous piecewise linear elements for the pressure. This is known to be a stable mixed method and gives second order convergence in upper H Superscript 1 for the velocity and upper L squared for the pressure. If one were to define the pressure space instead by using the construction discussed above, namely by composing linear functions on the reference square with bilinear mappings, then the approximation properties of mapped script upper P 1 discussed above would imply that the method could be at most first order accurate, at least for the pressures. Hence, although the use of mapped script upper P 1 as an alternative to unmapped script upper P 1 pressure elements is sometimes proposed Reference8, it is probably not advisable.

Another place where mapped script upper P Subscript r spaces arise is for approximating the scalar variable in mixed finite element methods for second order elliptic equations. Although the scalar variable is discontinuous, in order to prove stability it is generally necessary to define the space for approximating it by composition with the mapping to the reference element (while the space for the vector variable is defined by a contravariant mapping associated with the mapping to the reference element). In the case of the Raviart–Thomas rectangular elements, the scalar space on the reference square is script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , which maintains full upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis approximation properties under bilinear mappings. By contrast, the scalar space used with the Brezzi-Douglas-Marini and the Brezzi-Douglas-Fortin-Marini spaces is script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . This necessarily results in a loss of approximation order when mapped to quadrilaterals by bilinear mappings.

Another type of element which shares this difficulty is the simplest nonconforming quadrilateral element, which generalizes to quadrilaterals the well-known piecewise linear nonconforming element on triangles, with degrees of freedom at the midpoints of edges. On the square, a bilinear function is not well-defined by giving its value at the midpoint of edges (or its average on edges), because these quantities do not comprise a unisolvent set of degrees of freedom (the function left-parenthesis ModifyingAbove x With caret minus 1 slash 2 right-parenthesis left-parenthesis ModifyingAbove y With caret minus 1 slash 2 right-parenthesis vanishes at the four midpoints of the edges of the unit square). Hence, various definitions of nonconforming elements on rectangles replace the basis function ModifyingAbove x With caret ModifyingAbove y With caret by some other function, such as ModifyingAbove x With caret squared minus ModifyingAbove y With caret squared . Consequently, the reference space contains script upper P 1 left-parenthesis ModifyingAbove upper K With caret right-parenthesis , but does not contain script upper Q 1 left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and so there is a degradation of convergence on quadrilateral meshes. This is discussed and analyzed in the context of the Stokes problem in Reference7.

As a final application, we remark that many of the finite element methods proposed for the Reissner–Mindlin plate problem are based on mixed methods for the Stokes equations and/or for second order elliptic problems. As a result, many of them suffer from the same sort of degradation of convergence on quadrilateral meshes. An analysis of a variety of these elements will appear in forthcoming work by the present authors.

In Section 3, we prove our main result, the necessity of the condition that the reference space contain script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis in order to obtain upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis approximation on quadrilateral meshes. The proof relies on an analogous result for affine approximation on rectangular meshes, where the space script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis enters rather than script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . While this is a special case of known results, for the convenience of the reader we include an elementary proof in Section 2. Also in Section 3, we consider the case of asymptotically parallelogram meshes and show that in this situation, an upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis approximation is obtained if the reference space only contains script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . In the final section, we illustrate the results with numerical computations.

2. Approximation theory of rectangular elements

In this section, we prove some results concerning approximation by rectangular elements which will be needed to prove the main results in Section 3. The results in this section are essentially known, and many are true in far greater generality than stated here.

If upper K is any square with edges parallel to the axes, then upper K equals upper F Subscript upper K Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , where upper F Subscript upper K Baseline left-parenthesis ModifyingAbove x With caret right-parenthesis colon equals x Subscript upper K plus h Subscript upper K Baseline ModifyingAbove x With caret with x Subscript upper K Baseline element-of double-struck upper R squared and h Subscript upper K Baseline greater-than 0 the side length. For any function u element-of upper L Superscript 1 Baseline left-parenthesis upper K right-parenthesis , we define ModifyingAbove u With caret Subscript upper K Baseline equals u ring upper F Subscript upper K Baseline element-of upper L Superscript 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , i.e., ModifyingAbove u With caret Subscript upper K Baseline left-parenthesis ModifyingAbove x With caret right-parenthesis equals u left-parenthesis x Subscript upper K Baseline plus h Subscript upper K Baseline ModifyingAbove x With caret right-parenthesis . Given a subspace ModifyingAbove upper S With caret of upper L Superscript 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , we define the associated subspace on an arbitrary square upper K by

upper S left-parenthesis upper K right-parenthesis equals StartSet u colon upper K right-arrow double-struck upper R vertical-bar ModifyingAbove u With caret Subscript upper K Baseline element-of ModifyingAbove upper S With caret EndSet period

Finally, let normal upper Omega denote the unit square ( normal upper Omega and ModifyingAbove upper K With caret both denote the unit square, but we use the notation normal upper Omega when we think of it as a fixed domain, while we use ModifyingAbove upper K With caret when we think of it as a reference element). For n equals 1 comma 2 comma period period period , let script upper T Subscript h be the uniform mesh of normal upper Omega into n squared subsquares when h equals 1 slash n , and define

upper S Subscript h Baseline equals StartSet u colon normal upper Omega right-arrow double-struck upper R StartAbsoluteValue u EndAbsoluteValue Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis for all upper K element-of script upper T Subscript h Baseline EndSet period

In this definition, when we write u vertical-bar Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis we mean only that u vertical-bar Subscript upper K Baseline agrees with a function in upper S left-parenthesis upper K right-parenthesis almost everywhere, and so do not impose any interelement continuity.

