APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS

Transcription

1 MATHEMATICS OF COMPUTATION Volume 71, Number 239, Pages S (02) Article electronically publised on Marc 22, 2002 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS DOUGLAS N. ARNOLD, DANIELE BOFFI, AND RICHARD S. FALK Abstract. We consider te approximation properties of finite element spaces on quadrilateral meses. Te finite element spaces are constructed starting wit a given finite dimensional space of functions on a square reference element, wic is ten transformed to a space of functions on eac convex quadrilateral element via a bilinear isomorpism of te square onto te element. It is known tat for affine isomorpisms, a necessary and sufficient condition for approximation of order r +1 inl p and order r in Wp 1 is tat te given space of functions on te reference element contain all polynomial functions of total degree at most r. In te case of bilinear isomorpisms, it is known tat te same estimates old if te function space contains all polynomial functions of separate degree r. We sow, by means of a counterexample, tat tis latter condition is also necessary. As applications, we demonstrate degradation of te convergence order on quadrilateral meses as compared to rectangular meses for serendipity finite elements and for various mixed and nonconforming finite elements. 1. Introduction Finite element spaces are often constructed starting wit a finite dimensional space ˆV of sape functions given on a reference element ˆK and a class S of isomorpic mappings of te reference element. If F S, we obtain a space of functions V F (K) on te image element K = F ( ˆK) as te compositions of functions in ˆV wit F 1. Ten, given a partition T of a domain Ω into images of ˆK under mappings in S, we obtain a finite element space as a subspace 1 of te space V T of all functions on Ω wic restrict to an element of V F (K) oneack T. For example, if te reference element ˆK is te unit triangle, te reference space ˆV is te space P r ( ˆK) of polynomials of degree at most r on ˆK, and te mapping class S is te space Aff( ˆK) of affine isomorpisms of ˆK into R 2,tenV T is te familiar space of all piecewise polynomials of degree at most r on an arbitrary triangular mes T. WenS = Aff( ˆK), as in tis case, we speak of affine finite elements. If te reference element ˆK is te unit square, ten it is often useful to take S equal to a larger space tan Aff( ˆK), namely te space Bil( ˆK) of all bilinear isomorpisms of ˆK into R 2. Indeed, if we allow only affine images of te unit square, ten we Received by te editor Marc 10, Matematics Subject Classification. Primary 65N30, 41A10, 41A25, 41A27, 41A63. Key words and prases. Quadrilateral, finite element, approximation, serendipity, mixed finite element. 1 Te subspace is typically determined by some interelement continuity conditions. Te imposition of suc conditions troug te association of local degrees of freedom is an important part of te construction of finite element spaces, but, not being directly relevant to te present work, will not be discussed. 909 c 2002 American Matematical Society

