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arXiv:math/0005036v1 [math.NA] 3 May 2000

APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS

DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

Abstract. We consider the approximation properties of finite element spaces on quadri-

lateral meshes. The finite element spaces are constructed starting with a given finite di-

mensional 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 suffi-

cient condition for approximation of order r + 1 in L2and order r in H1is 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 spaceˆV of

shape functions given on a reference elementˆK and a class S of isomorphic mappings of the

reference element. If F ∈ S we obtain a space of functions VF(K) on the image element

K = F(ˆK) as the compositions of functions inˆV with F−1. Then, given a partition T of

a domain Ω into images ofˆK under mappings in S, we obtain a finite element space as a

subspace1of the space VTof all functions on Ω which restrict to an element of VF(K) on

each K ∈ T.

For example, if the reference elementˆK is the unit triangle, and the reference spaceˆV

is the space Pr(ˆK) of polynomials of degree at most r onˆK, and the mapping class S is

the space Aff(ˆK) of affine isomorphisms ofˆK into R2, then VTis the familiar space of

all piecewise polynomials of degree at most r on an arbitrary triangular mesh T. When

S = Aff(ˆK), as in this case, we speak of affine finite elements.

If the reference elementˆK is the unit square, then it is often useful to take S equal to a

larger space than Aff(ˆK), namely the space Bil(ˆK) of all bilinear isomorphisms ofˆK into R2.

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

Date: February 25, 2000.

1991 Mathematics Subject Classification. 65N30, 41A10, 41A25, 41A27, 41A63.

Key words and phrases. quadrilateral, finite element, approximation, serendipity, mixed finite element.

1The 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.

1

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2DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

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. See, e.g., [2] and [1]. A fundamental result holds in the case of affine finite

elements: S = Aff(ˆK). Under the assumption that the reference spaceˆV ⊇ Pr(ˆK), the

following result is well known: if T1, T2, ... is any shape-regular sequence of triangulations

of a domain Ω and u is any smooth function on Ω, then the L2error in the best approximation

of u by functions in VTnis O(hr+1) and the piecewise H1error is O(hr), where h = h(Tn) is

the maximum element diameter. It is also true, even if less well-known, that the condition

thatˆV ⊇ Pr(ˆK) is necessary if these estimates are to hold.

The above result does not restrict the choice of reference elementˆK, so it applies to

rectangular and parallelogram meshes by takingˆK to be the unit square. But it does not

apply to general quadrilateral meshes, since to obtain them we must choose S = Bil(ˆK),

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, [2], [1], [4, Section I.A.2]. Namely,

ifˆV ⊇ Qr(ˆK), then for any shape-regular sequence of quadrilateral partitions of a domain Ω

and any smooth function u on Ω, we again obtain that the error in the best approximation

of u by functions in VTnis O(hr+1) in L2and O(hr) in (piecewise) H1. It turns out, as

we shall show in this paper, that the hypothesis thatˆV ⊇ Qr(ˆK) is strictly necessary for

these estimates to hold. In particular, ifˆV ⊇ Pr(ˆK) butˆV ? Qr(ˆK), 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 Ω and a sequence, T1, T2, ... of

quadrilateral meshes of it, and prove that whenever V (ˆK) ? Qr(ˆK), then there is a function

u on Ω such that

inf

v∈VTn?u − v?L2(Ω)?= o(hr),

A similar result holds for H1approximation. (and so, a fortiori, is ?= O(hr+1)).

counterexample is far from pathological. Indeed, the domain Ω is as simple as possible,

namely a square; the mesh sequence Tnis 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.

The use of a reference space which contains Pr(ˆK) but not Qr(ˆK) 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. It has been

observed in special cases, often as a result of numerical experiments, cf. [7, Section 8.7].

We finish this introduction by considering some examples. Henceforth we shall always

useˆK to denote the unit square. First, consider finite elements with the simple polynomial

spaces as shape functions:ˆV = Pr(ˆK). These of course yield O(hr+1) approximation in L2

for rectangular meshes. However, since Pr(ˆK) ⊇ Q⌊r/2⌋(ˆK) but Pr(ˆK) ? Q⌊r/2⌋+1(ˆK), on

general quadrilateral meshes they only afford O(h⌊r/2⌋+1) 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 Sr(ˆK) which is the span of Pr(ˆK) together with the two monomials

The

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APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS3

ˆ xrˆ y and ˆ yˆ xr. (The purpose of the additional two functions is to allow local degrees of freedom

which can be used to ensure interelement continuity.) For r = 1, S1(ˆK) = Q1(ˆK), but for

r > 1 the situation is similar to that for Pr(ˆK), namely Sr(ˆK) ⊇ Q⌊r/2⌋(ˆK) but Sr(ˆK) ?