The following theorem gives a set of equivalent conditions for optimal order approximation of a smooth function u by elements of upper S Subscript h .

Theorem 1

Suppose 1 less-than-or-equal-to p less-than-or-equal-to normal infinity . Let ModifyingAbove upper S With caret be a finite dimensional subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and r a nonnegative integer. The following conditions are equivalent:

(1)

There is a constant upper C such that inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis .

(2)

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r Baseline right-parenthesis for all u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis .

(3)

script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of ModifyingAbove upper S With caret .

Proof.

For the proof we assume that p less-than normal infinity , but the argument carries over to the case p equals normal infinity with obvious modifications. The first condition implies that

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals 0 for u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis comma

and so implies the second condition. The fact that the third condition implies the first is a well-known consequence of the Bramble–Hilbert lemma. So we need only show that the second condition implies the third.

The proof is by induction on r . First consider the case r equals 0 . We have

StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Superscript p Baseline equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript v Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis Endscripts double-vertical-bar u minus v Subscript upper K Baseline double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p Baseline equals h squared sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar ModifyingAbove u With caret Subscript upper K Baseline minus w double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline comma EndLayout

where we have made the change of variable w equals ModifyingAbove v With caret Subscript upper K in the last step.

In particular, for u identical-to 1 on normal upper Omega , ModifyingAbove u With caret Subscript upper K Baseline identical-to 1 on ModifyingAbove upper K With caret for all upper K , so the quantity

c colon equals inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar ModifyingAbove u With caret Subscript upper K Baseline minus w double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p

is independent of upper K . Thus

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Superscript p Baseline equals h squared sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts c equals c period

The hypothesis that this quantity is o left-parenthesis 1 right-parenthesis implies that c equals 0 , i.e., that the constant function belongs to ModifyingAbove upper S With caret .

Now we consider the case r greater-than 0 . We again apply Equation1, this time for u an arbitrary homogeneous polynomial of degree r . Then

StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel ModifyingAbove u With caret Subscript upper K Baseline left-parenthesis ModifyingAbove x With caret right-parenthesis equals u left-parenthesis x Subscript upper K Baseline plus h ModifyingAbove x With caret right-parenthesis equals u left-parenthesis h ModifyingAbove x With caret right-parenthesis plus q left-parenthesis ModifyingAbove x With caret right-parenthesis equals h Superscript r Baseline u left-parenthesis ModifyingAbove x With caret right-parenthesis plus q left-parenthesis ModifyingAbove x With caret right-parenthesis comma EndLayout

where q element-of script upper P Subscript r minus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . Substituting in Equation1, and invoking the inductive hypothesis that ModifyingAbove upper S With caret superset-of-or-equal-to script upper P Subscript r minus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , we get that

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Superscript p Baseline equals h Superscript 2 plus p r Baseline sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar u minus w double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline equals h Superscript p r Baseline inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar u minus w double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline comma

where the last equality follows from the fact that the previous infimum is independent of upper K . Since the last expression is o left-parenthesis h Superscript p r Baseline right-parenthesis , we immediately deduce that u belongs to ModifyingAbove upper S With caret . Thus ModifyingAbove upper S With caret contains all homogeneous polynomials of degree r and all polynomials of degree less than r (by induction), so it indeed contains all polynomials of degree at most r .

A similar theorem holds for gradient approximation. Since the finite elements are not necessarily continuous, we write nabla Subscript h for the gradient operator applied piecewise on each element.

Theorem 2

Suppose 1 less-than-or-equal-to p less-than-or-equal-to normal infinity . Let ModifyingAbove upper S With caret be a finite dimensional subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and r a nonnegative integer. The following conditions are equivalent:

(1)

There is a constant upper C such that inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis .

(2)

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r minus 1 Baseline right-parenthesis for all u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis .

(3)

script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis plus ModifyingAbove upper S With caret .

Proof.

Again, we need only prove that the second condition implies the third. In analogy to Equation1, we have

StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel StartLayout 1st Row 1st Column inf Underscript v element-of upper S Subscript h Endscripts sigma-summation Underscript upper K element-of script upper T Subscript h Endscripts double-vertical-bar nabla left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p 2nd Column equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript v Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis Endscripts double-vertical-bar nabla left-parenthesis u minus v Subscript upper K Baseline right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p Baseline 2nd Row 1st Column Blank 2nd Column equals h Superscript 2 minus p Baseline sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar nabla left-parenthesis ModifyingAbove u With caret Subscript upper K Baseline minus w right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline comma EndLayout EndLayout

where we have made the change of variable w equals ModifyingAbove v With caret Subscript upper K in the last step.