2 910 D. N. ARNOLD, D. BOFFI, AND R. S. FALK obtain only parallelograms, and we are quite limited as to te domains tat we can mes (e.g., it is not possible to mes a triangle wit parallelograms). On te oter and, wit bilinear images of te square we obtain arbitrary convex quadrilaterals, wic can be used to mes arbitrary polygons. Te above framework is also well suited to studying te approximation properties of finite element spaces (e.g., see [2] and [1]). A fundamental result olds in te case of affine finite elements: S = Aff( ˆK). Under te assumption tat te reference space ˆV P r ( ˆK), te following result is well known: if T 1, T 2,... is any saperegular sequence of triangulations of a domain Ω and u is any smoot function on Ω, ten te L p error in te best approximation of u by functions in V Tn is O( ) and te piecewise Wp 1 error is O( r ), were = (T n )istemaximum element diameter. (Here, and trougout te paper, p can take any value between 1and, inclusive.) It is also true, even if less well-known, tat te condition tat ˆV P r ( ˆK) is necessary if tese estimates are to old. In fact, te problem of caracterizing wat is needed for optimal order approximation arises naturally in te study of te finite element metod and as been investigated for some time (see [9]). Te above result does not restrict te coice of reference element ˆK, so it applies to rectangular and parallelogram meses by taking ˆK to be te unit square. But it does not apply to general quadrilateral meses, since to obtain tem we must coose S = Bil( ˆK), and te result only applies to affine finite elements. In tis case tere is a standard result analogous to te positive result in te previous paragrap ([2], [1], [4, Section I.A.2]). Namely, if ˆV Q r ( ˆK), ten for any sape-regular sequence of quadrilateral partitions of a domain Ω and any smoot function u on Ω, we again obtain tat te error in te best approximation of u by functions in V Tn is O( ) in L p and O( r ) in (piecewise) Wp 1. It turns out, as we sall sow in tis paper, tat te ypotesis tat ˆV Q r ( ˆK) is strictly necessary for tese estimates to old. In particular, if ˆV P r ( ˆK) but ˆV Q r ( ˆK), ten te rate of approximation acieved on general sape-regular quadrilateral meses will be strictly lower tan is obtained using meses of rectangles or parallelograms. More precisely, we sall exibit in Section 3 a domain Ω and a sequence, T 1, T 2,... of quadrilateral meses of it, and prove tat wenever V ( ˆK) Q r ( ˆK), ten tere is a function u on Ω suc tat inf u v L p (Ω) o( r ) v V Tn (and so, a fortiori, is O( )). A similar result olds for Wp 1 approximation. Te counterexample is far from patological. Indeed, te domain Ω is as simple as possible, namely a square; te mes sequence T n is simple and as sape-regular as possible in tat all elements at all mes levels are similar to a single fixed trapezoid; and te function u is as smoot as possible, namely a polynomial of degree r. 2 Te use of a reference space wic contains P r ( ˆK) but not Q r ( ˆK) is not unusual, but te degradation of convergence order tat tis implies on general quadrilateral meses in comparison to rectangular (or parallelogram) meses is not widely appreciated. 2 However, as discussed at te end of Section 3 and illustrated numerically in Section 4, for a sequence of meses wic is asymptotically parallelogram (as defined at te end of Section 3), te ypotesis V ( ˆK) P r( ˆK) is sufficient for optimal order approximation.

3 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 911 We finis tis introduction by considering some examples. Hencefort we sall always use ˆK to denote te unit square. First, consider finite elements wit te simple polynomial spaces as sape functions: ˆV = Pr ( ˆK). Tese of course yield O( ) approximation in L p for rectangular meses. However, since P r ( ˆK) Q r/2 ( ˆK) but P r ( ˆK) Q r/2 +1 ( ˆK), on general quadrilateral meses tey only afford O( r/2 +1 ) approximation. A similar situation olds for serendipity finite element spaces, wic ave been popular in engineering computation for tirty years. Tese spaces are constructed using as reference sape functions te space S r ( ˆK) wic is te span of P r ( ˆK) togeter wit te two monomials ˆx r ŷ and ŷˆx r. (Te purpose of te additional two functions is to allow local degrees of freedom wic can be used to ensure interelement continuity.) For r =1,S 1 ( ˆK) =Q 1 ( ˆK), but for r>1tesituationis similar to tat for P r ( ˆK), namely S r ( ˆK) Q r/2 ( ˆK) but S r ( ˆK) Q r/2 +1 ( ˆK). So, again, te asymptotic accuracy acieved for general quadrilateral meses is only about alf tat acieved for rectangular meses: O( r/2 +1 )inl p and O( r/2 ) in Wp 1. In Section 4 we illustrate tis wit a numerical example. Te fact tat te eigt-node serendipity space S 2 does not perform as well as te nine-node space Q 2 on distorted meses as been noted previously by several autors, often as a result of numerical experiments. See, for example, [11, Section 8.7], [6], [5], [10]. Wile te serendipity elements are commonly used for solving second order differential equations, te pure polynomial spaces P r canonlybeusedonquadrilaterals wen interelement continuity is not required. Tis is te case in several mixed metods. For example, a popular element coice to solve te stationary Stokes equations is bilinearly mapped piecewise continuous Q 2 elements for te two components of velocity, and discontinuous piecewise linear elements for te pressure. Tis is known to be a stable mixed metod and gives second order convergence in H 1 for te velocity and L 2 for te pressure. If one were to define te pressure space instead by using te construction discussed above, namely by composing linear functions on te reference square wit bilinear mappings, ten te approximation properties of mapped P 1 discussed above would imply tat te metod could be at most first order accurate, at least for te pressures. Hence, altoug te use of mapped P 1 as an alternative to unmapped P 1 pressure elements is sometimes proposed [8], it is probably not advisable. Anoter place were mapped P r spaces arise is for approximating te scalar variable in mixed finite element metods for second order elliptic equations. Altoug te scalar variable is discontinuous, in order to prove stability it is generally necessary to define te space for approximating it by composition wit te mapping to te reference element (wile te space for te vector variable is defined by a contravariant mapping associated wit te mapping to te reference element). In te case of te Raviart Tomas rectangular elements, te scalar space on te reference square is Q r ( ˆK), wic maintains full O( ) approximation properties under bilinear mappings. By contrast, te scalar space used wit te Brezzi-Douglas-Marini and te Brezzi-Douglas-Fortin-Marini spaces is P r ( ˆK). Tis necessarily results in a loss of approximation order wen mapped to quadrilaterals by bilinear mappings. Anoter type of element wic sares tis difficulty is te simplest nonconforming quadrilateral element, wic generalizes to quadrilaterals te well-known piecewise linear nonconforming element on triangles, wit degrees of freedom at te midpoints of edges. On te square, a bilinear function is not well-defined by