Q⌊r/2⌋+1(ˆK). So, again, the asymptotic accuracy achieved for general quadrilateral meshes

is only about half that achieved for rectangular meshes: O(h⌊r/2⌋+1) in L2and O(h⌊r/2⌋) in

H1. In Section 4 we illustrate this with a numerical example.

While the serendipity elements are commonly used for solving second order differential

equations, the pure polynomial spaces Prcan only be used on quadrilaterals when interele-

ment 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 piece-

wise continuous Q2elements for the two components of velocity, and discontinuous piecewise

linear elements for the pressure. Typically the pressure space is taken to be functions which

belong to P1(K) on each element K. This is known to be a stable mixed method and gives

second order convergence in H1for the velocity and L2for the pressure. If one were to define

the pressure space instead by using the construction discussed above, namely by composing

linear functions on reference square with bilinear mappings, then the approximation prop-

erties of mapped P1discussed above would imply that method could be at most first order

accurate, at least for the pressures. Hence, although the use of mapped P1as an alternative

to unmapped P1pressure elements is sometimes proposed [6], it is probably not advisable.

Another place where mapped Prspaces 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 Qr(ˆK), which maintains full O(hr+1) approximation

properties under bilinear mappings. By contrast, the scalar space used with the Brezzi-

Douglas-Marini and the Brezzi-Douglas-Fortin-Marini spaces is Pr(ˆK). 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 quadri-

lateral element, which generalizes to quadrilaterals the well-known piecewise linear non-

conforming 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 (ˆ x − 1/2)(ˆ y − 1/2) vanishes at the four midpoints of the edges of the

unit square). Hence, various definitions of nonconforming elements on rectangles replace the

basis function ˆ xˆ y by some other function such as ˆ x2− ˆ y2. Consequently, the reference space

contains P1(ˆK), but does not contain Q1(ˆK), and so there is a degradation of convergence

on quadrilateral meshes. This is discussed and analyzed in the context of the Stokes problem

in [5].

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

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4DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

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 Qr(ˆK) in order to obtain O(hr+1) approximation on quadrilateral meshes. The

proof relies on an analogous result for affine approximation on rectangular meshes, where

the space Pr(ˆK) enters rather than Qr(ˆK). While this is a special case of known results,

for the convenience of the reader we include an elementary proof in Section 2. 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 in the next section where the main results are proved. The results in

this section are essentially known, and many are true in far greater generality than stated

here.

If K is any square with edges parallel to the axes, then K = FK(ˆK) where FK(ˆ x) :=

xK+hKˆ x with xK∈ R2and hK> 0 the side length. For any function u ∈ L2(K), we define

ˆ uK= u ◦ FK∈ L2(ˆK), i.e., ˆ uK(ˆ x) = u(xK+ hKˆ x). Given a subspaceˆS of L2(ˆK) we define

the associated subspace on an arbitrary square K by

S(K) = {u : K → R| ˆ uK∈ˆS }.

Finally, let Ω denote the unit cube (Ω andˆK both denote the unit cube, but we use the

notation Ω when we think of it as a fixed domain, while we useˆK when we think of it as

a reference element). For n = 1,2,..., let Thbe the uniform mesh of Ω into ndsubcubes

when h = 1/n, and define

Sh= {u : Ω → R|u|K∈ S(K) for all K ∈ Th}.

In this definition, when we write u|K∈ S(K) we mean only that u|Kagrees with a function

in SKalmost 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 Sh.

Theorem 1. LetˆS be a finite dimensional subspace of L2(ˆK), r a non-negative integer.

The following conditions are equivalent:

1. There is a constant C such that inf

v∈Sh?u − v?L2(Ω)≤ Chr+1|u|r+1for all u ∈ Hr+1(Ω).

2. inf

v∈Sh?u − v?L2(Ω)= o(hr) for all u ∈ Pr(Ω).