The proof proceeds by induction on r , the case r equals 0 being trivial. For r greater-than 0 , apply Equation3 with u an arbitrary homogeneous polynomial of degree r . Substituting Equation2 in Equation3, and invoking the inductive hypothesis that script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis plus ModifyingAbove upper S With caret superset-of-or-equal-to script upper P Subscript r minus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , we get that

StartLayout 1st Row 1st Column inf Underscript v element-of upper S Subscript h Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Superscript p 2nd Column equals h Superscript 2 minus p plus p r Baseline sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar nabla left-parenthesis u minus w right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline 2nd Row 1st Column Blank 2nd Column equals h Superscript p left-parenthesis r minus 1 right-parenthesis Baseline inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar nabla left-parenthesis u minus w right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline period EndLayout

Since we assume that this quantity is o left-parenthesis h Superscript p left-parenthesis r minus 1 right-parenthesis Baseline right-parenthesis , the last infimum must be 0 , so u differs from an element of ModifyingAbove upper S With caret by a constant. Thus script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis plus ModifyingAbove upper S With caret contains all homogeneous polynomials of degree r and all polynomials of degree less than r (by induction), so it indeed contains all polynomials of degree at most r .

Remarks

1. If ModifyingAbove upper S With caret contains script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis , which is usually the case, then the third condition of Theorem 2 reduces to that of Theorem 1.

2. A similar result holds for higher derivatives (replace nabla Subscript h by nabla Subscript h Superscript m in the first two conditions, and script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis by script upper P Subscript m minus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis in the third).

3. Approximation theory of quadrilateral elements

In this, the main section of the paper, we consider the approximation properties of finite element spaces defined with respect to quadrilateral meshes using bilinear mappings starting from a given finite dimensional space of polynomials ModifyingAbove upper V With caret on the unit square ModifyingAbove upper K With caret equals left-bracket 0 comma 1 right-bracket times left-bracket 0 comma 1 right-bracket . For simplicity, we assume that ModifyingAbove upper V With caret superset-of-or-equal-to script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis . For example, ModifyingAbove upper V With caret might be the space script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis of polynomials of total degree at most r , or the space script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis of polynomials of degree at most r in each variable separately, or the serendipity space script upper S Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis spanned by script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis together with the monomials ModifyingAbove x With caret Subscript 1 Superscript r Baseline ModifyingAbove x With caret Subscript 2 and ModifyingAbove x With caret Subscript 1 Baseline ModifyingAbove x With caret Subscript 2 Superscript r . Let upper F be a bilinear isomorphism of ModifyingAbove upper K With caret onto a convex quadrilateral upper K equals upper F left-parenthesis ModifyingAbove upper K With caret right-parenthesis . Then for u element-of upper L Superscript 1 Baseline left-parenthesis upper K right-parenthesis we define ModifyingAbove u With caret Subscript upper K comma upper F Baseline element-of upper L Superscript 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis by ModifyingAbove u With caret Subscript upper K comma upper F Baseline equals u ring upper F , and set

upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis equals StartSet u colon upper K right-arrow double-struck upper R vertical-bar ModifyingAbove u With caret Subscript upper K comma upper F Baseline element-of ModifyingAbove upper V With caret EndSet period

(Note that we have changed notation slightly from Section 2 to include the mapping upper F , since various definitions depend on the particular choice of the bilinear isomorphism upper F of ModifyingAbove upper K With caret onto upper K . Whenever the space ModifyingAbove upper V With caret is invariant under the symmetries of the square, which is usually the case in practice, this will not be so.) We also note that the functions in upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis need not be polynomials if upper F is not affine, i.e., if upper K is not a parallelogram.

Given a quadrilateral mesh script upper T of some domain, normal upper Omega , we can then construct the space of functions upper V Superscript script upper T consisting of functions on the domain which when restricted to a quadrilateral upper K element-of script upper T belong to upper V Subscript upper F Sub Subscript upper K Baseline left-parenthesis upper K right-parenthesis , where upper F Subscript upper K is a bilinear isomorphism of ModifyingAbove upper K With caret onto upper K . (Again, if ModifyingAbove upper V With caret is not invariant under the symmetries of the square, the space upper V Superscript script upper T will depend on the specific choice of the maps upper F Subscript upper K .)

It follows from the results of the previous section that if we consider the sequence of meshes of the unit square into congruent subsquares of side length h equals 1 slash n , then each of the approximation estimates

StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel inf Underscript v element-of upper V Superscript script upper T Super Subscript h Superscript Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis comma 2nd Row with Label left-parenthesis 5 right-parenthesis EndLabel inf Underscript v element-of upper V Superscript script upper T Super Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis EndLayout

holds if and only if script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of ModifyingAbove upper V With caret . It is not hard to extend these estimates to shape-regular sequences of parallelogram meshes as well. However, in this section we show that for these estimates to hold for more general quadrilateral mesh sequences, a stronger condition on ModifyingAbove upper V With caret is required, namely that ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis .

The positive result, that when ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , then the estimates Equation4 and Equation5 hold for any shape-regular sequence of quadrilateral meshes script upper T Subscript h , is known. See, e.g., Reference2, Reference1, or Reference4, Section I.A.2. We wish to show the necessity of the condition ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis .

