Report Number: CS-TR-65-31
Institution: Stanford University, Department of Computer Science
Title: On the approximation of weak solutions of linear parabolic equations by a class of multistep difference methods
Author: Raviart, Pierre Arnaud
Date: December 1965
Abstract: We consider evolution equations of the form (1) du(t)/dt + A(t)u(t) = f(t), $0 \leq\ t \leq\ T$, f given, with the initial condition (2) u(o) = $u_o$, $u_o$ given, where each A(t) is an unbounded linear operator in a Hilbert space H, which is in practice an ellilptic partial differential operator subject to appropriate boundary conditions. Let $V_h$ be a Hilbert space which depends on the parameter h. Let k be the time-step such that m = $\frac{T}{k}$ is an integer. We approximate the solution u of (1), (2) by the solution $u_{h,k}$ ($u_{h,k}$ = {$u_{h,k}(rk) \in V_{h}$, r = 0,1,...,m-1}) of the multistep difference scheme (3) $\frac{u_{h,k}(rk) - u_{h,k}((r-1)k)}{k} = \sum_{{\ell}=0}^{p} {\gamma}_{\ell} A_{h}((r-{\ell})k) u_{h,k}((r-{\ell}k) = \sum_{{\ell}=0}^{p} {\gamma}_{\ell} f_{h,k}((r-{\ell})k), r = p,...,m-1$ (4) $u_{h,k}(o),...,u_{h,k}((p-1)k)$ given, where each $A_{h}(rk) is a linear continuous operator from $V_h$ into $V_h$, $f_{h,k}(rk)$ (r = 0,1,...,m-1) are given, and ${\gamma}_{\ell}({\ell}=0,...,p) are given complex numbers. Our paper is mainly concerned by the study of the stability of the approximation. The methods used here are very closely related to those developed in the author's thesis and we shall refer to the thesis frequently. In Section 1,2, we define the continuous and approximate problems in precise terms. In Section 4, we find sufficient conditions for $u_{h,k}$ to satisfy some a priori estimates. The definition of the stability is given in Section 5 and we use the a priori estimates for proving a general stability theorem. In Section 6 we prove that the stability conditions may be weakened when A(t) is a self-adjoint operator (or when only the principal part of A(t) is self-adjoint). We give in Section 7 a weak convergence theorem. Section 8 is concerned with regularity properties. We apply our abstract analysis to a class of parabolic partial differential equations with variable coefficients in Section 9. Strong convergence theorems can be obtained as in the author's thesis (via compactness arguments) or as in the thesis of J.P. Aubin. We do not study here the discretization error (see author's thesis). For the study of the stability of multistep difference methods in the case of the Cauchy problem for parabolic differential operators, we refer to Kreiss [1959], Widlund [1965].