EXPLICIT ERROR BOUNDS IN A CONFORMING FINITE ELEMENT METHOD

Transcription

1 MATHEMATICS OF COMPUTATION Volume 68, Number 228, Pages S (99) Article electronically publised on February 24, 1999 EXPLICIT ERROR BOUNDS IN A CONFORMING FINITE ELEMENT METHOD PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Abstract. Te goal of tis paper is to define a procedure for bounding te error in a conforming finite element metod. Te new point is tat tis upper bound is fully explicit and can be computed locally. Numerical tests prove te efficiency of te metod. It is presented ere for te case of te Poisson equation and a first order finite element approximation. 1. Introduction Let us consider te following problem: { find u H0 1(Ω) suc tat, for all v H1 0 (1) (Ω), Ω u v= Ω fv, were f is a function in te space L 2 (Ω) and denotes te gradient of a function. It is well known tat (1) as a unique solution. Furtermore, under classical assumptions, one can prove tat u is an element of te space H 2 (Ω) H0 1 (Ω) (no re-entrant angle on te boundary, wic sould be piecewise smoot enoug). Let us now consider a family of triangulations of Ω, assumed to be uniformly regular (see Girault and Raviart [16]). One triangulation is denoted by T,were denotes te size of te mes. Te approximation space of H0 1 (Ω), based on te triangulation T, is denoted by V and is, for instance, defined by (2) V = {v H0 1 (Ω), T, v P 1 ()}, were P 1 () is te first degree polynomial space. Ten te approximation of u, denoted by u, is defined by { find u V suc tat for all v V (3) Ω u v= Ω fv. Te classical error estimate between u and u is derived from te a priori inequality (4) u u 1,Ω inf v u 1,Ω, v V were 1,Ω is te classical H 1 seminorm. From te interpolation results (see Ciarlet [8] or Raviart and Tomas [22]), one can deduce tat tere exists a constant c wic Received by te editor June 5, 1996 and, in revised form, February 19, Matematics Subject Classification. Primary 65N30, 65R20, 73C50. As a matter of fact, te internal angles on te boundary sould be smaller tan π c 1999 American Matematical Society

2 1380 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET is independent of bot and u, andsuctat (5) u u 1,Ω c u 2,Ω. But, unfortunately, tis estimate is not explicit because u 2,Ω is not. Let us explain anoter way to derive an error bound, wic was introduced by P. Ladevèze in is tesis [17] and wic is a particular implication of Prager and Synge s identity. Let us first introduce te following set of vector fields in Ω: (6) H f (div, Ω) = {p (L 2 (Ω)) 2, div p + f = 0 in Ω}. Ten one as te inequality, used first by Ladevèze [17], (7) u u 1,Ω inf p H f (div,ω) p u 0,Ω, te proof of wic is a straigtforward consequence of te following identity (Prager and Synge [20]). Let p and v be arbitrary elements in te sets H f (div, Ω) and V, respectively. Ten, if u is a solution of te Poisson model, one as u v 2 1,Ω + p u 2 0,Ω = p v 2 0,Ω. Because of its simplicity te proof is left to te reader. Te inequality (7) is obtained wit v = u. Te goal of te metod tat we develop is ten to define an element p in te set H f (div, Ω) suc tat p u is as small as possible. In tis paper we suggest acoiceforp, and we prove tat te term p u 0,Ω is O(), provided tat u is in te space H 2 (Ω) and tat te mes family satisfies a uniform regularity assumption. Let us point out te differences between (4) and (7). Te first is a so-called a priori estimate, and te second is a posteriori. In te first case te exact solution u is involved, but in te second case only u is necessary. Te error bound deduced from (4) requires us to define an element v in te case V suc tat v u 1,Ω will be as small as possible. Te space V is a conforming approximation of H0 1 (Ω). Wen (7) is used, te infimum is taken over vector fields cosen in te admissible set H f (div, Ω) for te problem dual to (1). Let us recall tat tis dual problem consists in minimizing in H f (div, Ω) te function p 1 (8) p 2. 2 Ω Te numerical approximation of tis problem is very difficult, and one prefers to use a mixed formulation. It enables one to avoid requiring exact satisfaction of te condition (9) div p + f = 0 in Ω. As a matter of fact it is, for instance, replaced by div p + 1 f =0, T (p being te solution of a first order mixed finite element). Sucanelementp cannot be used in (7) because p / H f (div, Ω). Furter details concerning mixed finite elements can be found in Roberts and Tomas [23].

