RAIRO ANALYSE NUMÉRIQUE REINHARD SCHOLZ A mixed method for 4th order problems using linear finite elements RAIRO Analyse numérique, tome 12, n o 1 (1978), p. 85-90. <http://www.numdam.org/item?id=m2an_1978 12_1_85_0> AFCET, 1978, tous droits réservés. L accès aux archives de la revue «RAIRO Analyse numérique» implique l accord avec les conditions générales d utilisation (http://www.numdam.org/ legal.php). Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
R.AJ.R.O. Analyse numérique/numerical Analysis (vol. 12,n l, 1978, p. 85 à 90) A MIXED METHOD FOR 4TH ORDER PROBLEMS USING LINEAR FINITE ELEMENTS (*) by Reinhard SCHOLZ (*) Communiqué par J. A. Nitsche Abstract. The solution ofthefourth order probiem À 2 u = f in Cl, u = öujdv = 0 on dcl, Q, boundedin R 2, andits Laplacian are approximated by linear finite éléments. L 2 -and h^-error estimâtes are given. Let Q ç R 2 be a bounded domain with sufficiently smooth boxmdary. We consider the fourth order boundary value probiem A 2 w = ƒ in Q, 1 u» ôu/dv = 0 on ÔQ J with fel 2. The basic idea of the mixed method considered hère due to Ciarlet- Raviart [3]-is to write the équation (1) as a System -AM 2 = ƒ in tî,, } (2) u t = dujdv ==0 on 3Q to approxirnatë^/^ and^2-^î2i u ltâneousl3lby suitably chosen subspaces. (Another mixed method can be used if one is interested to approximate u and ail second dcrivatives of u, ïn this context we refer to Brezzi-Raviart [1] and the références given there.) Using finite element spaces of piecewise polynomials of degree r^2 as approximating subspaces the first L 2 -error estimâtes were given by Ciarlet- Raviart [3], In [9] improved Z^-estimates have been obtained, and in the case of quadratic finite éléments Rannacher [8] proved an L^-estimate. In this note we show that also in the case of linear finite éléments the mixed method approximations are convergent, and we dérive an error estimate in the L 2 - as well as in the L - norm. (*) Reçu juillet 1977. (*) Institut fur Angewandte Mathematik, Albert-Ludwigs-Universitât, Freiburg, Fédéral Republic of Germany. R.A.LR.O. Analyse numérique/numerical Analysis, vol. 12, n 1, 1978 (1)
86 R. SCHOLZ For h > 0 let T h be a x-regular partition of Q in generalized triangles, that means: (i) À e T h is a triangle if À and d l have at most one common point, otherwise one of the sides of A may be curved; (ii) there is a fixed x > 0 such that for each À e T h two circles K and K exists with radii x~ x h and x h and K ^ A ç ÜT. Let S,, = 5 ft (F A ) be the space of continuous functions which are linear in each triangle of T h with the usual modification for the curved éléments (see Ciarlet-Raviart [2], Zlamal [10]). The space S h is the intersection of S h and W\ 9 The approximation properties of the spaces S h and S h are wellknown; confer Ciarlet-Raviart [2] for example. Further we dénote with (.,.) and D (.,.) the inner product in L 2 and the Dirichlet intégral. Finally the Ritz operators R h : W\ -> S h and R h : W\ -> S h are defined by D(v-R h v > y\) = 0 for all (v R h v)ds = O and D(u-R h u, ) = 0 for all qes h respectively. The following lemma is fundamental for the dérivation of our estimâtes. The proof rests on L^-error estimâtes for the Ritz approximation of second order problems (see Nitsche [7], confer also Frehse-Rannacher [4], Nitsche [6]). LEMMA: For all u e W\ n W\ and for all x\ e S h we have D(u R h u, r\) j ^ C h 112 In h 11 j Au Loo j TJ \ L2, (3) with C independent of u, t and h. Proof: Let Q h be the union of all triangles A with A n dû, # 0. With >e S h we dénote the function which interpolâtes TJ at the interior grid-points of the triangulation T h. For cp : = r\ Ç it follows Setting : = u R h u and using the inverse estimate R.A.I.R.O. Analyse numérique/numerical Analysis
A MIXED METHOD 87 we therefore get D(e, T ) = D(s, 9) r grad s. grad cp ds k(a), (4) where the sum has to be taken over all triangles A ç Q h. Now it is easy to see that also in the curved triangles q> Loo(A)^ C max \y(s k )\, fc=l,2, 3 where s k dénotes the vertices of À. Therefore we find and with the inverse estimate we get from (4) r ioo (A> è Ch' 1 \\i\ since the number of triangles in Q h is of order h~ x. With the L^-estimate for the Ritz-approximation (Nitsche [7]) the lemma is proven, The mixed finite element approximation (u\ 9 u h 2) e S h x S h for the solution (w l5 u 2 ) of the probiem (2) is given by D(ul %)=(/,$ for auçes* (5) D(u\, T)) = (u h 2y r ) for ail r\ e S h J (see Ciarlet-Raviart [3], Scholz [9].) Since S h S h holds, the System (5) is uniquely solvable, and with e t : = u t u\ i = 1, 2, we can rewrite (5) in the foliowing form E) = 0 for ail stes h, ) - ^ *' [ > "H) = ( e 2-> T l) ^Or a U "H e S fc. In the L 2 -norm we get the following estimâtes, vol. 12, n 1, 1978 (5')