4 912 D. N. ARNOLD, D. BOFFI, AND R. S. FALK giving its value at te midpoint of edges (or its average on edges), because tese quantities do not comprise a unisolvent set of degrees of freedom (te function (ˆx 1/2)(ŷ 1/2) vanises at te four midpoints of te edges of te unit square). Hence, various definitions of nonconforming elements on rectangles replace te basis function ˆxŷ by some oter function, suc as ˆx 2 ŷ 2. Consequently, te reference space contains P 1 ( ˆK), but does not contain Q 1 ( ˆK), and so tere is a degradation of convergence on quadrilateral meses. Tis is discussed and analyzed in te context of te Stokes problem in [7]. As a final application, we remark tat many of te finite element metods proposed for te Reissner Mindlin plate problem are based on mixed metods for te Stokes equations and/or for second order elliptic problems. As a result, many of tem suffer from te same sort of degradation of convergence on quadrilateral meses. An analysis of a variety of tese elements will appear in fortcoming work by te present autors. In Section 3, we prove our main result, te necessity of te condition tat te reference space contain Q r ( ˆK) in order to obtain O( ) approximation on quadrilateral meses. Te proof relies on an analogous result for affine approximation on rectangular meses, were te space P r ( ˆK) enters rater tan Q r ( ˆK). Wile tis is a special case of known results, for te convenience of te reader we include an elementary proof in Section 2. Also in Section 3, we consider te case of asymptotically parallelogram meses and sow tat in tis situation, an O( ) approximation is obtained if te reference space only contains P r ( ˆK). In te final section, we illustrate te results wit numerical computations. 2. Approximation teory of rectangular elements In tis section, we prove some results concerning approximation by rectangular elements wic will be needed to prove te main results in Section 3. Te results in tis section are essentially known, and many are true in far greater generality tan stated ere. If K is any square wit edges parallel to te axes, ten K = F K ( ˆK), were F K (ˆx) :=x K + K ˆx wit x K R 2 and K > 0 te side lengt. For any function u L 1 (K), we define û K = u F K L 1 ( ˆK), i.e., û K (ˆx) =u(x K + K ˆx). Given a subspace Ŝ of L1 ( ˆK), we define te associated subspace on an arbitrary square K by S(K) ={ u : K R û K Ŝ }. Finally, let Ω denote te unit square (Ω and ˆK bot denote te unit square, but we use te notation Ω wen we tink of it as a fixed domain, wile we use ˆK wen we tink of it as a reference element). For n =1, 2,...,letT be te uniform mes of Ω into n 2 subsquares wen =1/n, and define S = { u :Ω R u K S(K) for all K T }. In tis definition, wen we write u K S(K) we mean only tat u K agrees wit a function in S(K) almost everywere, and so do not impose any interelement continuity. Te following teorem gives a set of equivalent conditions for optimal order approximation of a smoot function u by elements of S.