As a first step, we show that the condition upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K right-parenthesis is necessary and sufficient to have that ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis whenever upper F is a bilinear isomorphism of ModifyingAbove upper K With caret onto a convex quadrilateral. This is proven in the following two theorems.

Theorem 3

Suppose that ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . Let upper F be any bilinear isomorphism of ModifyingAbove upper K With caret onto a convex quadrilateral. Then upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K right-parenthesis .

Proof.

The components of upper F left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis are linear functions of ModifyingAbove x With caret and ModifyingAbove y With caret , so if p is a polynomial of total degree at most r , then p left-parenthesis upper F left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis right-parenthesis is of degree at most r in ModifyingAbove x With caret and ModifyingAbove y With caret separately, i.e., p ring upper F element-of script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of ModifyingAbove upper V With caret . Therefore p element-of upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis .

The reverse implication holds even under the weaker assumption that upper V Subscript upper F Baseline left-parenthesis upper K right-parenthesis contains script upper P Subscript r Baseline left-parenthesis upper K right-parenthesis just for the two specific bilinear isomorphisms

ModifyingAbove upper F With tilde left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis equals left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret left-parenthesis ModifyingAbove x With caret plus 1 right-parenthesis right-parenthesis comma ModifyingAbove upper F With bar left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis equals left-parenthesis ModifyingAbove y With caret comma ModifyingAbove x With caret left-parenthesis ModifyingAbove y With caret plus 1 right-parenthesis right-parenthesis comma

both of which map ModifyingAbove upper K With caret isomorphically onto the quadrilateral upper K prime with vertices left-parenthesis 0 comma 0 right-parenthesis , left-parenthesis 1 comma 0 right-parenthesis , left-parenthesis 0 comma 1 right-parenthesis , and left-parenthesis 1 comma 2 right-parenthesis . This fact is established below.

Theorem 4

Let ModifyingAbove upper V With caret be a vector space of functions on ModifyingAbove upper K With caret . Suppose that upper V Subscript upper F overTilde Baseline left-parenthesis upper K prime right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K prime right-parenthesis and upper V Subscript upper F overbar Baseline left-parenthesis upper K prime right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K prime right-parenthesis . Then ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis .

Proof.

We prove that ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis by induction on r . The case r equals 0 being true by assumption, we consider r greater-than 0 and show that the monomials ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript s and ModifyingAbove x With caret Superscript s Baseline ModifyingAbove y With caret Superscript r belong to ModifyingAbove upper V With caret for s equals 0 comma 1 comma period period period comma r . From the identity

StartLayout 1st Row with Label left-parenthesis 6 right-parenthesis EndLabel StartLayout 1st Row 1st Column ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript s 2nd Column equals ModifyingAbove x With caret Superscript r minus s Baseline left-bracket ModifyingAbove y With caret left-parenthesis ModifyingAbove x With caret plus 1 right-parenthesis right-bracket Superscript s Baseline minus sigma-summation Underscript t equals 1 Overscript s Endscripts StartBinomialOrMatrix s Choose t EndBinomialOrMatrix ModifyingAbove x With caret Superscript r minus t Baseline ModifyingAbove y With caret Superscript s Baseline 2nd Row 1st Column Blank 2nd Column equals upper F overTilde Subscript 1 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript r minus s Baseline upper F overTilde Subscript 2 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript s Baseline minus sigma-summation Underscript t equals 1 Overscript s Endscripts StartBinomialOrMatrix s Choose t EndBinomialOrMatrix ModifyingAbove x With caret Superscript r minus t Baseline ModifyingAbove y With caret Superscript s Baseline comma EndLayout EndLayout

we see that for 0 less-than-or-equal-to s less-than r , the monomial ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript s is the sum of a polynomial which clearly belongs to ModifyingAbove upper V With caret (since upper F overTilde Subscript 1 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript r minus s Baseline upper F overTilde Subscript 2 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript s Baseline equals x Superscript r minus s Baseline y Superscript s Baseline element-of script upper P Subscript r Baseline left-parenthesis upper K prime right-parenthesis subset-of upper V Subscript upper F overTilde Baseline left-parenthesis upper K prime right-parenthesis ) and a polynomial in upper Q Subscript r minus 1 Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , which belongs to ModifyingAbove upper V With caret by induction. Thus each of the monomials ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript s with 0 less-than-or-equal-to s less-than r belongs to ModifyingAbove upper V With caret , and, using upper F overbar , we similarly see that all the monomials ModifyingAbove x With caret Superscript s Baseline ModifyingAbove y With caret Superscript r , 0 less-than-or-equal-to s less-than r , belong to ModifyingAbove upper V With caret . Finally, from Equation6 with s equals r , we see that ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript r is a linear combination of an element of ModifyingAbove upper V With caret and monomials ModifyingAbove x With caret Superscript s Baseline ModifyingAbove y With caret Superscript r with s less-than r , so it too belongs to ModifyingAbove upper V With caret .