3. Pr(ˆK) ⊂ˆS.

Proof. The first condition implies that infv∈Sh?u−v?L2(Ω)= 0 for u ∈ Pr(Ω), 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 = 0. We have

inf

v∈Sh?u − v?2

L2(Ω)=

?

K∈Th

inf

vK∈S(K)?u − vK?2

L2(K)= h2?

K∈Th

inf

w∈ˆS?ˆ uK− w?2

L2(ˆ K),(1)

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where we have made the change of variable w = ˆ vKin the last step.

In particular, for u ≡ 1 on Ω, ˆ uK≡ 1 onˆK for all K, so the quantity

c := inf

w∈ˆS?ˆ uK− w?2

L2(ˆ K)

is independent of K. Thus

inf

v∈Sh?u − v?2

L2(Ω)= h2?

K∈Th

c = c

The hypothesis that this quantity is o(1) implies that c = 0, i.e., that the constant function

belongs toˆS.

Now we consider the case r > 0.We again apply (1), this time for u an arbitrary

homogeneous polynomial of degree r. Then

ˆ uK(ˆ x) = u(xK+ hˆ x) = u(hˆ x) + p(ˆ x) = hru(ˆ x) + p(ˆ x),(2)

where p ∈ Pr−1(ˆK). Substituting in (1), and invoking the inductive hypothesis thatˆS ⊇

Pr−1(ˆK), we get that

inf

v∈Sh?u − v?2

L2(Ω)= h2+2r?

K∈Th

inf

w∈ˆS?u − w?2

L2(ˆ K)= h2rinf

w∈ˆS?u − w?2

L2(ˆ K).

Again the last infimum is independent of K so we immediately deduce that u belongs toˆS.

ThusˆS 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 ∇h for the gradient operator applied piecewise on each

element.

Theorem 2. LetˆS be a finite dimensional subspace of L2(ˆK), r a non-negative integer.

The following conditions are equivalent:

1. There is a constant C such that inf

v∈Sh?∇h(u−v)?L2(Ω)≤ Chr|u|r+1for all u ∈ Hr+1(Ω).

2. inf

v∈Sh?∇h(u − v)?L2(Ω)= o(hr−1) for all u ∈ Pr(Ω).

3. Pr(ˆK) ⊂ P0(ˆK) +ˆS.

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

(1), we have

inf

v∈Sh

?

K∈Th

?∇(u − v)?2

L2(K)=

?

K∈Th

?

K∈Th

inf

vK∈S(K)?∇(u − vK)?2

L2(K)

=inf

w∈ˆS?∇(ˆ uK− w)?2

L2(ˆ K),

(3)

where we have made the change of variable w = ˆ vKin the last step.

The proof proceeds by induction on r, the case r = 0 being trivial. For r > 0, apply

(3) with u an arbitrary homogeneous polynomial of degree r. Substituting (2) in (3), and

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6DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

invoking the inductive hypothesis that P0(ˆK) +ˆS ⊇ Pr−1(ˆK), we get that

inf

v∈Sh?∇h(u − v)?2

L2(Ω)= h2r?

K∈Th

inf

w∈ˆS?∇(u − w)?2

L2(ˆ K)= h2r−2inf

w∈ˆS?∇(u − w)?2

L2(ˆ K).

Since we assume that this quantity is o(h2r−2), the last infimum must be 0, so u differs

from an elementˆS by a constant. Thus P0(ˆK)+ˆS 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. IfˆS contains P0(ˆK), 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 ∇hby ∇m

and P0(ˆK) by Pm−1(ˆK) in the third).

hin the first two conditions,

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 polynomialsˆV on the unit squareˆK = [0,1]×[0,1].

For simplicity we assume thatˆV ⊇ P0(ˆK). For exampleˆV might be the space Pr(ˆK) of

polynomials of degree at most r, or the space Qr(ˆK) of polynomials of degree at most r in

each variable separately, or the serendipity space Sr(ˆK) spanned by Pr(ˆK) together with the

monomials ˆ xr

K = F(ˆK). Then for u ∈ L2(K) we define ˆ uF∈ L2(ˆK) by ˆ uF(ˆ x) = u(Fˆ x), and set

1ˆ x2and ˆ x1ˆ xr

2. Let F be a bilinear isomorphism ofˆK onto a convex quadrilateral

VF(K) = {u : K → R| ˆ uF∈ˆV }.

