A Weak Discrete Maximum Principle and Stability

Let ÍÍ be a polygonal domain in the plane and Sy(£l) denote the finite element space of continuous piecewise polynomials of degree < r — 1 (r > 2) defined on a quasi-uniform triangulation of ii (with triangles roughly of size h). It is shown that if un e Sy(Sl) is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form ""ftHi^n) < CII"/illL„.(3iï) holds. Now let u be a continuous function on £2 and un be the usual finite element projection of u into Sy(f2) (with un interpolating u at the boundary nodes). It is shown that for any xESf (ii) ii" "hhjitn < c(ln l)r ii" xir^ii). where '" = {„ ¡f \ > 3' This says that (modulo a logarithm for r = 2) the finite element method is bounded in L„, on plane polygonal domains. 0. Introduction and Statement of Results. The purpose of this paper is to discuss some estimates for the finite element method on polygonal domains. In particular, we shall consider the validity of (for want of a better terminology) a "discrete weak maximum principle" for discrete harmonic functions and then use this result to discuss the boundedness in L^ of the finite element projection. In this part we shall discuss the case of a quasi-uniform mesh. In Part II we shall concern ourselves with meshes which are refined near points. Let us first formulate the problems we wish to consider and state our results. References to other work in the literature which are relevant to our considerations will be given as we go along. For simplicity let S2 be a simply connected (this is not essential) polygonal domain in R2 with boundary 3fi and maximal interior angle a, 0 < a < 27r, where we emphasize that in general Í2 is not convex. On £2 we define a family of finite element spaces. For simplicity of presentation we shall restrict ourselves to a special but important class of piecewise polynomials. For each 0 < h < 1, let Th denote a triangulation of Í2 with triangles having straight edges. We shall assume that each triangle r is contained in a sphere of radius h and contains a sphere of radius yh for some positive constant y. We shall also assume that the family {Tn } of triangulations Received September 19, 1978. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. •This work was supported in part by the National Science Foundation. © 1980 American Mathematical Society 0025-5718/80/0000-0004/$04.75 77 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use