3 EXPLICIT ERROR BOUND 1381 Hence we add a complementary element, call it δp, defined by (10) δp = δu, in, δu = f 1 f, in, δu ν =0on and δu =0. Te term p = p + δp is ten in H f (div, Ω). It can be proved tat, wit tis coice, p u is O() intel 2 (Ω) norm. But unfortunately te computation of p is not local and requires te solution of a global linear system (over te wole triangulation T ). Hence our goal is to construct a local approximation of p tat does not require many computations. One application of te metod is te adaptive mes refinement, but let us point out tat te true new point is tat te error bound is explicit. Ten we also discuss te asymptotic exactness of our error estimator. Tere is a well-developed literature on a posteriori error estimates and adaptive mes refinement for te elliptic equations. It seems quite impossible to list eac contribution in a single paper. But let us try to mention some of te papers tat are closest to our formulation. Te closest idea is due to Ladevèze [17]. But it appears tat tis autor did not use an exact construction of te dual variable, wic we need in our formulation. From a matematical point of view, Ainswort and Oden [1] ave underlined te interest in a coupling between a conformal finite element approximation and a ybrid one. Tey suggest using te Lagrange multiplier, wic is defined in order to prescribe te inter-element continuity, in order to construct an error estimator by solving a local (i.e., element by element) problem. Te way tey do it is close to but different from te one we suggest in tis paper. Te idea of comparing te finite element solution wit te dual problem is also te origin of te metod developed by Zienkiewicz and Zu in [26] and [27], but tey did not require te dual variable (te stress field in teir mecanical applications) to satisfy te equilibrium equation. Moreover, tey used a wole continuity of tis dual variable at te interelement, instead of only te one of te normal component. Terefore, te strategy seems to be azardous in case of singularities like a discontinuity of coefficients in te operator (bimaterial). From te matematical point of view, let us mention tree oter strategies wic are well founded and seem to be very promising. To our best knowledge te first is due to Babuška and Reinboldt [4]. Te basic trick consists in bounding te error between te exact and approximate solution by a constant times te so-called residual terms. Tere are two. One of tem is te jump between te normal derivatives of te finite element solution across te inter-elements, and te oter one is te lack of equilibrium inside te elements. Ten Verfürt [25] and Bernardi, Métivet and Verfürt [7] proved tat tis error bound is also up to a multiplicative constant a lower bound on te error. Te metod, wic can be extended to elasticity and te Stokes model [25], [7], seems to be very efficient in numerical applications. Tis is wy we used it to compare wit our formulation in te numerical tests in te last section of tis paper.

4 1382 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Te second strategy is certainly te most promising for te near future. It is based on superconvergence results. For particular meses tis strategy was developed by Babuška and Rodriguez [5], but te most important step for defining superconvergence points was acieved by Scatz, Sloan and Walbin [24]. Te mes refinement sould be defined using tese points. Te advantage is tat te metod would lead to a local contribution to te error. Te tird metod is quite close to ours from a teoretical point of view. It was developed by Bank and Weiser [6]. Te basic point seems to be to solve a local Neumann problem in order to construct an a posteriori error estimator. Te main advantage of te metod, compared to oters, is tat it gives (in an appropriate norm) an asymptotically exact estimate of te error as te mes size tends to zero. 2. Organisation of te paper First we will recall a few properties of te approximation model. Ten we find an element, call it p, wose construction can be performed locally (i.e., in te vicinity of one vertex of te mes). Tis element satisfies div p + 1 f =0 T, and p H(div, Ω). Te next step consists in finding a solution δu of (10) and in proving tat δu can be small if f is smoot enoug. Using te Green kernel, tis term is explicit as te solution of a local boundary integral equation. Te numerical solution can ten be found wit a predefined accuracy. Te last step, but not te least, is to prove tat te error bound p u,were p=p + δu in eac element of T, is itself bounded by O(). 3. Properties of te conforming finite element solution Let us denote by S te set of all te internal vertices of T. For eac vertex S i we introduce te basis function λ i of V, wic is equal to 1 at S i and 0 at all te oter vertices. From te definition of u we ave (11) u λ i fλ i =0, C i were Ci is te so-called cluster around S i te collection of elements of T wic ave S i as a vertex (see Figure 1). We denote by γi k te sides of C i tat ave S i as one of teir two extremities. Te number of elements in Ci is κ. Using te Stokes formula, one can transform (11) into te following relationsip: (12) κ k=1 { meas(γ k i ) 2 [ u ν ] k i C i } fλ i =0, k were [ ] k i is te jump of a quantity across te side γk i. Tis relation can be interpreted from a mecanical (for instance) point of view. Te first terms represent te moment of u at te vertex S i and along te side γi k. Te second term is te moment of te external forces acting in and expressed at te vertex S i. Ten (12) gives a global equilibrium of tese moments at S i. Te basic idea in te construction of p mentioned in te introduction is to equilibrate separately on eac element of Ci te moments at te vertex S i and