(Note that the definition of this space depends on the particular choice of the bilinear iso-

morphism F ofˆK onto K, but whenever the spaceˆV 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 VF(K) need not be polynomials if F is not affine, i.e., if K is not a parallelogram.

Given a quadrilateral mesh T of some domain, Ω, we can then construct the space of

functions VTconsisting of functions on the domain which when restricted to a quadrilateral

K ∈ T belong to VFK(K) where FKis a bilinear isomorphism ofˆK onto K. (Again, ifˆV is

not invariant under the symmetries of the square, the space VTwill depend on the specific

choice of the maps FK.)

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 = 1/n, then each of

the approximation estimates

inf

v∈VTh?u − v?L2(Ω)≤ Chr+1|u|r+1for all u ∈ Hr+1(Ω),

inf

(4)

v∈VTh?∇h(u − v)?L2(Ω)≤ Chr|u|r+1for all u ∈ Hr+1(Ω),(5)

holds if and only Pr(ˆK) ⊂ˆV . 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

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estimates to hold for more general quadrilateral mesh sequences, a stronger condition onˆV

is required, namely thatˆV ⊇ Qr(ˆK).

The positive result, that whenˆV ⊇ Qr(ˆK), then the estimates (4) and (5) hold for any

shape regular sequence of quadrilateral meshes Th, is known. See, e.g., [2], [1], or [4, Section

I.A.2]. We wish to show the necessity of the conditionˆV ⊇ Qr(ˆK).

As a first step we show that the condition VF(K) ⊇ Pr(K) is necessary and sufficient to

have thatˆV ⊇ Qr(ˆK) whenever F is a bilinear isomorphism ofˆK onto a convex quadrilateral.

This is proven in the following two theorems.

Theorem 3. Suppose thatˆV ⊇ Qr(ˆK). Let F be any bilinear isomorphism ofˆK onto a

convex quadrilateral. Then VF(K) ⊇ Pr(K).

Proof. The components of F(ˆ x, ˆ y) are linear functions of ˆ x and ˆ y, so if p is a polynomial of

degree at most r, then p(F(ˆ x, ˆ y)) is of degree at most r in ˆ x and ˆ y, i.e., p◦F ∈ Qr(ˆK) ⊂ˆV .

Therefore p ∈ VF(K).

The reverse implication holds even under the weaker assumption that VF(K) contains

Pr(K) just for the two specific bilinear isomorphism

˜F(ˆ x, ˆ y) = (ˆ x, ˆ y(ˆ x + 1)),

¯F(ˆ x, ˆ y) = (ˆ y, ˆ x(ˆ y + 1)),

both of which mapˆK isomorphically onto the quadrilateral K′with vertices (0,0), (1,0),

(0,1), and (1,2). This fact is established below.

Theorem 4. LetˆV be a vectorspace of functions onˆK. Suppose that Qr(ˆK) ?ˆV . Then

either V˜F(K′) ? Pr(K′) or V ¯ F(K′) ? Pr(K′).

Remark. If the spaceˆV is invariant under the symmetries of the square, then V˜F(K′) =

V ¯F(K′) so neither contains Pr(K′).

Proof. Assume to the contrary that V˜F(K′) ⊇ Pr(K′) and V ¯F(K′) ⊇ Pr(K′). We prove

thatˆV ⊇ Qr(ˆK) by induction on r. The case r = 0 being true by assumption, we consider

r > 0 and show that the monomials ˆ xrˆ ysand ˆ xsˆ yrbelong toˆV for s = 0,1,... ,r. From the

identity

s

?

t=1

t

we see that for 0 ≤ s < r, the monomial ˆ xrˆ ysis the sum of a polynomial which clearly

belongs toˆV (since˜F1(ˆ x, ˆ y)r−s˜F2(ˆ x, ˆ y)s= xr−sys∈ Pr(K′) ⊂ V˜F(K′)) and a polynomial in

Qr−1(ˆK), which belongs toˆV by induction. Thus each of the monomials ˆ xrˆ yswith 0 ≤ s < r

belongs toˆV , and, using¯F, we similarly see that all the monomials ˆ xsˆ yr, 0 ≤ s < r belong

toˆV . Finally, from (6) with s = r, we see that ˆ xrˆ yris a linear combination of an element of

ˆV and monomials ˆ xsˆ yrwith s < r, so it too belongs toˆV .

ˆ xrˆ ys= ˆ xr−s[ˆ y(ˆ x + 1)]s−

?s

?

