Interior error estimates for semi-discrete Galerkin approximations for parabolic equations

The initial boundary value problemfor the heat équation in a domain Q and the corresponding standard Galerkin method is consideied A certain regularity of the initial data in some subdomain Q1 leads to the same regularity of the solution in Q± and for ail times It is shown that the error between the exact solution and the Galerkm approximation is also of (almost) optimal order m the intenor ofQ.^ Of course certain properties ofthe underlying approximation spaces are needed, they are typical for finite éléments Résume — On considère le problème aux limites avec conditions initiales pour Véquation de la chaleur dans un domaine Q, ainsi que l approximation habituelle de Galerkin correspondante Une certaine régulante des données initiales dans un sous-domaine Ql conduit a la même régulante de la solution dans Ql pour tous les temps On montre que l'erreur entre la solution exacte et Vapproximation de Galerkin evt aussi à ordrp (presque) optiwnl da^s I^nter'evr deCix ^aîu e^emevt^ certaines propriétés des espaces d approximations sont utilisées, qui sont caractéristiques des espaces d éléments finis 1. In order to avoid technical details we restnet ourselves to the model problem M = Au m Q x (0, T], u = 0 on SQx (0, T], (1) ut=0 = v in Q The boundary of Q e u is assumed to be sufficiently smooth With the help of a finite element spaceS^ e H^Q) the Galerkin approximation uh = uh(t)eSh is defîned by + D(Mh> X) = 0 f o r %€Sh A t > 0 , (*) Reçu en janvier 1980 (**) Presented at the Conference on Progress m the Theory and Practice of the Finite Element Method, Goteborg, Sweden, August 27-29, 1979 O Institut fur Angewandte Mathematik, Albert-Ludwigs-Universitat, Freiburg, RFA R A I R O Analyse numerique/Numencal Analysis, 0399 0516/1981/171/8 5 00 © Bordas-Dunod 172 J. A. NiTsCHE Hère (., .) resp. D(M .) is the L2-inner product resp. the Dirichlet intégral and (for simplicity) Ph is the L2-projector onto Sh. For the corresponding elliptic problem Au = ƒ in Q , u 0 on dQ ( 3 ) interior estimâtes of the error e = u — uh of the Ritz approximation uh e Sh defmed by % X ) = ( / , X ) for %eSh (4) were derived in [1], [5], [6], [7]. They are of the following type (*) : Assume f e L2(Q) and in addition f e Hfe_2(^i) f° r some domain £lx ç Q and k > 2. Further let Q2 be contained properly inQ1.lïSh is of degree r with r ^ k then the error e is of order k in Q2, Le. \\e\\n* 0 fixed. Corresponding to (5) we would expect in the parabolic case an estimate of the type \\e\\L„{L2in2)) 0 arbitrary small. This problem was already treated in Thomee [9]. There the local error in the Lo0(L2(Q>2)) norm is bounded by the L2{L2(Qi)) norm besides of a remainder. Although this result does not give the final answer it turns out to be the main step. With respect to the notations as already mentioned as well as to the (*) We use the notations of [7] resp. [9]. R.A.LR.O. Analyse numérique/Numerical Analysis lia, < < ^ || « lia, + fe II » II + I II M* U n , ' M (14) ƒ.'*«•*}• GALERKIN APPROXIMATIONS FOR PARABOLIC EQUATIONS 173 assumptions on Sh we refer to [9]. Since we do not extend our resuit to différence quotients the uniformity of the subdivisions in defining Sh is not necessary of course. 2. We start with an interior estimate for uh, see lemma 3.3 in [9] : LEMMA 1 (Thomée) : Let uh be the solution of (2). Further assume and let q > 0 befixed. Then 1 «JLt) là , ̂ c | || Ph v \\ 2 ni + j [|| uh \\ 2 ni + h" II ùh | | â j dx 1 . (9)