Negative norm estimates and superconvergence in Galerkin methods for parabolic problems

Negative norm error estimates for semidiscrete Galerkin-finite element methods for parabolic problems are derived from known such estimates for elliptic problems and applied to prove superconvergence of certain procedures for evaluating point values of the exact solution and its derivatives. Our first purpose in this paper is to show how known negative norm error estimates for Galerkin-finite element type methods applied to the Dirichlet problem for second order elliptic equations can be carried over to initial-boundary value problems for nonhomogeneous parabolic equations. We then want to describe how such estimates may be used to prove superconvergence of a number of procedures for evaluating point values of the exact solution and its derivatives. These applications include in particular the case of one space dimension with continuous, piecewise polynomial approximating subspaces, where we analyze methods proposed by Douglas, Dupont and Wheeler [3]. Further, in higher dimensions we discuss the application of an averaging procedure by Bramble and Schatz [11 for elements which are uniform in the interior and in the nonuniform case a method employing a local Green's function considered by Louis and Natterer [4]. The error analysis of this paper takes place in the general framework introduced in Bramble, Schatz, Thome'e and Wahlbin [21 allowing approximating subspaces which do not necessarily satisfy the homogeneous boundary conditions of the exact solution. These subspaces are assumed to permit approximation to order O(h') in L2 (r > 2) and to yield O(h2r2) error estimates for the elliptic problem in norms of order -(r 2). The superconvergent order error estimates which we aim for in the parabolic problem are then of this higher order. In [2], estimates of the type considered here were obtained for homogeneous parabolic equations by spectral representation; our basic results in this paper are derived by the energy method. 1. Preliminaries. We shall be concerned with the approximate solution of the initial-boundary value problem (ut = au/at, R+ = {t; t > O}) Lu-ut +Au-f in Q x R+, ;( .I) u(x, t) O on a x R , u(x, 0) =v(x) on Q2. Received September 12, 1978. AMS (MOS) subject classifications (1970). Primary 65N15, 65N30. ? 1980 American Mathematical Society 0025-571 8/80/0000-0005/$06.25 93 This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 03:53:30 UTC All use subject to http://about.jstor.org/terms 94 VIDAR THOMEE Here Q2 is a bounded domain in RN with sufficiently smooth boundary M, Au N (aj auk + au, j, k=l 1 i k) with alk and ao sufficiently smooth time-independent functions, the matrix (alk) symmetric and uniformly positive definite and ao nonnegative in Q2. In order to introduce some notation, we consider first the corresponding elliptic problem (1.2) Au =f in 2, u = 0 on a2, and denote by T: L2(Q) Hol) n H2(Q) its solution operator, defined by u = Tf. Notice that by the symmetry of A, T is selfadjoint and positive definite in L2(Q). Recall also the elliptic regularity estimate 1I Tflls+2 6 Cllf 11, for s > 0, where 11 Il denotes the norm in Hs(Q). Set now for s a nonnegative integer and v, w E L2(Q), with (,) the inner product in L2(Q), (1.3) (v, w)_S = (TSv, w), IIvIL_ = (TSv, v)l 2. Since T is positive definite, (, *) is an inner product. One can show that 11 IIis equivalent to the norm