5 EXPLICIT ERROR BOUND 1383 S i γ i k ν γ i k+1 k Figure 1. Cluster C i, S i S to ensure te equilibrium between two neigbour elements. Hence we look for p suc tat p i H(div, C i ), p i ν =0 on C i, div p i = 1 { (13) u λ i fλ i }, wic implies tat an assumed solution satisfies, for all Ci, p i ν = u λ i fλ i u = γi 1 ν λ u i + γi 2 ν λ i fλ i = meas(γ1 i ) ( ) u + meas(γ2 i ) ( ) u fλ i 2 ν 2 ν (γi 1 and γ2 i are te two sides of wic ave S i as an extremity). Hence te term p i appears as te complementary system of forces wic could be applied in order to equilibrate separately eac triangle of te cluster Ci for te test variable λ i.te existence of p i is proved in te next section. 4. Definition of an equilibrium vector field on Ω Let us consider an arbitrary vertex S i of te triangulation T. We associate to S i te cluster Ci, wic is te set of elements of T suc tat S i is a vertex of. ButS i can be a point on te boundary of Ω. In bot cases te cluster Ci can be defined as sown in Figure 2. It is wort noting tat te definition of te boundary Ci in tis second situation does not include te vertices wic belong to te boundary Ω ofω. In order to approac te vector fields of te space H(div, Ω), we make use of te finite elements introduced by Raviart and Tomas in [21]. Teir restriction to te

6 1384 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET C i for internal point S i C i Ω S i C i C i for a boundary point Figure 2. Te two types of clusters cluster Ci is denoted by H RT1 (div, Ci ), and we use te definition { H RT1 (div, Ci )= p (L 2 (Ci)) 2,p ν =0 on Ci, Ci,p = a } +b x. c +b y Te index RT1 means Raviart-Tomas, degree 1. One remarkable property of te vectors in te space H RT1 (div, Ci )istatp νis constant and continuous across te sides of te elements, because p is in te space H(div, Ci ). Ten we introduce te following problem: Find p i H RT1(div, Ci ) suc tat for all C i, div p i = 1 { } (14) u λ i fλ i, in, and we prove te next result. Teorem 1. Tere exists a solution to (14) tat is defined up to an element of te kernel of te linear system. More precisely, p i = p i + α i rot λ i, α i R, were p i is a particular solution of (14) Proof. a) Let us begin wit te case were S i is an internal vertex of te triangulation T. Ten te linear system (14) as te same number of unknowns and of equations (tere are as many sides from S i as elements in Ci ). Let us terefore analyze te omogeneous system associated to (14). Tis problem consists in finding an element δi in H RT1 (div, Ci ) suc tat for all C i, (15) div δi =0 in.

7 EXPLICIT ERROR BOUND 1385 Because te cluster Ci suc tat is a simply connected open set, tere exists a function ψ i δ i = rot ψ i, ψ i H 1 (C i ). Furtermore, te condition δi ν =0on C i implies tat ψ i is constant along Ci. As ψ i is defined up to a constant, witout any loss of generality we can coose ψ i =0on Ci. But δ i is in te space H RT1 (div, Ci ). Hence on eac element of Ci one as rot ψ i = a + b x c + b y = a (b is zero because div δ i = 0). Finally, ψ i is piecewise linear and terefore is proportional to te basis function λ i. In tis situation (internal vertex), te kernel of (14), wic is one dimensional, is generated by te vector rot λ i. b) If now we consider S i on te boundary of Ω, everyting we did in te previous situation is still valid concerning te kernel. It as been proved tat te linear system (14) is singular and tat te kernel is one dimensional. Wen te vertex S i is internal to te triangulation T,tematrix of te linear system (14) is a square matrix and terefore te rigt-and side must be ortogonal to te cokernel (i.e., te kernel of te transposed matrix). Wen S i is on te boundary Ω, tere is no compatibility requirement because te matrix associated to (14) is rectangular and we ave one more unknown tan equations. Let us caracterize te cokernel of te matrix associated to te linear system (14). We already know tat it is one dimensional. An element X =(X i ), i =1,...,κ, of te cokernel satisfies c q H RT1 (div, C i ), κ X j (div q) j =0 ( j Ci ), j=1 but as X j (div q) j or else is constant on eac triangle j of Ci, one as κ j=1 κ j=1 γ j i 1 X j (div q) j =0, j j ( Xj j X ) j 1 q ν j 1 γ j =0, i were γ j i denotes te sides of C i tat ave S i as an extremity, as sown in Figure 2. Tus te quantity ξ = X j / j, j =1,...,κ, is constant for any j. Te cokernel is finally spanned by te vector X =(X j )= ( j )(measureof j ).