5 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 913 Teorem 1. Suppose 1 p.letŝbe a finite dimensional subspace of Lp ( ˆK), and r a nonnegative integer. Te following conditions are equivalent: 1. Tere is a constant C suc tat inf u v L v S p (Ω) C u W p (Ω) for all u Wp (Ω). 2. inf u v L v S p (Ω) = o( r ) for all u P r (Ω). 3. P r ( ˆK) Ŝ. Proof. For te proof we assume tat p<, but te argument carries over to te case p = wit obvious modifications. Te first condition implies tat inf u v L v S p (Ω) =0 foru P r (Ω), and so implies te second condition. Te fact tat te tird condition implies te first is a well-known consequence of te Bramble Hilbert lemma. So we need only sow tat te second condition implies te tird. Te proof is by induction on r. First consider te case r =0. Weave (1) inf u v p v S L p (Ω) = K T inf u v K p v L p (K) = 2 K S(K) w Ŝ K T inf û K w p L p ( ˆK), were we ave made te cange of variable w =ˆv K in te last step. In particular, for u 1onΩ,û K 1on ˆK for all K, sotequantity c := inf K w w Ŝ û p L p ( ˆK) is independent of K. Tus inf v S u v p L p (Ω) = 2 K T c = c. Te ypotesis tat tis quantity is o(1) implies tat c = 0, i.e., tat te constant function belongs to Ŝ. Now we consider te case r>0. We again apply (1), tis time for u an arbitrary omogeneous polynomial of degree r. Ten (2) û K (ˆx) =u(x K + ˆx) =u(ˆx)+q(ˆx) = r u(ˆx)+q(ˆx), were q P r 1 ( ˆK). Substituting in (1), and invoking te inductive ypotesis tat Ŝ P r 1( ˆK), we get tat inf u v p v S L p (Ω) = 2+pr inf u w p L p ( ˆK) = pr inf u w p L p ( ˆK), w Ŝ K T were te last equality follows from te fact tat te previous infimum is independent of K. Since te last expression is o( pr ), we immediately deduce tat u belongs to Ŝ. TusŜ contains all omogeneous polynomials of degree r and all polynomials of degree less tan r (by induction), so it indeed contains all polynomials of degree at most r. A similar teorem olds for gradient approximation. Since te finite elements are not necessarily continuous, we write for te gradient operator applied piecewise on eac element. w Ŝ

6 914 D. N. ARNOLD, D. BOFFI, AND R. S. FALK Teorem 2. Suppose 1 p.letŝbe a finite dimensional subspace of Lp ( ˆK), and r a nonnegative integer. Te following conditions are equivalent: 1. Tere is a constant C suc tat inf (u v) L v S p (Ω) C r u W p (Ω) for all u Wp (Ω). 2. inf (u v) Lp (Ω) = o( r 1 ) for all u P r (Ω). v S 3. P r ( ˆK) P 0 ( ˆK)+Ŝ. Proof. Again, we need only prove tat te second condition implies te tird. In analogy to (1), we ave inf (u v) p v S L p (K) = inf (u v K) p v L K S(K) p (K) K T K T (3) = 2 p inf (û K w) p L p ( ˆK), w Ŝ K T were we ave made te cange of variable w =ˆv K in te last step. Te proof proceeds by induction on r, tecaser = 0 being trivial. For r>0, apply (3) wit u an arbitrary omogeneous polynomial of degree r. Substituting (2) in (3), and invoking te inductive ypotesis tat P 0 ( ˆK)+Ŝ P r 1( ˆK), we get tat inf v S (u v) p L p (Ω) = 2 p+pr K T inf w) w Ŝ (u p L p ( ˆK) = p(r 1) inf (u w) p w Ŝ L p ( ˆK). Since we assume tat tis quantity is o( p(r 1) ), te last infimum must be 0, so u differs from an element of Ŝ by a constant. Tus P 0( ˆK) +Ŝ contains all omogeneous polynomials of degree r and all polynomials of degree less tan r (by induction), so it indeed contains all polynomials of degree at most r. Remarks. 1. If Ŝ contains P 0( ˆK), wic is usually te case, ten te tird condition of Teorem 2 reduces to tat of Teorem A similar result olds for iger derivatives (replace by m in te first two conditions, and P 0 ( ˆK) byp m 1 ( ˆK) in te tird). 3. Approximation teory of quadrilateral elements In tis, te main section of te paper, we consider te approximation properties of finite element spaces defined wit respect to quadrilateral meses using bilinear mappings starting from a given finite dimensional space of polynomials ˆV on te unit square ˆK =[0, 1] [0, 1]. For simplicity, we assume tat ˆV P 0 ( ˆK). For example, ˆV migt be te space Pr ( ˆK) of polynomials of total degree at most r, or te space Q r ( ˆK) of polynomials of degree at most r in eac variable separately, or te serendipity space S r ( ˆK) spanned by P r ( ˆK) togeter wit te monomials ˆx r 1ˆx 2 and ˆx 1ˆx r 2.LetF be a bilinear isomorpism of ˆK onto a convex quadrilateral K = F ( ˆK). Ten for u L 1 (K) we define û K,F L 1 ( ˆK) byû K,F = u F,andset V F (K) ={ u : K R û K,F ˆV }.