We now combine this result with those of the previous section to show the necessity of the condition ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis for optimal order approximation. Let ModifyingAbove upper V With caret be some fixed finite dimensional subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis which does not include upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis . Consider the specific division of the unit square ModifyingAbove upper K With caret into four quadrilaterals shown on the left in Figure 1. For definiteness we place the vertices of the quadrilaterals at left-parenthesis 0 comma 1 slash 3 right-parenthesis , left-parenthesis 1 slash 2 comma 2 slash 3 right-parenthesis and left-parenthesis 1 comma 1 slash 3 right-parenthesis and the midpoints of the horizontal edges and the corners of ModifyingAbove upper K With caret .

The four quadrilaterals are mutually congruent and affinely related to the specific quadrilateral upper K prime defined above. Therefore, by Theorem 4, we can define for each of the four quadrilaterals upper K double-prime shown in Figure 1 an isomorphism upper F double-prime from the unit square so that upper V Subscript upper F double-prime Baseline left-parenthesis upper K double-prime right-parenthesis neither-a-superset-of-nor-equal-to script upper P Subscript r Baseline left-parenthesis upper K double-prime right-parenthesis . If we let ModifyingAbove upper S With caret be the subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis consisting of functions which restrict to elements of upper V Subscript upper F double-prime Baseline left-parenthesis upper K double-prime right-parenthesis on each of the four quadrilaterals upper K double-prime , then certainly ModifyingAbove upper S With caret does not contain script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , since even its restriction to any one of the quadrilaterals upper K double-prime does not contain upper P Subscript r Baseline left-parenthesis upper K double-prime right-parenthesis .

Next, for n equals 1 comma 2 comma period period period consider the mesh script upper T prime Subscript h of the unit square normal upper Omega shown in Figure 1b, obtained by first dividing it into a uniform n times n mesh of subsquares, n equals 1 slash h , and then dividing each subsquare as in Figure 1a. Then the space of functions u on normal upper Omega whose restrictions on each subsquare upper K element-of script upper T Subscript h satisfy ModifyingAbove u With caret Subscript upper K Baseline left-parenthesis ModifyingAbove x With caret right-parenthesis equals u left-parenthesis x Subscript upper K Baseline plus h ModifyingAbove x With caret right-parenthesis with ModifyingAbove u With caret Subscript upper K Baseline element-of ModifyingAbove upper S With caret is precisely the same as the space upper V left-parenthesis script upper T prime Subscript h right-parenthesis constructed from the initial space ModifyingAbove upper V With caret and the mesh script upper T prime Subscript h . In view of Theorems 1 and 2 and the fact that ModifyingAbove upper S With caret neither-a-superset-of-nor-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , the estimates Equation4 and Equation5 do not hold. In fact, neither of the estimates

inf Underscript v element-of upper V left-parenthesis script upper T Subscript h Baseline right-parenthesis Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r Baseline right-parenthesis

nor

inf Underscript v element-of upper V left-parenthesis script upper T Subscript h Baseline right-parenthesis Endscripts double-vertical-bar nabla left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r minus 1 Baseline right-parenthesis

holds, even for u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis .

While the condition ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis is necessary for upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis approximation on general quadrilateral meshes, the condition ModifyingAbove upper V With caret superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis suffices for meshes of parallelograms. Naturally, the same is true for meshes whose elements are sufficiently close to parallelograms. We conclude this section with a precise statement of this result and a sketch of the proof. If ModifyingAbove upper V With caret superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis and upper K equals upper F left-parenthesis ModifyingAbove upper K With caret right-parenthesis with upper F element-of sans-serif upper B times sans-serif i times sans-serif l left-parenthesis ModifyingAbove upper K With caret right-parenthesis , then by standard arguments, as in Reference1, we get

double-vertical-bar v minus pi Subscript upper K Baseline v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Baseline less-than-or-equal-to upper C double-vertical-bar upper J Subscript upper F Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript 1 slash p Baseline StartAbsoluteValue v ring upper F EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline comma

where upper J Subscript upper F is the Jacobian determinant of upper F and pi Subscript upper K denotes any convenient projection or interpolant satisfying left-parenthesis pi Subscript upper K Baseline v right-parenthesis ring upper F equals pi Subscript ModifyingAbove upper K With caret Baseline left-parenthesis v ring upper F right-parenthesis , e.g., the upper L squared projection. Now, using the formula for the derivative of a composition (as in, e.g., Reference3, p. 222), and the fact that upper F is quadratic, and so its third and higher derivatives vanish, we get that

StartAbsoluteValue v ring upper F EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline less-than-or-equal-to upper C double-vertical-bar upper J Subscript upper F Sub Superscript negative 1 Subscript Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper K right-parenthesis Superscript 1 slash p Baseline double-vertical-bar v double-vertical-bar Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline sigma-summation Underscript i equals 0 Overscript left floor left-parenthesis r plus 1 right-parenthesis slash 2 right floor Endscripts StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 1 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript r plus 1 minus 2 i Baseline StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 2 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript i Baseline period