8 1386 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Te compatibility condition for te system (14) can ten be formulated as [ κ ] u λ i fλ i =0, j=1 j j or else (16) u λ i = fλ i, C i wic is precisely (for internal vertices) one of te equations caracterizing u. Terefore te rigt-and side of (14) is ortogonal to te cokernel, and Teorem 1 is proved. (17) From te elements p i C i defined in Teorem 1 we introduce te term p = p i, i S were S is te set of all vertices of triangulation T, including tose on te boundary of Ω (S is restricted to te internal points). From te definition of p i, and because p i =0onΩ C i, one as div p = div p i = { } 1 u λ i fλ i, i S i S C i and because on eac triangle we ave i S λ i = 1, we conclude tat, for all T, div p + 1 (18) f =0. As te element p defined in (18) is not in te set H f (div, Ω), we add a local term δu (defined on eac triangle of T ) suc tat δp = δu on, wit δu H 1 (), δu = f 1 (19) f on, δu =0on and δu =0. ν Te existence and uniqueness of a solution to (19) is very classical, and finally we set, on eac of T, (20) p = p + δp. It is wort noting tat δp H(div, Ω), because of te omogeneous Neumann boundary condition tat we cose on. Ten a simple compilation of te previous results sows tat (21) div p + f =0 onω. As a matter of fact, te term δu is only dependent of te rigt-and side f of te problem (1). It is obvious tat δu =0iff=0on. More precisely, we can upper bound δu depending on te regularity (local) of f. Te result is made explicit in te following teorem.

9 EXPLICIT ERROR BOUND 1387 Teorem 2. Assume tat f is in L 2 (Ω) and tat te triangulation family is regular. Ten tere exists a constant c, independent of bot f and, suctat δu 1, c f 0,. Furtermore, if f is in H 1 (), ten, under te same assumptions, δu 1, c 2 f 1,. Proof. From te definition of δu, and letting Π 0 denote te L 2 () projection onto te constants, one obtains ( δu 2 1, = (δu )δu = f 1 ) f δu f Π 0 f 0, δu 0, and, by Lemmas 1 and 2 (see te Appendix), δu 2 1, c f Π 0 f 0, δu 1, { c f 0, δu 1, if f L 2 (), c 2 f 1, δu 1, if f H 1 (). Tis completes te proof of Teorem 2. Remark. In te definition of p i te coefficient α i (see Teorem 1) is not yet defined. Let us mention one possibility. Consider one side of a cluster wit te center S i as an extremity. Ten on tis side, call it γi 1, one as α i rot λ i ν + p i ν 1 u γi 1 γi 1 γi 1 2 ν ( = α i + p i ν 1 u ), γi 1 2 ν and we can coose α i suc tat tis quantity is zero. Hence ( α i = p i ν 1 u ) γi 1 2 ν ) (22) = (p i ν u γi 1 ν λ i [ = meas(γi 1 ) p i ν 1 u ] 2 ν γi 1 We sall prove below tat suc a coice leads to a consistent error bound. 5. Asymptotic beaviour of te explicit error bound between u and u wen tends to zero Let us consider te element p defined in (20). From te a posteriori inequality, we ave u u 1,Ω p u 0,Ω =ε. Te main result of tis section is to prove tat ε is bounded by O(). Tis will justify tat tis explicit error bound is consistent wit respect to te classical results known in finite element metods.