7 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 915 (Note tat we ave canged notation sligtly from Section 2 to include te mapping F, since various definitions depend on te particular coice of te bilinear isomorpism F of ˆK onto K. Wenever te space ˆV is invariant under te symmetries of te square, wic is usually te case in practice, tis will not be so.) We also note tat te functions in V F (K) need not be polynomials if F is not affine, i.e., if K is not a parallelogram. Given a quadrilateral mes T of some domain, Ω, we can ten construct te space of functions V T consisting of functions on te domain wic wen restricted to a quadrilateral K T belong to V FK (K), were F K is a bilinear isomorpism of ˆK onto K. (Again, if ˆV is not invariant under te symmetries of te square, te space V T will depend on te specific coice of te maps F K.) It follows from te results of te previous section tat if we consider te sequence of meses of te unit square into congruent subsquares of side lengt =1/n, ten eac of te approximation estimates (4) (5) inf u v L p (Ω) C u W v V T p (Ω) inf (u v) L p (Ω) C r u W v V T p (Ω) for all u Wp (Ω), for all u Wp (Ω) olds if and only if P r ( ˆK) ˆV. It is not ard to extend tese estimates to saperegular sequences of parallelogram meses as well. However, in tis section we sow tat for tese estimates to old for more general quadrilateral mes sequences, a stronger condition on ˆV is required, namely tat ˆV Q r ( ˆK). Te positive result, tat wen ˆV Q r ( ˆK), ten te estimates (4) and (5) old for any sape-regular sequence of quadrilateral meses T, is known. See, e.g., [2], [1], or [4, Section I.A.2]. We wis to sow te necessity of te condition ˆV Q r ( ˆK). As a first step, we sow tat te condition V F (K) P r (K) is necessary and sufficient to ave tat ˆV Q r ( ˆK) wenever F is a bilinear isomorpism of ˆK onto a convex quadrilateral. Tis is proven in te following two teorems. Teorem 3. Suppose tat ˆV Q r ( ˆK). Let F be any bilinear isomorpism of ˆK onto a convex quadrilateral. Ten V F (K) P r (K). Proof. Te components of F (ˆx, ŷ) are linear functions of ˆx and ŷ, so if p is a polynomial of total degree at most r, tenp(f (ˆx, ŷ)) is of degree at most r in ˆx and ŷ separately, i.e., p F Q r ( ˆK) ˆV. Terefore p V F (K). Te reverse implication olds even under te weaker assumption tat V F (K) contains P r (K) just for te two specific bilinear isomorpisms F (ˆx, ŷ) =(ˆx, ŷ(ˆx +1)), F (ˆx, ŷ) =(ŷ, ˆx(ŷ +1)), bot of wic map ˆK isomorpically onto te quadrilateral K wit vertices (0, 0), (1, 0), (0, 1), and (1, 2). Tis fact is establised below. Teorem 4. Let ˆV be a vector space of functions on ˆK. Suppose tat V F (K ) P r (K ) and V F (K ) P r (K ).TenˆV Q r ( ˆK). Proof. We prove tat ˆV Q r ( ˆK) by induction on r. Tecaser =0beingtrueby assumption, we consider r>0 and sow tat te monomials ˆx r ŷ s and ˆx s ŷ r belong

