Error Estimates for Spatially Discrete Approximations of Semilinear Parabolic Equations with Initial Data of Low Regularity

Semidiscrete finite element methods for a semilinear parabolic equation in Rd, d < 3, were considered by Johnson, Larsson, Thomée, and Wahlbin. With h the discretization parameter, it was proved that, for compatible and bounded initial data in Ha, the convergence rate is essentially 0(h2+a) for t positive, and for a = 0 this was seen to be best possible. Here we shall show that for 0 < a < 2 the convergence rate is, in fact, essentially 0(h2+2a), which is sharp. 0. Introduction. The aim of this paper is to improve certain results from Johnson, Larsson, Thomée, and Wahlbin [2]. In this introduction we shall describe these results and motivate and state our new findings. The investigations in [2] are concerned with nonsmooth data error estimates for spatially discrete approximations to the solution of the initial-boundary value problem ut Au = f(u) in fi x 7, I = (0, t*\, (0.1) u = 0 on dfi x J, u(0) = v in fi, where fi is a bounded domain in Rd, d = 1,2, or 3, with smooth boundary <3fi, and / is a smooth function on R which is bounded together with an appropriate number of its derivatives. (For a discussion of this assumption, see [2, Section 3].) It is assumed throughout that v, and hence u, is bounded. The spatially discrete approximation Uh(t) is sought in a finite-dimensional space Sh C Hq (fi) and is defined by (uh,t, X) + (Vu„, Vx) = (f(uh), x) for X € Sh, tel, uh(0) = Pqv, where (v, w) is the standard inner product in ¿2 = L2(fi), and Pq is the orthogonal projection in ¿2 onto ShIt is assumed that the family {Sh} is such that the elliptic projection Pi, the orthogonal projection onto Sh with respect to the Dirichlet inner product (Vt), Viu), has an error of order hr, r > 2 integer, or, more precisely, ||Piu>-u>|| <Chr\\w\\r forweHrnHr), where || • || and || • ||r denote the standard norms in ¿2 and Hr = Hr(U), respectively. It was first proved that (cf. also Helfrich [1]) \\uh(t) u(t)\\ < C(R)h2 \og(l/h)r1 for ||v|| < R, tel. Received April 20, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 65M20, 65M60. ©1989 American Mathematical Society 0025-5718/89 $1.00 + $.25 per page