mathematics of computation volume 38,number 158 april 1982, pages 401-413 Optimal Error Properties of Finite Element Methods for Second Order Elliptic Dirichlet Problems By Arthur G. Werschulz Abstract. We use the informational approach of Traub and Wozniakowski [9] to study the variational form of the second order elliptic Dirichlet problem Lu = f on ü C RN. For /e Hr(Q), where r> -1, a quasi-uniform finite element method using n linear functional Jaf^i nas T7'(ß)-norm error 0(n~<r+1)/'v). We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if / € Hr(ü), where r 3» -m, then there is a finite element method whose //"(fi)-norm error is %(n~(2m+r~a)/N) for 0 « a « m, and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals jaf'p, are approxi- mated by using n evaluations off, then there is a finite element method with quadrature with 7i'(ß)-norm error 0(n~r/N) where r > N/2. We show that when N = 1, there is no method using n function evaluations whose error is better than ñ(n~r); thus for N = 1, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of /. 1. Introduction. This paper deals with the optimal solution of second order elliptic partial differential equations. We wish to consider the variational form of the problem (1.1) Lu=f infiCR", u = 0 on3fi, (see Section 2). Suppose that we evaluate information of the form <"> [u*.//*•}• If / E Hr(Q), where r > -1, there exists a finite element method using (1.2) whose error is 0(n~(r+ X)/N) when measured in the i/'(ß) norm. We first wish to answer two questions. First, is there another method using the information (1.2) whose error is better than that of this finite element method? Second, is the information (1.2) the best possible information using n linear functionals? That is, is there another set of n linear functionals such that the best algorithm using this new information is better than the best algorithm using (1.2)? In Section 3, we asymptotically answer these questions in the negative. Thus,this finite element method is of asymptotically optimal error among all algorithms, linear or nonlinear, using any n linear functionals whatsoever. We also report some results on 2mth order elliptic problems which indicate that as m increases, the same number of evaluations yields smaller Hm(ü) error. Received May 20, 1980; revised June 15, 1981. 1980 Mathematics Subject Classification. Primary 65N30, 68C25; Secondary 65J10, 65N05, 65N15. 401 ©1982 American Mathematical Society 0025-5718/81/0000-0127/S04.25 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use