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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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2 DOUGLAS 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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4 DOUGLAS 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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6 DOUGLAS 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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8 DOUGLAS 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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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 ELEMENTS 11
Table 1.
mial solution.
Errors and rates of convergence for the test problem with polyno-
Mapped biquadratic 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
37.253
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% rate err.% 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 ELEMENTS 13
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/
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