ˆ xr−tˆ ys=˜F1(ˆ x, ˆ y)r−s˜F2(ˆ x, ˆ y)s−

s

?

t=1

?s

t

?

ˆ xr−tˆ ys,(6)

We now combine this result with the those of the previous section to show the necessity

of the conditionˆV ⊇ Qr(ˆK) for optimal order approximation. LetˆV be some fixed finite

dimensional subspace of L2(ˆK) which does not include Qr(ˆK). Consider the specific division

of the unit squareˆK into four quadrilaterals shown on the left in Figure 1. For definiteness we

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8DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

place the vertices of the quadrilaterals at (0,1/3), (1/2,2/3) and (1,1/3) and the midpoints

of the horizontal edges and the corners ofˆK.

Figure 1.

composed of translated dilates of this partition.

a. A partition of the square into four trapezoids. b. A mesh

The four quadrilaterals are mutually congruent and affinely related to the specific quadri-

lateral K′defined above. Therefore, by Theorem 4, we can define for each of the four

quadrilaterals K′′shown in Figure 1 an isomorphism F′′from the unit square so that

VF′′(K′′) ? Pr(K′′). If we letˆS be the subspace of L2(ˆK) consisting of functions which

restrict to elements of VF′′(K′′) on each of the four quadrilaterals K′′, then certainlyˆS does

not contain Pr(ˆK), since even its restriction to any one of the quadrilaterals K′′does not

contain Pr(K′′).

Next, for n = 1,2,... consider the mesh T′

obtained by first dividing it into a uniform n × n mesh of subsquares, n = 1/h, and then

dividing each subsquare as in Figure 1a. Then the space of functions u on Ω whose restrictions

on each subsquare K ∈ Thsatisfy ˆ uK(ˆ x) = u(xK+hˆ x) with ˆ uK∈ˆS is precisely the same as

the space V (T′

and 2 and the fact thatˆS ? Pr(ˆK), the estimates (4) and (5) do not hold. In fact, neither

of the estimates

hof the unit square Ω shown in Figure 1b,

h) constructed from the initial spaceˆV and the mesh T′

h. In view of Theorems 1

inf

v∈V (Th)?u − v?L2(Ω)= o(hr),

nor

inf

v∈V (Th)?∇(u − v)?L2(Ω)= o(hr−1),

holds, even for u ∈ Pr(Ω).

While the conditionˆV ⊇ Qr(ˆK) is necessary for O(hr+1) on general quadrilateral meshes,

the conditionsˆV ⊇ Pr(ˆK) 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. IfˆV ⊇ Pr(ˆK) and

K = F(ˆK) with F ∈ Bil(ˆK), then by standard arguments, as in [1], we get

?v − πv?L2(K)≤ C?JF?1/2

L∞(ˆ K)|v ◦ F|Hr+1(ˆ K),

where JF is the Jacobian determinant of F. Now, using the formula for the derivative of a

composition (as in, e.g., [3, p. 222]), and the fact that F is quadratic, and so its third and

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APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS9

higher derivatives vanish, we get that

|v ◦ F|Hr+1(ˆ K)≤ C?JF−1?1/2

L∞(K)?v?Hr+1(K)

⌊(r+1)/2⌋

?

i=0

|F|r+1−2i

W1

∞(ˆ K)|F|i

W2

∞(ˆ K).

Now,

?JF?L∞(ˆ K)≤ Ch2

K,?JF−1?L∞(ˆ K)≤ Ch−2

K,|F|W1

∞(ˆ K)≤ ChK,

where hKis the diameter of K and C depends only on the shape-regularity of K. We thus

get

?v − πv?L2(K)≤ C?v?Hr+1(K)

?

i

hr+1−2i

K

|F|i

W2

∞(ˆ K).

It follows that if |F|W2

∞(ˆ K)= O(h2

K), we get the desired estimate

?v − πv?L2(K)≤ Chr+1

K?v?Hr+1(K).

Following [5], we measure the deviation of a quadrilateral from a parallelogram, by the

quantity σK:= max(|π − θ1|,|π − θ2|), where θ1is the angle between the outward normals

of two opposite sides of K and θ2is the angle between the outward normals of the other two

sides. Thus 0 ≤ σK< π, with σK= 0 if and only if K is a parallelogram. As pointed out in

[5], |F|W2

meshes is asymptotically parallelogram if σK= O(hK), i.e., if σK/hKis uniformly bounded

for all the elements in all the meshes. From the foregoing considerations, if the reference

space contains Pr(ˆK) we obtain O(hr+1) convergence for asymptotically parallelogram, shape

regular meshes.