8 916 D. N. ARNOLD, D. BOFFI, AND R. S. FALK Figure 1. a. A partition of te square into four trapezoids. b. A mes composed of translated dilates of tis partition. to ˆV for s =0, 1,...,r.Fromteidentity s ( ) s ˆx r ŷ s =ˆx r s [ŷ(ˆx +1)] s ˆx r t ŷ s t t=1 (6) = F s ( ) s 1 (ˆx, ŷ) r s F2 (ˆx, ŷ) s ˆx r t ŷ s, t we see tat for 0 s<r, te monomial ˆx r ŷ s is te sum of a polynomial wic clearly belongs to ˆV (since F 1 (ˆx, ŷ) r s F2 (ˆx, ŷ) s = x r s y s P r (K ) V F (K )) and a polynomial in Q r 1 ( ˆK), wic belongs to ˆV by induction. Tus eac of te monomials ˆx r ŷ s wit 0 s<rbelongs to ˆV, and, using F, we similarly see tat all te monomials ˆx s ŷ r,0 s<r,belongto ˆV. Finally, from (6) wit s = r, we see tat ˆx r ŷ r is a linear combination of an element of ˆV and monomials ˆx s ŷ r wit s<r,soittoobelongsto ˆV. We now combine tis result wit tose of te previous section to sow te necessity of te condition ˆV Q r ( ˆK) for optimal order approximation. Let ˆV be some fixed finite dimensional subspace of L p ( ˆK) wic does not include Q r ( ˆK). Consider te specific division of te unit square ˆK into four quadrilaterals sown on te left in Figure 1. For definiteness we place te vertices of te quadrilaterals at (0, 1/3), (1/2, 2/3) and (1, 1/3) and te midpoints of te orizontal edges and te corners of ˆK. Te four quadrilaterals are mutually congruent and affinely related to te specific quadrilateral K defined above. Terefore, by Teorem 4, we can define for eac of te four quadrilaterals K sown in Figure 1 an isomorpism F from te unit square so tat V F (K ) P r (K ). If we let Ŝ be te subspace of Lp ( ˆK) consisting of functions wic restrict to elements of V F (K ) on eac of te four quadrilaterals K, ten certainly Ŝ does not contain P r( ˆK), since even its restriction to any one of te quadrilaterals K does not contain P r (K ). Next, for n = 1, 2,... consider te mes T of te unit square Ω sown in Figure 1b, obtained by first dividing it into a uniform n n mes of subsquares, n =1/, and ten dividing eac subsquare as in Figure 1a. Ten te space of t=1

9 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 917 functions u on Ω wose restrictions on eac subsquare K T satisfy û K (ˆx) = u(x K + ˆx) witû K Ŝ is precisely te same as te space V (T ) constructed from te initial space ˆV and te mes T. In view of Teorems 1 and 2 and te fact tat Ŝ P r ( ˆK), te estimates (4) and (5) do not old. In fact, neiter of te estimates inf u v L p (Ω) = o( r ) v V (T ) nor inf (u v) L p (Ω) = o( r 1 ) v V (T ) olds, even for u P r (Ω). Wile te condition ˆV Q r ( ˆK) is necessary for O( ) approximation on general quadrilateral meses, te condition ˆV P r ( ˆK) suffices for meses of parallelograms. Naturally, te same is true for meses wose elements are sufficiently close to parallelograms. We conclude tis section wit a precise statement of tis result and a sketc of te proof. If ˆV P r ( ˆK) andk = F ( ˆK) witf Bil( ˆK), ten by standard arguments, as in [1], we get v π K v L p (K) C J F 1/p L ( ˆK) v F Wp ( ˆK), were J F is te Jacobian determinant of F and π K denotes any convenient projection or interpolant satisfying (π K v) F = π ˆK(v F ), e.g., te L 2 projection. Now, using te formula for te derivative of a composition (as in, e.g., [3, p. 222]), and te fact tat F is quadratic, and so its tird and iger derivatives vanis, we get tat Now, v F W p ( ˆK) C J F 1 1/p L (K) v Wp (K) ()/2 i=0 F 2i W 1 ( ˆK) F i W 2 ( ˆK). J F L ( ˆK) C2 K, J F 1 L ( ˆK) C 2 K, F W 1 ( ˆK) C K, were K is te diameter of K and C depends only on te sape-regularity of K. We tus get v π K v Lp (K) C v W p (K) i 2i K F i W 2 ( ˆK). It follows tat if F W 2 ( ˆK) = O(2 K ), we get te desired estimate v π K v Lp (K) C K v Wp (K). Following [7], we measure te deviation of a quadrilateral from a parallelogram by te quantity σ K := max( π θ 1, π θ 2 ), were θ 1 is te angle between te outward normals of two opposite sides of K and θ 2 is te angle between te outward normals of te oter two sides. Tus 0 σ K <π,witσ K =0ifandonlyifK is a parallelogram. As pointed out in [7], F W 2 ( ˆK) C K( K + σ K ). Tis motivates te definition tat a family of quadrilateral meses is asymptotically parallelogram if σ K = O( K ), i.e., if σ K / K is uniformly bounded for all te elements in all te meses. From te foregoing considerations, if te reference space contains P r ( ˆK) weobtaino( ) convergence for asymptotically parallelogram, saperegular meses.