10 1388 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Teorem 3. Assume tat f is in L 2 (Ω) and tat te triangulation family is uniformly regular. Ten tere exists a constant c, independent of bot (te mes size) and f, suc tat p u O,Ω c[ f 0,Ω + u 2,Ω ]. Proof. First of all, on eac element of te triangulation T we set k = p + δu u. Note tat on we ave curl k =0anddivk+f= 0. Furtermore, on te boundary of, k satisfies ( k ν = p ν u because δu ) =0on. ν ν Terefore, we can deduce tat tere exists a function ϕ suc tat k = ϕ and ϕ =0, ϕ = H 1 (); in addition ϕ is a solution of ϕ = f in, ϕ =0, (23) ϕ ν = p ν u on, ϕ H 1 (). ν Te previous model defines ϕ uniquely. But one also as (24) and, from (23), ε 2 def ε 2 = p + δu u 2 = ϕ 2 1, fϕ + ) (p ν u ϕ. ν Our goal is now to prove tat ε is O(). First of all, ε 2 = fϕ + p i νϕ i S u ν ϕ. But te summation over te index i ere is restricted to te tree vertices of. Let us introduce te element ˆq in te space H RT1 (div,) defined (see Raviart and Tomas in [21]) by 1 u γ, ˆq ν γ = meas(γ) γ ν, were γ is a side of and u is te solution of te initial problem, and we assume tat u is in H 2 (Ω). Te error estimates proved by Raviart and Tomas [21] lead to (te triangulation family is assumed to be regular) ˆq u 0, c u 2,, (25) and div ˆq div( u) 0, = div ˆq + f 0, c[ f 0, + u 2, ].

11 EXPLICIT ERROR BOUND 1389 Ten one obtains te following equality: ε 2 = fϕ + + i {i 1,i 2,i 3} (ˆq ν u ν ) ϕ, (p i ν λ i ˆq ν)ϕ or else, using te Stokes formula (i 1,i 2 and i 3 are te tree vertices of ), ε 2 = fϕ + (ˆq u ) ϕ + div ˆqϕ + (p i ν λ i ˆq ν)ϕ. Hence (26) i {i 1,i 2,i 3} ε 2 f+ div ˆq 0, ϕ 0, + ˆq u 0, ϕ 1, + (p i ν λ i ˆq ν)ϕ. i {i 1,i 2,i 3} From te triangular inequality and Lemma 1 in te Appendix, we deduce tat ε 2 c[ f 0, + u 2, + u u 1, ] ϕ 1, (27) + (p i ν λ i ˆq ν)ϕ. i {i 1,i 2,i 3} Butoneacsideγof only two terms p i (for i = i 1 and i 2, for instance) are different from zero. Terefore (p i ν λ i ˆq ν)ϕ i {i 1,i 2,i 3} = (p i ν 1 2 ˆq ν)ϕ γ i {i γ 1,i 2} 2 { p i ν 1 2 ˆq ν } meas(γ) ϕ γ 0,γ, γ were γi k is one side of te cluster Ci wit te center S i as an extremity. First of all, if we define Xi k def = (p i ν λ i ˆq ν) =meas(γi)[r k i ν 1 2 ˆq ν] γ, k γi k i ten X k+1 i Xi γ k = (p i ν λ i ˆq ν) (p i ν λ i ˆq ν) k+1 i γi (28) k = div p i div ˆq λ i ˆq λ i k k k and, because of te definition p i (see Figure 3),

12 1390 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET i k k i k+1 Figure 3. A triangle k of C i X k+1 i Xi k = ( u ˆq) λ i k (f+ div ˆq)λ i, k wic is bounded by c[ u 2, k + u u 1, k +[ f 0, k + u 2 k] λ i 0, k] or else (29) Xi k+1 Xi k c( f 0, + u k 2, k)+ u u 1, k. But from te definition of te coefficient α i in te expression of p i (22)): Xi 1 = (p i ν λ i ˆq ν) γ 1 i = α i + = 1 2 γ 1 i γ 1 i ( u p i ν 1 2 ν ˆq ν γ 1 i ). u ν + 1 ( ) u 2 γi 1 ν ˆq ν one as (see Let us denote by ξ te second degree polynomial function on te triangle i 1 equal to 1 on te middle of γi 1 and zero on te two oter sides. Ten, setting ( ξ =meas(γ 1 γi 1 i )2 3 ), we ave Xi 1 = 3 ( ) u 4 γi 1 ν ˆq ν ξ, and, from te Stokes formula, Xi 1 = ( u ˆq) ξ div ˆqξ 1 1 c( u u 1, 1 + u ˆq 0, 1 + div ˆq 0, 1). Finally, te inequalities u ˆq 0, 1 c u 2, 1, div ˆq 0, 1 c u 2, 1 enable one to obtain te estimate (30) Xi 1 c[ f 0, 1 + u 2, 1]+ u u 1, 1,

