Maximum Norm Estimates in the Finite Element Method

The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Rate of convergence estimates in the maximum norm, up to the boundary, are given locally. The rate of convergence may vary from point to point and is shown to depend on the local smoothness of the solution and on a possible pollution effect. In one of the applications given, a method is proposed for calculating the first few coefficients (stress intensity factors) in an expansion of the solution in singular functions at a corner from the finite element solution. In a second application the location of the maximum error is determined. A rather general class of non-quasi-uniform meshes is allowed in our present investigations. In a subsequent paper, Part 2 of this work, we shall consider meshes that are refined in a systematic fashion near a corner and derive sharper results for that case. 0. Introduction. Let D be a bounded simply connected domain in the plane with boundary dD consisting of a finite number of straight line segments meeting at vertices v-, j = 1, . . . , M, of interior angles 0 < a1 < • • ■ < aM < 2n (in a suitable ordering). We shall consider the Dirichlet problem !-Au = f in Í2, u = 0 on dD, where / is a given function, which for simplicity we assume to be smooth. To solve the problem (0.1) numerically, let Sh = Sh(D), 0 < h < 1, denote a o. . one-parameter family of finite dimensional subspaces of H (D) O Wl,(D). We have in mind piecewise polynomials of a fixed degree on a sequence of partitions of Í2. In our considerations the partitions do not have to be quasi-uniform, not even locally (cf. examples in Section 9). Let un E S be the approximate solution of (0.1) defined by the relation (0.2) A(un,x) = (f,X) for all x Giriere A(v, w) = fn Vu • Vw dx, and (v, w) = invw dx. We wish to obtain local estimates up to the boundary in the maximum norm for the error u uh. Although our present assumptions allow meshes that are refined near a corner, in the subsequent paper, Part 2, we shall investigate the error in more detail in that case, and obtain sharper results. The general results derived in the present paper will be essential in those investigations. Received April 2 5, 1977. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. •This work was supported in part by the National Science Foundation. Copyright © 1978, American Mathematical Society