Interior estimates for

x2 ; 0<t T ;
(2:1)
1
with initial condition
u(x; 0) = u0(x) ; x 2 : The boundary conditions that we will consider are either { Dirichlet b.c.: u = 0 on a Lipschitz boundary ? = @ , or { periodic b.c. for the case that is a square or a cube. For Neumann b.c. or mixed b.c. we would obtain results similar to the Dirichlet case. We assume that the matrices ( ij (x; t))d =1 are symmetric and uniformly positive de nite, i;j and that the coe cients ij , j , are smooth functions, on the closure of (0; T ) or together with their space-periodic extension. The inhomogeneity f and the initial data u0 are also assumed to be smooth, but not necessarily compatible. We write u(t) = u( ; t) and f (t) = f ( ; t), and we interpret the parabolic initial-boundary value problem as an initial value problem in the Hilbert space setting of analytic semigroups that we used also in our earlier paper LuO2] (based on Ka], Li1]): u0 + A(t)u = f (t) ; u(0) = u0 ;
M for j arg j ? ; uniformly in 0 t T ; (2:3) 1+j j or at least this is satis ed for A(t) + I instead of A(t), with a suitable > 0. For ease of presentation, we assume henceforth that A(t) itself satis es (2.3). (For > 0, we would have to consider fractional powers of A(t) + I instead of A(t) in the sequel.) We further assume the existence of a second Hilbert space V , independent of t, such that
Abstract: It is known that Runge-Kutta time discretizations of parabolic initial-boundary
September 1994
Christian Lubich, Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany
1. Introduction
2. Preparations
We consider time discretization of a parabolic initial-boundary value problem over a bounded domain Rd ,
d @u = X @ @t i;j=1 @xi d X @u @u (x; t) @x + ij j (x; t) @x + (x; t)u + f (x; t) j j =1 j
Interior estimates for time discretizations of parabolic equations
Ch. Lubich and A. Ostermann
value problems su er an order reduction caused by the boundary conditions. Here we show that the full nonsti order of temporal convergence is attained in the interior of the domain. This holds even in the case of incompatible initial/boundary data or non-smooth boundaries. 1991 Mathematics Subject Classi cation: 65M12, 65M15, 65M20
alex@mat1.uibk.ac.at
In numerical experiments with discretizations of parabolic initial-boundary value problems, using implicit Runge-Kutta time-stepping and su ciently ne spatial grids, the following error behaviour is observed: (1) Order reduction: The numerically observed order of temporal convergence towards a smooth solution is non-integer, and smaller than the nonsti order p of the method, when the error is measured e.g. in the L2 or L1 norm over the whole domain. (2) Superconvergence in the interior: Away from the boundary, the numerical results show the full order of convergence p. (3) Error localization: Even if the data are incompatible, or nonsmooth in some spatial region, there is the full order of convergence p in subdomains bounded away from that region and from the boundary. Observation (1) was apparently rst communicated by Verwer Ve]. Related remarks about the local error were previously made on theoretical grounds by Crouzeix Cr]. The precise, non-integer orders of convergence were subsequently derived in OsR], LuO1], LuO2] for various classes of linear and nonlinear parabolic equations. These orders depend on the norms chosen and on the type of boundary conditions. In the present article we explain observations (2) and (3) for linear parabolic problems, ignoring space discretization. After describing the general framework in Section 2, we show in Section 3, Theorem 1, that the full order p of temporal convergence is attained for parabolic problems with periodic boundary conditions and smooth data. This theorem generalizes a result of Crouzeix Cr] and, in its abstract setting, serves as a basis for the interior estimates given in Section 4. There, our main result Theorem 3 provides the theoretical background for observations (2) and (3). Its proof is based on a boot-strap procedure that bounds strong norms of the error in the interior subdomain by weaker norms over larger domains, and on a full-order error estimate in very weak norms over the whole domain which follows from Theorem 1. We consider only Runge-Kutta time discretization in this paper. For BDF methods, there is no order reduction as in (1) above, but observation (3) is still valid. There is again a result like Theorem 3 for BDF methods. This can be proved by combining the techniques of the present paper with the stability bounds of Savare Sa].
lubich@na.uni-tuebingen.de
Alexander Ostermann, Institut fur Mathematik und Geometrie, Universitat Innsbruck, Tensbruck, Austria
(2:2) on a Hilbert space H with scalar product ( ; ) and norm j j. Here we have typically H = L2( ) (other choices of H will also be useful below), and ?A(t) : D(A(t)) H ! H is de ned as the di erential operator on the right-hand side of Eq. (2.1) equipped with the appropriate boundary conditions. This is the generator of an analytic semigroup on H . It is thus a densely de ned closed operator on H whose resolvent satis es, with some < 2 ,