Maximum Norm Estimates in the Finite Element Method on Plane Polygonal Domains

The finite element method is considered when applied to a model Dirichlet problem on a plane polygonal domain. Local error estimates are given for the case when the finite element partitions are refined in a systematic fashion near corners. 0. Introduction. We assume that the reader is familiar with Part 1, [2], of this paper; some notation is briefly recollected in Section 1. General references to the literature were given in the Bibliography of Part 1. Of these references, the following are particularly relevant to our present situation: Babuska [1], Babuska and Aziz [2], Babuska and Rheinboldt [4], Babuska and Rosenzweig [5], Eisenstat and Schultz [11], Thatcher [36]. Let Í2 be a bounded simply connected plane polygonal domain with interior angles 0 < a. < • • • < aM < 2rr, and consider the Dirichlet problem (O!) -Au=f in Í2, u = 0 on 9Í2, where /is a given, sufficiently smooth, function. To solve this problem numerically, let SH = Sh(Çl), 0 < h < 1, be a one param° i i eter family of finite element spaces, all subspaces of 77 (£2) n W„(£l). Define the approximate solution uh £ Sh by the relation (0.2) A(uh,x) = (fX) for all x SS", where A(v, w) = /n Vu • Vw dx and (v, w) = f^-w dx. We now describe briefly a representative result from Part 1 concerning the local rate of convergence for the finite element solution. Let r > 2 denote the optimal order of the parameter h to which the spaces Sh can approximate smooth functions in L norms. Furthermore, let ÍL, / = 1.M, he the intersection of Í2 with a disc of radius 7?. centered at the /th vertex and such that Í2ycontains no other vertex, and set i_0 = -7\(Ut^iH). Also, put ßj = ../a,-. In Part 1 we showed that with e > 0 arbitrarily small (see Part 1, Theorem 4.1 Received March 1, 1978. AMS (MOS) subject classifications (1970). Primary 6SN30, 65N15. * This work was supported in part by the National Science Foundation. © 1979 American Mathematical Society 0025-5718/79/0000-0051/$08.00 465 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 466 A. H. SCHATZ AND L. B. WAHLBIN for the precise hypotheses), ^-uhh00(ilj) »"-"r.ii¿oo(nM)/ = {x G fi: \x vM\ <2~kMRM}. Recall (Part 1, Section 1) that \Dau(x)\ < C|jc uM|^M"|a|_e, and thus the fiM k are regions where the bound for derivatives of u is roughly constant. Employing an interpolant x leadto, by (0.3), \\u-X\\L„ r, we may take hM k — h (i.e., no refinement is necessary); whereas if ßM < r, we should have (0.7) hM,*n} for any positive integer p. Thus, apart from the rightmost term in (0.9), the finite element solution mimics the pure approximation properties of Ylh and Sh. Actually, we shall need a slightly stronger refinement than the one described in (0.7), (0.8) in order to prove (0.9), viz., hMk < h(2~k)(1~ßMlr+S) for some positive S. This is due to technicalities in our proof. We refer the reader to Theorem 2.1 for the exact hypotheses. The second term on the right of (0.9) needs to be estimated. It contains the socalled "pollution effects" from other corners, and if no refinements were done at the remaining comers, the best we could say is that with p large, lll«-",IH-P,n r/2, no refinement is necessary at that vertex. If ßj < r/2, introduce the domains fi/>k, / = 1, . . . , M 1, k = k0 j, . . . , k¡, and fi;j as in (0.5), (0.6) but with / replacing M. Choose k0j such that (0.10) 2-fc°-/' =■ h(r'2~1 )l(r~ '"^ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 468 A. H. SCHATZ AND L. B. WAHLBIN and let the local meshsize hj k on fi. k satisfy (0-11) hhk < ri* 2) the refinement process can be taken to start fairly close to the corner according to (0.10), and is less stringent than at the Mih vertex (even if 0. = fttf)The conditions (0.10)—(0.12) can also be motivated from simple approximation considerations, see Section 4. Let us remark that if an hr~e rate of convergence is desired only on the interior domain fi0, then the weaker kind of refinement described in (0.10)—(0.12) suffices at each comer. To elucidate the above, let us give three examples. Example 0.1. A procedure for placing the nodes in the radial direction near vM. Consider the problem of how to place N + 1 nodes over [0, 1] so as to obtain an efficient approximation of the function xß (0 = ßM) with piecewise polynomials of degree r 1. This problem was solved by Rice [1], who explicitly prescribed the location of the nodes so as to obtain a good approximation, asymptotically as TV —► °°. Essentially, the TV + 1 nodes x¡, i = 0, . . . , TV, were taken as x¡ = (ilN)rlß. In the two dimensional situation, one can, e.g., construct a triangular mesh near vM in the following fashion, Figure 1. Draw TV + 1 radial lines (including the boundaries) from vM; along each of these mark down the TV + 1 points x¡. Then connect the rth points on the successive radial lines, thus obtaining a cobweb-like set of quadrilaterals. Now triangulate those by drawing one diagonal in each. The family of triangulations obtained in this simple way will, as TV—► °°, satisfy a maximum angle condition, but not a minimum angle one. In order to satisfy the latter, a more complicated construction would be necessary.