As a final note, we remark that any polygon can be meshed by an asymptotically paral-

lelogram, 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.

∞(ˆ K)≤ ChK(hK+ σK). This motivates the definition that a family of quadrilateral

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 Q2(ˆK) and S2(ˆK) 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

−∆u = f in Ω,u = g on ∂Ω,(7)

where the domain Ω is the unit square. In the first problem, f and g are taken so that the

exact solution is the quartic polynomial

u(x,y) = x3+ 5y2− 10y3+ y4.

Table 1 shows 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 subdi-

visions of the domain into n × n subsquares, n = 2,4,8,.... Meshes in the second sequence

Page 10

10 DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

are partitions of the domain into n × n congruent trapezoids, all similar to the trapezoid

with vertices (0,0), (1/2,0), (1/2,2/3), and (0,1/3). In Table 1 we report the errors in L2

for the finite element solution and its gradient both in absolute terms and as a percentage of

the L2norm 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 con-

verge 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 its gradient.

Figure 2.

asymptotically parallelogram. Each is shown for n = 2 ,4, 8, and 16.

Three sequences of meshes of the unit square: square, trapezoidal, and

As a second test example we again solved the Dirichlet problem (7), but this time choosing

the data so that the solution is the sharply peaked function

u(x,y) = exp?−100[(x − 1/4)2+ (y − 1/3)2]?.

As seen in Table 2, in this case the loss of convergence order 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.

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 3.

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 × 16 trapezoids, the

Page 11

APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS11

Table 1.

mial solution.

Errors and rates of convergence for the test problem with polyno-

Mapped biquadratic elements

square meshestrapezoidal meshes

?u − uh?L2

err.

?∇(u − uh)?L2

err.

?u − uh?L2

err.

?∇(u − uh)?L2

err.

n

% rate% rate%rate%rate

2

4

8

3.5e−02

4.4e−03

5.5e−04

6.9e−05

8.6e−06

1.1e−06

2.877

0.360

0.045

0.006

0.001

0.000

4.5e−01

1.1e−01

2.8e−02

7.1e−03

1.8e−03

4.4e−04

37.253

9.333

2.329

0.583

0.146

0.036

4.8e−02

5.8e−03

7.1e−04

8.7e−05

1.1e−05

1.3e−06

3.951

0.475

0.058

0.007

0.001

0.000

5.9e−01

1.5e−01

3.7e−02

9.2e−03

2.3e−03

5.7e−04

48.576

12.082

3.017

0.753

0.188

0.047

3.0

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

2.0

3.1

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

2.0

16

32

64

Serendipity elements

square meshes trapezoidal meshes

?u − uh?L2

err.

?∇(u − uh)?L2

err.

?u − uh?L2

err.

?∇(u − uh)?L2

err.

n

% rate%rate%rate% rate

2

4

8

3.5e−02

4.4e−03

5.5e−04

6.9e−05

8.6e−06

1.1e−06

2.877

0.360

0.045

0.006

0.001

0.000

4.5e−01

1.1e−01

2.8e−02

7.1e−03

1.8e−03

4.4e−04

37.252

9.333

2.329

0.583

0.146

0.036

5.0e−02

6.7e−03

9.7e−04

1.6e−04

3.3e−05

7.4e−06

4.066

0.548

0.080

0.013

0.003

0.001

6.2e−01

1.8e−01

5.9e−02

2.3e−02

1.0e−02

4.9e−03

51.214

14.718

4.836

1.890

0.842

0.401

3.0

3.0

3.0

3.0

3.0

2.0

2.0

2.0

2.0

2.0

2.9

2.8

2.6

2.3

2.1

1.8

1.6

1.4

1.2

1.1

16

32

64

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 × 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 non-parallelogram

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.

References

1. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978.

2. P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements with applications to fiite element

methods, Comput. Methods Appl. Mech. Engrg. 1 (1972), 217–249.

3. H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969.

4. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations, Springer-Verlag, New

York, 1986.

Page 12

12 DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK

Table 2.

nential solution.

Errors and rates of convergence for the test problem with expo-

Mapped biquadratic elements

square meshestrapezoidal meshes

?u − uh?L2

?∇(u − uh)?L2

err.