13 EXPLICIT ERROR BOUND 1391 and from (29) we obtain (31) X k i c[ f 0,C i + u 2,C i ]+c u u 1,C i for all k =1,...,κ (all te sides γ k i of te cluster C i wit S i as an extremity). Te proof of Teorem 3 is ten a consequence of (5), (27), (31), and Lemmas 1 and 3 in te Appendix. Obviously it requires tat te number of triangles in a cluster must be bounded above. Remark. As we proved in Teorem 2 tat δu 1, c 2 f 1,, it can be suggested tat if f is smoot enoug, tis term can be omitted. Remark. Wen δu must be computed, it is interesting to use a subgrid on. We point out tat tis computation is igly parallel, or can even be vectorized. Terefore, te computational time is very muc reduced. 6. Numerical tests Let us now suppose tat u is a solution of te classical conforming finite element metod defined by (3). Let us recall tat in order to apply te Prager-Synge relation, one as to construct a vector field p lying in te set H f (div, Ω). Te Raviart-Tomas finite element is used. For clarity, we recall briefly te basic principles of our strategy. At eac node S i of a mes T, we define te cluster of elements Ci, wic is te union of elements aving S i as a vertex. Ten for eac S i (even on te boundary of Ω), we set p i H RT1(div, Ci ), div p i = 1 { (32) u λ i fλ i }, Ci (λ i is te continuous piecewise linear function equal to one at S i and 0 at all te oter nodes). Here we ave put H RT1 (div, Ci {p H(div, )= Ci ),p ν =0 on C i and Ci p = a } +b x. c +b y Te existence of a solution to (32) as been proved in 4. But te solution is not unique. More precisely, we proved tat te general solution is were p i (33) p i = p i + α i rot λ i, is a particular solution of (32) and α i is an arbitrary constant. We set p = p i = p i + α i rot λ i, i S i S i S were S denotes te set of all te nodes of T (including tose on te boundary).

14 1392 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Figure 4.1. Examples of a regular mes Figure 4.2. Examples of a mes obtained by a mes-generator A nice coice for te coefficients α i is obtained by minimizing te error bound (assuming tat div p + f = 0, or else tat f is piecewise constant, for simplicity): α R L p + (34) α i rot λ i u, i S 0,Ω were L = card(s ). Two strategies can be ten discussed. One consists in replacing (34) by a local minimization (one iteration of te Jacobi algoritm, for instance, even if te matrix is not diagonal dominant). Te second one is more well founded, and it consists in adding to te former one iteration of te SSOR algoritm. Tese two strategies ave been cecked on te test model presented in tis paper. One can see tat te second one is more reliable for irregular meses (see Figures 5 and 6). As te additional cost is negligible, it as to be recommended for general applications. Te open set used is a square and two different kinds of meses are used. Tey are represented in Figures 4.1 and 4.2. In order to compare te metod described ere and te error indicator strategy of Bernardi, Métivet and Verfürt [7], we ave plotted tis quantity (denoted by B ) in Figures 5 and 6. Let us recall tat it is defined by B = f + u 0, + 1 [ ] 2 1/2 u γ 1/2 (35). 2 ν T γ Te indicator B is larger (a ratio of 6 wit te exact error instead of 1.3 for te metod developed ere). Anoter advantage of our error bound is tat one can improve te approximation of u by a local minimization problem, for instance, by adding degrees of freedom on te sides between elements. For example, we ave added degrees of freedom at te midpoint of te inner edges of te meses. Te 0,γ

15 EXPLICIT ERROR BOUND Mes size Present estimate, local minimization Present estimate, one iteration of Gauss-Seidel Bernardi-Metivet-Verfürt estimate Exact error Figure 5. Te error bounds, te exact errors, and te error indicators for regular meses Nonregular mes Present estimate, local minimization Present estimate, one iteration of Gauss-Seidel Bernardi-Metivet-Verfürt estimate Exact error Figure 6. Te error bounds, te exact errors, and te error indicators for irregular meses