Now,

double-vertical-bar upper J Subscript upper F Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline less-than-or-equal-to upper C h Subscript upper K Superscript 2 Baseline comma double-vertical-bar upper J Subscript upper F Sub Superscript negative 1 Subscript Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline less-than-or-equal-to upper C h Subscript upper K Superscript negative 2 Baseline comma StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 1 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline less-than-or-equal-to upper C h Subscript upper K Baseline comma

where h Subscript upper K is the diameter of upper K and upper C depends only on the shape-regularity of upper K . We thus get

double-vertical-bar v minus pi Subscript upper K Baseline v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Baseline less-than-or-equal-to upper C double-vertical-bar v double-vertical-bar Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline sigma-summation Underscript i Endscripts h Subscript upper K Superscript r plus 1 minus 2 i Baseline StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 2 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript i Baseline period

It follows that if StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 2 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline equals upper O left-parenthesis h Subscript upper K Superscript 2 Baseline right-parenthesis , we get the desired estimate

double-vertical-bar v minus pi Subscript upper K Baseline v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Baseline less-than-or-equal-to upper C h Subscript upper K Superscript r plus 1 Baseline double-vertical-bar v double-vertical-bar Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis upper K right-parenthesis Baseline period

Following Reference7, we measure the deviation of a quadrilateral from a parallelogram by the quantity sigma Subscript upper K Baseline colon equals max left-parenthesis StartAbsoluteValue pi minus theta 1 EndAbsoluteValue comma StartAbsoluteValue pi minus theta 2 EndAbsoluteValue right-parenthesis , where theta 1 is the angle between the outward normals of two opposite sides of upper K and theta 2 is the angle between the outward normals of the other two sides. Thus 0 less-than-or-equal-to sigma Subscript upper K Baseline less-than pi , with sigma Subscript upper K Baseline equals 0 if and only if upper K is a parallelogram. As pointed out in Reference7, StartAbsoluteValue upper F EndAbsoluteValue Subscript upper W Sub Subscript normal infinity Sub Superscript 2 Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Baseline less-than-or-equal-to upper C h Subscript upper K Baseline left-parenthesis h Subscript upper K Baseline plus sigma Subscript upper K Baseline right-parenthesis . This motivates the definition that a family of quadrilateral meshes is asymptotically parallelogram if sigma Subscript upper K Baseline equals upper O left-parenthesis h Subscript upper K Baseline right-parenthesis , i.e., if sigma Subscript upper K Baseline slash h Subscript upper K is uniformly bounded for all the elements in all the meshes. From the foregoing considerations, if the reference space contains script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis we obtain upper O left-parenthesis h Superscript r plus 1 Baseline right-parenthesis convergence for asymptotically parallelogram, shape-regular meshes.

As a final note, we remark that any polygon can be meshed by an asymptotically parallelogram, shape-regular family of meshes with mesh size tending to zero. Indeed, if we begin with any mesh of convex quadrilaterals, and refine it by dividing each quadrilateral in four by connecting the midpoints of the opposite edges, and continue in this fashion, as in the last row of Figure 2, the resulting mesh is asymptotically parallelogram and shape-regular.

4. Numerical results

In this section, we report on results from a numerical study of the behavior of piecewise continuous mapped biquadratic and serendipity finite elements on quadrilateral meshes (i.e., the finite element spaces are constructed starting from the spaces upper Q 2 left-parenthesis ModifyingAbove upper K With caret right-parenthesis and upper S 2 left-parenthesis ModifyingAbove upper K With caret right-parenthesis on the reference square and then imposing continuity). We present the results of two test problems. In both we solve the Dirichlet problem for Poisson’s equation

StartLayout 1st Row with Label left-parenthesis 7 right-parenthesis EndLabel minus normal upper Delta u equals f in normal upper Omega comma u equals g on partial-differential normal upper Omega comma EndLayout

where the domain normal upper Omega is the unit square. In the first problem, f and g are taken so that the exact solution is the quartic polynomial

u left-parenthesis x comma y right-parenthesis equals x cubed plus 5 y squared minus 10 y cubed plus y Superscript 4 Baseline period

Tables 1 and 2 show results for both types of elements using meshes from each of the first two mesh sequences shown in Figure 2. The first sequence of meshes consists of uniform square subdivisions of the domain into n times n subsquares, n equals 2 comma 4 comma 8 comma period period period . Meshes in the second sequence are partitions of the domain into n times n congruent trapezoids, all similar to the trapezoid with vertices left-parenthesis 0 comma 0 right-parenthesis , left-parenthesis 1 slash 2 comma 0 right-parenthesis , left-parenthesis 1 slash 2 comma 2 slash 3 right-parenthesis , and left-parenthesis 0 comma 1 slash 3 right-parenthesis . In Tables 1 and 2 we report the errors in upper L squared and upper L Superscript normal infinity , respectively, for the finite element solution and its gradient both in absolute terms and as a percentage of the norm of the exact solution and its gradient, and we also report the apparent rate of convergence based on consecutive meshes in a sequence. For this test problem, the rates of convergence are very clear: for either mesh sequence, the mapped biquadratic elements converge with the expected order 3 for the solution and 2 for its gradient. The same is true for the serendipity elements on the square meshes, but, as predicted by the theory given above, for the trapezoidal mesh sequence the order of convergence for the serendipity elements is reduced by 1 both for the solution and for its gradient.