10 918 D. N. ARNOLD, D. BOFFI, AND R. S. FALK Figure 2. Tree sequences of meses of te unit square: square, trapezoidal, and asymptotically parallelogram. Eac is sown for n =2,4,8,and16. As a final note, we remark tat any polygon can be mesed by an asymptotically parallelogram, sape-regular family of meses wit mes size tending to zero. Indeed, if we begin wit any mes of convex quadrilaterals, and refine it by dividing eac quadrilateral in four by connecting te midpoints of te opposite edges, and continue in tis fasion, as in te last row of Figure 2, te resulting mes is asymptotically parallelogram and sape-regular. 4. Numerical results In tis section, we report on results from a numerical study of te beavior of piecewise continuous mapped biquadratic and serendipity finite elements on quadrilateral meses (i.e., te finite element spaces are constructed starting from te spaces Q 2 ( ˆK) ands 2 ( ˆK) on te reference square and ten imposing continuity). We present te results of two test problems. In bot we solve te Diriclet problem for Poisson s equation (7) u = f in Ω, u = g on Ω, were te domain Ω is te unit square. In te first problem, f and g are taken so tat te exact solution is te quartic polynomial u(x, y) =x 3 +5y 2 10y 3 + y 4. Tables 1 and 2 sow results for bot types of elements using meses from eac of te first two mes sequences sown in Figure 2. Te first sequence of meses consists of uniform square subdivisions of te domain into n n subsquares, n =2, 4, 8,...

11 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 919 Table 1. L 2 errors and rates of convergence for te test problem wit polynomial solution. square meses Mapped biquadratic elements trapezoidal meses u u L 2 (u u ) L 2 u u L 2 (u u ) L e e e e e e e e e e e e e e e e e e e e e e e e square meses Serendipity elements trapezoidal meses u u L 2 (u u ) L 2 u u L 2 (u u ) L e e e e e e e e e e e e e e e e e e e e e e e e Meses in te second sequence are partitions of te domain into n n congruent trapezoids, all similar to te trapezoid wit vertices (0, 0), (1/2, 0), (1/2, 2/3), and (0, 1/3). In Tables 1 and 2 we report te errors in L 2 and L, respectively, for te finite element solution and its gradient bot in absolute terms and as a percentage of te norm of te exact solution and its gradient, and we also report te apparent rate of convergence based on consecutive meses in a sequence. For tis test problem, te rates of convergence are very clear: for eiter mes sequence, te mapped biquadratic elements converge wit te expected order 3 for te solution and 2 for its gradient. Te same is true for te serendipity elements on te square meses, but, as predicted by te teory given above, for te trapezoidal mes sequence te order of convergence for te serendipity elements is reduced by 1 bot for te solution and for its gradient. As a second test example we again solved te Diriclet problem (7), but tis time coosing te data so tat te solution is te sarply peaked function u(x, y) =exp ( 100[(x 1/4) 2 +(y 1/3) 2 ] ). As seen in Table 3, in tis case te loss of convergence order in te L 2 norm for te serendipity elements on te trapezoidal mes is not nearly as clear. Some loss is evident, but apparently very fine meses (and very ig precision computation) would be required to see te final asymptotic orders. Similar results old wen te error in te L norm is considered, as sown in Table 4. Finally we return to te first test problem, and consider te beavior of te serendipity elements on te tird mes sequence sown in Figure 2. Tis mes sequence begins wit te same mes of four quadrilaterals as in previous case, and continues wit systematic refinement as described at te end of te last section, and so is asymptotically parallelogram. Terefore, as explained tere, te rate of