88 R. SCHOLZ THEOREM 1 : The différences e t = MJ MJ, / = 1, 2, between the exact solution of the problem (2) and the mixed finite element approximation can be estimated by \ \ ^ \ \ \ \ \ ^ \ \ (6) where C is independent of {u u u 2 ) and h. COROLLARY: AS a conséquence of (6) combined with the second part of (5') >v get for e x in the W\-norm H^II^^C^Ilnftl^ll^lk. (?) Proof: Let cpj e S h and <p 2 e S h be the Ritz approximations of u t respective u 2 as defined above. Using the équations (5') we find -(p u u h 2-<p 2 ) (8) With the standard error estimâtes for the Ritz approximations in the L 2 -norm the first term on the right-hand side can be estimated by For the second term we get with the help of the lemma and the third term is equal to zero by définition of <p 2. Combining these inequalities with (8) the estimate for e 2 follows. The other part of (6) is proven by a duality argument. Let w e W\ n W\ be the solution of A2. _ A w ~ e t in O, w = dw/dv = 0 on 3Q. Observing (5') we get e x \\t 2 = -!)(«!, Aw-R h Aw) + (e 2, Aw-R h Aw)+D(e 2, w-r k w). (9) Since u\ and ^ are éléments of S h, we obtain for the first term and the second term can be estimated by I (e 2, Au,-/?* Au,) ^ Cfc 2 1 e 2 t2 IHI M. R.A.LR.O. Analyse numérique/numerical Analysis
Finally we get again with the help of (3) A MIXED METHOD 89 \D(e 2, w-r h w)\^ch 2 \\u 2 \\ wl (\\w\\ wl + \\ Using Sobolev's embedding theorem we obtain Together with the estimate for e 2 we find \ D(e 2, w-r h w)\ S Ch\\nh\ 2 \\u 2 \\ wï \\w\\ w <. Combining these inequalities with (9) and observing, u ilhk^lhlk. the theorem is proven. The second resuit is an Z^-estimate for e t. THEOREM 2: The error u 1 u\ in the L^-norm can be estimated by II^II^^CfcllnfcHlutll^, (10) where C is independent of (u l9 u 2 ) and h. Proof: We choose À e T h such that With Standard arguments we get e l il = H e l luoo(a)- ^n^lio ai) and it suffices to estimate e x L2 (A). This again is done by a duality technique* Let wew\r\ W\ be the solution of tv = dw/dv 0 on /5Q. where % A is the characteristic function of the triangle A. With the same arguments as above we obtain and from a priori-estimates the inequality ^WAwlUWu.W^, (12) immediately follows* Further with the help of Sobolev's intégral identity (see for instance Mikhlin [5], pp. 60-66) we find for all real e > 0: vol. 12, n 1, 1978 d l l l ^ I l l l ^ l l l l l l ^ (14)
90 R. SCHOLZ with C independent of E. From Sobolev's imbedding theorem it foliows for all p > 2: û C p h 2 '" We.W^, (15) C p independent of h, and from Frehse-Rannacher [4], Theorem 4.B we Fixing/? > 2 and choosing : = h 2 ~ 2/p, we get from (13), (14), (15), and (16): ki il(a)gc/z 3 ln/z 3 j ei Loo Wl r 4. Together with (11) the last inequality gives the desired result. Remark: An analogue of the estimate (3) can be shown for finite element spaces of higher degree. Therefore the error estimâtes in this case can be improved too, especially in the case of quadratic finite éléments, provided that the solution is sufficiently smooth. REFERENCES 1. F. BREZZI and P. A. RAVIART, Mixed Finite Element Methods for Mh Order Elliptic Equations (to appear). 2. P. G. CIARLET and P. A. RAVIART, Interpolation Theory Over Curved Eléments, with Applications to Finite Element Methods, Computer Meth. AppL Mech. Engrg., 1, 1972, pp. 217-249. 3. P. G. CIARLET and P. A. RAVIART, A Mixed Finite Element Methodfor the Biharmonic Equation in Mathematical Aspects of Finite Eléments in Partial Differential Equations, C. DE BOOR, Ed., Proc. Symp. Math. Res. Center, Univ. Wisconsin, April 1-3, 1974, Academie Press, New York - San Francisco- London, 1974, pp. 125-145. 4. J. FREHSE and R. RANNACHER, Eine L x -Fehlerabschatzung für diskrete Grundlösungen in der Methode der finiten Elemente, Bonner Math. Schriften, 89, 1976, pp. 92-114. 5. S. G. MDCHLIN, The Problem of the Minimum of a Quadratic Functional, Holden- Day, Inc., San Francisco - London - Amsterdam, 1965. 6. J. NITSCHE, L^-Convergence of Finite Element Approximations, Rom, 1975 (to appear). 7. J. NITSCHE, L m -Convergence of the Ritz-Method with Linear Finite Eléments for Second Order Elliptic Boundary Value Problems (to appear in the anniversary volume dedicated to academician I. N. Vekua). 8. R.RANNACHER, Punktweise Konvergenz der Methode der finiten Elemente beim Plattenproblem, Manuscripta Math., 19, 1976, pp. 401-416. 9. R. SCHOLZ, Approximation von Sattelpunkten mit finiten Elementen, Bonner Math. Schriften, 89, 1976, pp. 53-66. 10. M. ZLAMAL, Curved Eléments in the Finite Element Method, I. S.LA.M. J. Numer. Anal., 10, 1973, pp. 229-240; II. S.I.A.M. J. Numer. Anal.,11, 1974, pp. 347-362. R.A.I.R.O. Analyse numérique/numerical Analysis