As a second test example we again solved the Dirichlet problem Equation7, but this time choosing the data so that the solution is the sharply peaked function

u left-parenthesis x comma y right-parenthesis equals exp left-parenthesis minus 100 left-bracket left-parenthesis x minus 1 slash 4 right-parenthesis squared plus left-parenthesis y minus 1 slash 3 right-parenthesis squared right-bracket right-parenthesis period

As seen in Table 3, in this case the loss of convergence order in the upper L squared norm for the serendipity elements on the trapezoidal mesh is not nearly as clear. Some loss is evident, but apparently very fine meshes (and very high precision computation) would be required to see the final asymptotic orders.

Similar results hold when the error in the upper L Superscript normal infinity norm is considered, as shown in Table 4.

Finally we return to the first test problem, and consider the behavior of the serendipity elements on the third mesh sequence shown in Figure 2. This mesh sequence begins with the same mesh of four quadrilaterals as in previous case, and continues with systematic refinement as described at the end of the last section, and so is asymptotically parallelogram. Therefore, as explained there, the rate of convergence for serendipity elements is the same as for affine meshes. This is clearly illustrated in Table 5.

While the asymptotic rates predicted by the theory are confirmed in these examples, it is worth noting that in absolute terms the effect of the degraded convergence rate is not very pronounced. For the first example, on a moderately fine mesh of 16 times 16 trapezoids, the solution error with serendipity elements exceeds that of mapped biquadratic elements by a factor of about 2, and the gradient error by a factor of 2.5. Even on the finest mesh shown, with 64 times 64 elements, the factors are only about 5.5 and 8.5, respectively. Of course, if we were to compute on finer and finer meshes with sufficiently high precision, these factors would tend to infinity. Indeed, on any quadrilateral mesh which contains a nonparallelogram element, the analogous factors can be made as large as desired by choosing a problem in which the exact solution is sufficiently close to—or even equal to—a quadratic function, which the mapped biquadratic elements capture exactly, while the serendipity elements do not (such a quadratic function always exists). However, it is not unusual that the serendipity elements perform almost as well as the mapped biquadratic elements for reasonable, and even for quite small, levels of error. This, together with their optimal convergence on asymptotically parallelogram meshes, provides an explanation of why the lower rates of convergence have not been widely noted.

Figures

Figure 1.

a. A partition of the square into four trapezoids. b. A mesh composed of translated dilates of this partition.

Figure 2.

Three sequences of meshes of the unit square: square, trapezoidal, and asymptotically parallelogram. Each is shown for n equals 2 , 4 , 8 , and 16 .

Table 1.

upper L squared errors and rates of convergence for the test problem with polynomial solution.

Table 2.

upper L Superscript normal infinity errors and rates of convergence for the test problem with polynomial solution.

Table 3.

upper L squared errors and rates of convergence for the test problem with exponential solution.

Table 4.

upper L Superscript normal infinity errors and rates of convergence for the test problem with exponential solution.

Table 5.

upper L squared errors and rates of convergence for the test problem with polynomial solution using serendipity elements on asymptotically affine meshes.

Mathematical Fragments

Theorem 1

Suppose 1 less-than-or-equal-to p less-than-or-equal-to normal infinity . Let ModifyingAbove upper S With caret be a finite dimensional subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and r a nonnegative integer. The following conditions are equivalent:

(1)

There is a constant upper C such that inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis .

(2)

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r Baseline right-parenthesis for all u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis .

(3)

script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of ModifyingAbove upper S With caret .

Equation (1)
StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Superscript p Baseline equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript v Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis Endscripts double-vertical-bar u minus v Subscript upper K Baseline double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p Baseline equals h squared sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar ModifyingAbove u With caret Subscript upper K Baseline minus w double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline comma EndLayout
Equation (2)
StartLayout 1st Row with Label left-parenthesis 2 right-parenthesis EndLabel ModifyingAbove u With caret Subscript upper K Baseline left-parenthesis ModifyingAbove x With caret right-parenthesis equals u left-parenthesis x Subscript upper K Baseline plus h ModifyingAbove x With caret right-parenthesis equals u left-parenthesis h ModifyingAbove x With caret right-parenthesis plus q left-parenthesis ModifyingAbove x With caret right-parenthesis equals h Superscript r Baseline u left-parenthesis ModifyingAbove x With caret right-parenthesis plus q left-parenthesis ModifyingAbove x With caret right-parenthesis comma EndLayout
Theorem 2

Suppose 1 less-than-or-equal-to p less-than-or-equal-to normal infinity . Let ModifyingAbove upper S With caret be a finite dimensional subspace of upper L Superscript p Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis , and r a nonnegative integer. The following conditions are equivalent:

(1)

There is a constant upper C such that inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis .

(2)

inf Underscript v element-of upper S Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline equals o left-parenthesis h Superscript r minus 1 Baseline right-parenthesis for all u element-of script upper P Subscript r Baseline left-parenthesis normal upper Omega right-parenthesis .