?u − uh?L2

?∇(u − uh)?L2

err.

n

err. %rate %rateerr.%rate%rate

2

4

8

2.8e−01 224.000

1.2e−01

1.7e−02

1.1e−03

1.3e−04

1.5e−05

1.9e−06

3.0e+00 169.630

1.5e+00

4.6e−01

1.0e−01

2.5e−02

6.3e−03

1.6e−03

2.6e−01

2.1e−01

2.3e−02

1.3e−03

1.5e−04

1.9e−05

2.4e−06

204.800

169.600

18.160

1.048

0.124

0.015

0.002

2.8e+00 159.208

1.8e+00

5.9e−01

1.2e−01

3.2e−02

7.9e−03

2.0e−03

93.600

13.520

0.920

0.101

0.012

0.002

1.3

2.8

3.9

3.2

3.1

3.0

87.322

25.809

5.860

1.424

0.354

0.088

1.0

1.8

2.1

2.0

2.0

2.0

0.3

3.2

4.1

3.1

3.0

3.0

99.305

33.185

6.819

1.794

0.448

0.112

0.7

1.6

2.3

1.9

2.0

2.0

16

32

64

128

Serendipity elements

square meshestrapezoidal meshes

?u − uh?L2

?∇(u − uh)?L2

err.

?u − uh?L2

?∇(u − uh)?L2

err.

n

err.% rate%rateerr.%rate%rate

2

4

8

2.0e−01 159.200

1.2e−01

1.7e−02

1.1e−03

1.3e−04

1.5e−05

1.9e−06

2.4e+00 133.372

1.4e+00

4.6e−01

1.1e−01

2.5e−02

6.3e−03

1.6e−03

2.1e−01

2.1e−01

2.4e−02

1.5e−03

2.0e−04

2.7e−05

3.7e−06

169.600

168.000

18.880

1.208

0.162

0.022

0.003

2.3e+00 130.340

1.7e+00

6.1e−01

1.4e−01

3.8e−02

1.1e−02

3.4e−03

92.000

13.520

0.920

0.101

0.012

0.002

0.8

2.8

3.9

3.2

3.1

3.0

80.531

26.293

5.948

1.432

0.354

0.088

0.7

1.6

2.1

2.1

2.0

2.0

0.0

3.2

4.0

2.9

2.9

2.9

93.819

34.564

7.737

2.156

0.597

0.191

0.5

1.4

2.2

1.8

1.9

1.6

16

32

64

128

Table 3.

mial solution using serendipity elements on asympotically affine meshes.

Errors and rates of convergence for the test problem with polyno-

?u − uh?L2

err.

?∇(u − uh)?L2

err.

n

%rate%rate

2

4

8

5.0e−02

6.2e−03

7.6e−04

9.4e−05

1.2e−05

1.5e−06

1.9e−07

4.066

0.510

0.062

0.008

0.001

0.000

0.000

6.2e−01

1.5e−01

3.6e−02

9.0e−03

2.2e−03

5.6e−04

1.4e−04

51.214

12.109

2.948

0.735

0.183

0.046

0.012

3.0

3.0

3.0

3.0

3.0

3.0

2.1

2.0

2.0

2.0

2.0

2.0

16

32

64

128

5. R. Rannacher and S. Turek, Simple nonconforming quadrilateral stokes element, Numer. Meth. Part. Diff.

Equations 8 (1992), 97–111.

6. P. Sharpov and Y. Iordanov, Numerical solution of Stokes equations with pressure and filtration boundary

conditions, J. Comp. Phys. 112 (1994), 12–23.

Page 13

APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS13

7. O. C. Zienkiewicz and R. L. Taylor, The finite element method, fourth edition, volume 1: Basic formulation

and linear problems, McGraw-Hill, London, 1989.

Department of Mathematics, Penn State University, University Park, PA 16802

E-mail address: dna@psu.edu

URL: http://www.math.psu.edu/dna/

Dipartimento di Matematica, Universit` e Pavia, 27100 Pavia, Italy

E-mail address: boffi@dimat.unipv.it

URL: http://dimat.unipv.it/~boffi/

Department of Mathematics, Rutgers University, Piscataway, NJ 08854

E-mail address: falk@math.rutgers.edu

URL: http://www.math.rutgers.edu/~falk/