12 920 D. N. ARNOLD, D. BOFFI, AND R. S. FALK Table 2. L errors and rates of convergence for te test problem wit polynomial solution. square meses Mapped biquadratic elements trapezoidal meses u u L (u u ) L u u L 2 (u u ) L 2 5.0e e e e e e e e e e e e e e e e e e e e e e e e square meses Serendipity elements trapezoidal meses u u L 2 (u u ) L 2 u u L 2 (u u ) L e e e e e e e e e e e e e e e e e e e e e e e e Table 3. L 2 errors and rates of convergence for te test problem wit exponential solution. square meses Mapped biquadratic elements trapezoidal meses u u L 2 (u u ) L 2 u u L 2 (u u ) L e e e e e e e e e e e e e e e e e e e e e e e e e e e e square meses Serendipity elements trapezoidal meses u u L 2 (u u ) L 2 u u L 2 (u u ) L e e e e e e e e e e e e e e e e e e e e e e e e e e e e convergence for serendipity elements is te same as for affine meses. Tis is clearly illustrated in Table 5. Wile te asymptotic rates predicted by te teory are confirmed in tese examples, it is wort noting tat in absolute terms te effect of te degraded convergence rate is not very pronounced. For te first example, on a moderately fine mes of

13 APPROXIMATION BY QUADRILATERAL FINITE ELEMENTS 921 Table 4. L errors and rates of convergence for te test problem wit exponential solution. square meses Mapped biquadratic elements trapezoidal meses u u L (u u ) L u u L (u u ) L 2 1.0e e e e e e e e e e e e e e e e e e e e e e e e e e e e square meses Serendipity elements trapezoidal meses u u L (u u ) L u u L (u u ) L 2 5.8e e e e e e e e e e e e e e e e e e e e e e e e e e e e Table 5. L 2 errors and rates of convergence for te test problem wit polynomial solution using serendipity elements on asymptotically affine meses. u u L 2 (u u ) L 2 n err. % rate err. % rate 2 5.0e e e e e e e e e e e e e e trapezoids, te solution error wit serendipity elements exceeds tat of mapped biquadratic elements by a factor of about 2, and te gradient error by a factor of 2.5. Even on te finest mes sown, wit elements, te factors are only about 5.5 and 8.5, respectively. Of course, if we were to compute on finer and finer meses wit sufficiently ig precision, tese factors would tend to infinity. Indeed, on any quadrilateral mes wic contains a nonparallelogram element, te analogous factors can be made as large as desired by coosing a problem in wic te exact solution is sufficiently close to or even equal to a quadratic function, wic te mapped biquadratic elements capture exactly, wile te serendipity elements do not (suc a quadratic function always exists). However, it is not unusual tat te serendipity elements perform almost as well as te mapped biquadratic elements for reasonable, and even for quite small, levels of error. Tis, togeter wit teir optimal convergence on asymptotically parallelogram meses, provides an explanation of wy te lower rates of convergence ave not been widely noted.

14 922 D. N. ARNOLD, D. BOFFI, AND R. S. FALK References 1. P. G. Ciarlet, Te finite element metod for elliptic problems, Nort-Holland, Amsterdam, MR 58: P. G. Ciarlet and P.-A. Raviart, Interpolation teory over curved elements wit applications to finite element metods, Comput. Metods Appl. Mec. Engrg. 1 (1972), MR 51: H. Federer, Geometric measure teory, Springer-Verlag, New York, MR 41: V. Girault and P.-A. Raviart, Finite element metods for Navier Stokes equations, Springer- Verlag, New York, MR 88b: F. Kikuci, M. Okabe, and H. Fujio, Modification of te 8-node serendipity element, Comp. Metods Appl. Mec. Engrg. 179 (1999), R. H. McNeal and R. L. Harder, Eigt nodes or nine?, Int.J.Numer.MetodsEngrg.33 (1992), R. Rannacer and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Met. Part. Diff. Equations 8 (1992), MR 92i: P. Sarpov and Y. Iordanov, Numerical solution of Stokes equations wit pressure and filtration boundary conditions, J. Comp. Pys. 112 (1994), G. Strang and G. Fix, A Fourier analysis of te finite element variational metod, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E. II Ciclo, 1971, pp J. Zang and F. Kikuci, Interpolation error estimates of a modified 8-node serendipity finite element, Numer. Mat. 85 (2000), no. 3, MR 2001f: O. C. Zienkiewicz and R. L. Taylor, Te finite element metod, fourt edition, volume 1: Basic formulation and linear problems, McGraw-Hill, London, Institute for Matematics and its Applications, University of Minnesota, Minneapolis, Minnesota address: arnold@ima.umn.edu URL: ttp:// Dipartimento di Matematica, Università di Pavia, Pavia, Italy address: boffi@dimat.unipv.it URL: ttp://dimat.unipv.it/~boffi/ Department of Matematics, Rutgers University, Piscataway, New Jersey address: falk@mat.rutgers.edu URL: ttp://