(3)

script upper P Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis subset-of script upper P 0 left-parenthesis ModifyingAbove upper K With caret right-parenthesis plus ModifyingAbove upper S With caret .

Equation (3)
StartLayout 1st Row with Label left-parenthesis 3 right-parenthesis EndLabel StartLayout 1st Row 1st Column inf Underscript v element-of upper S Subscript h Endscripts sigma-summation Underscript upper K element-of script upper T Subscript h Endscripts double-vertical-bar nabla left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p 2nd Column equals sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript v Subscript upper K Baseline element-of upper S left-parenthesis upper K right-parenthesis Endscripts double-vertical-bar nabla left-parenthesis u minus v Subscript upper K Baseline right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis upper K right-parenthesis Superscript p Baseline 2nd Row 1st Column Blank 2nd Column equals h Superscript 2 minus p Baseline sigma-summation Underscript upper K element-of script upper T Subscript h Baseline Endscripts inf Underscript w element-of ModifyingAbove upper S With caret Endscripts double-vertical-bar nabla left-parenthesis ModifyingAbove u With caret Subscript upper K Baseline minus w right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis ModifyingAbove upper K With caret right-parenthesis Superscript p Baseline comma EndLayout EndLayout
Equations (4), (5)
StartLayout 1st Row with Label left-parenthesis 4 right-parenthesis EndLabel inf Underscript v element-of upper V Superscript script upper T Super Subscript h Superscript Baseline Endscripts double-vertical-bar u minus v double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r plus 1 Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis comma 2nd Row with Label left-parenthesis 5 right-parenthesis EndLabel inf Underscript v element-of upper V Superscript script upper T Super Subscript h Baseline Endscripts double-vertical-bar nabla Subscript h Baseline left-parenthesis u minus v right-parenthesis double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Omega right-parenthesis Baseline less-than-or-equal-to upper C h Superscript r Baseline StartAbsoluteValue u EndAbsoluteValue Subscript upper W Sub Subscript p Sub Superscript r plus 1 Subscript left-parenthesis normal upper Omega right-parenthesis Baseline for all u element-of upper W Subscript p Superscript r plus 1 Baseline left-parenthesis normal upper Omega right-parenthesis EndLayout
Theorem 4

Let ModifyingAbove upper V With caret be a vector space of functions on ModifyingAbove upper K With caret . Suppose that upper V Subscript upper F overTilde Baseline left-parenthesis upper K prime right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K prime right-parenthesis and upper V Subscript upper F overbar Baseline left-parenthesis upper K prime right-parenthesis superset-of-or-equal-to script upper P Subscript r Baseline left-parenthesis upper K prime right-parenthesis . Then ModifyingAbove upper V With caret superset-of-or-equal-to script upper Q Subscript r Baseline left-parenthesis ModifyingAbove upper K With caret right-parenthesis .

Equation (6)
StartLayout 1st Row with Label left-parenthesis 6 right-parenthesis EndLabel StartLayout 1st Row 1st Column ModifyingAbove x With caret Superscript r Baseline ModifyingAbove y With caret Superscript s 2nd Column equals ModifyingAbove x With caret Superscript r minus s Baseline left-bracket ModifyingAbove y With caret left-parenthesis ModifyingAbove x With caret plus 1 right-parenthesis right-bracket Superscript s Baseline minus sigma-summation Underscript t equals 1 Overscript s Endscripts StartBinomialOrMatrix s Choose t EndBinomialOrMatrix ModifyingAbove x With caret Superscript r minus t Baseline ModifyingAbove y With caret Superscript s Baseline 2nd Row 1st Column Blank 2nd Column equals upper F overTilde Subscript 1 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript r minus s Baseline upper F overTilde Subscript 2 Baseline left-parenthesis ModifyingAbove x With caret comma ModifyingAbove y With caret right-parenthesis Superscript s Baseline minus sigma-summation Underscript t equals 1 Overscript s Endscripts StartBinomialOrMatrix s Choose t EndBinomialOrMatrix ModifyingAbove x With caret Superscript r minus t Baseline ModifyingAbove y With caret Superscript s Baseline comma EndLayout EndLayout
Equation (7)
StartLayout 1st Row with Label left-parenthesis 7 right-parenthesis EndLabel minus normal upper Delta u equals f in normal upper Omega comma u equals g on partial-differential normal upper Omega comma EndLayout

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Article Information

MSC 2000
Primary: 65N30 (Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods), 41A10 (Approximation by polynomials), 41A25 (Rate of convergence, degree of approximation), 41A27 (Inverse theorems), 41A63 (Multidimensional problems)
Keywords
  • Quadrilateral
  • finite element
  • approximation
  • serendipity
  • mixed finite element
Author Information
Douglas N. Arnold
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota 55455
arnold@ima.umn.edu
Homepage
Daniele Boffi
Dipartimento di Matematica, Università di Pavia, 27100 Pavia, Italy
boffi@dimat.unipv.it
Homepage
Richard S. Falk
Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
falk@math.rutgers.edu
Homepage
Journal Information
Mathematics of Computation, Volume 71, Issue 239, ISSN 0025-5718, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on and published on .
Copyright Information
Copyright 2002 American Mathematical Society
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