16 1394 PHILIPPE DESTUYNDER AND BRIGITTE MÉTIVET Nonregular mes Mes size Exact error for te f.e.m. solution Error estimator for te f.e.m. solution Exact error for te upgraded f.e.m. solution Error estimator for te upgraded f.e.m. solution Figure 7. Local upgrade of te finite element solution using midpoint degrees of freedom error estimator obtained wit tis upgraded solution as been plotted in Figure 7. We ave also represented te exact error between te solution u and tis new term. Te former results ave been recalled in order to evaluate te improvement due to tis trick. 7. Conclusion Te metod tat we ave developed in tis paper is a new strategy for explicitly bounding te error in a finite element metod using conforming approximation. Te extension to elliptical problems does not require any new tricks. For instance, te case of 2D-elasticity can be andled. Te difficulty is ten to construct a symmetrical equilibrium finite element. Te way we know consists in replacing a Raviart-Tomas element by one of te family suggested by Arnold, Douglas and Gupta in [3]. In anoter respect te contribution to te error ε defined at (24) can be used as an error indicator in an automatic mes refinement algoritm. Te restricted numerical discussion given in 6 enables one to observe a few of te advantages of te metod developed in tis paper. A more extensive presentation of te numerical tests is given by Destuynder, Collot and Salaün in [13], and also in M. Collot s tesis [9], were te extension of te metod to adaptive mes refinements is discussed.

17 EXPLICIT ERROR BOUND 1395 Appendix In tis paper we used several classical results wic are quite well known. Tey are recalled ere. Lemma 1. Let be a triangle of a mes family assumed to be regular (see Ciarlet [8], and Girault and Raviart [16]). Ten let be te maximum lengt of te sides of. For any function ϕ H 1 (), satisfying ϕ =0, tere exists a constant (say c) wic is independent of bot and ϕ and suc tat i) ϕ 0, c ϕ 1,. Lemma 2 (Same ypotesis as in Lemma 1). For all ϕ L 2 () and T, let Π 0 ϕ be defined by 1 Π 0 ϕ = ϕ, meas() ten tere exists a constant c wic is independent of bot and ϕ and suc tat ϕ Π 0 ϕ 0, c ϕ 1,. Lemma 3. (Same ypotesis as in Lemma 1 but we also assume tat te family of triangulations is uniformly regular, as described by Girault and Raviart in [16]). Tere exists a constant c wic is independent of bot and ϕ and suc tat ϕ 0, c ϕ 1,. References 1. M. Ainswort and J. T. Oden [1993], A unified approac to a posteriori error estimation using element residual metods, Numer. Mat. 65, MR 95a: R. Arcangéli and J. L. Gout [1976], Sur l évaluation de l erreur d interpolation de Lagrange dans un ouvert de R n,rev.française Automat. Inform. Rec. Opér. Sér. Rouge Anal. Numér. 10, MR 58:29586 (preprint) 3. D. Arnold, J. Douglas, and C. P. Gupta [1984], A family of iger order mixed finite element metods for plane elasticity, Numer. Mat. 45, MR 86a: I. Babuška and W. C. Reinboldt [1978], Error estimates for adaptive finite element computation, SIAM J. Numer. Anal. 15, MR 58: I. Babuška and R. Rodriguez [1993], Te problem of te selection of an a posteriori error indicator based on smooting tecniques, Internat. Numer. Metods Engrg. 36, MR 93k: R. E. Bank and A. Weiser [1985], Some a posteriori error estimators for elliptic partial differential equations, Mat. Comp. 44, MR 86g: C. Bernardi, B Métivet and R. Verfürt [1993], Analyse numérique d indicateurs d erreur, note EDF-M172/ P. G. Ciarlet [1978], Te finite element metod for elliptic problems, Nort-Holland, Amsterdam. MR 58: M. Collot [1998], Analysis and implementation of explicit error bounds for finite element metods; Static and dynamic aspects, CNAM, Paris. 10. P. Destuynder and B. Métivet [1998], Explicit error bounds for a non-conforming finite element metod, Finite Element Metods (Jyväskylä, 1997), Lecture Notes Pure Appl. Mat., vol. 196, Marcel Dekker, New York, 1998, pp CMP 98: [1996a], Estimation de l erreur explicite pour une métode d éléments finis non conformes, C. R. Acad. Sci. Paris Sér. I. Mat. 322, MR 97e: [1996b], Estimation de l erreur explicite pour une métode d éléments finis conformes, C. R. Acad. Sci. Paris Sér. I. Mat. 323, MR 97:65138

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