Linear Multistep Methods and Galerkin Procedures for Initial Boundary Value Problems

Abstract

The present paper is concerned with the numerical solution of parabolic and hyperbolic initial boundary value problems ($k = 1$ and $k = 2$). For the discretization in the space variables, Galerkin procedures are used and the resulting semidiscrete system is solved by linear multistep methods with a nonempty interval of absolute stability. For hyperbolic problems such methods are derived. Denoting by $u^ * $ the Ritz projection of the exact solution u onto the subspace S introduced by the Galerkin procedure and by $u_S $ the solution of the full discretized system. The propagation of $e = u^ * - u_S $ is studied in unbounded time intervals under the assumption that a bound for $u - u^ * $ is known from approximation theory. For time-independent problems it is proved that $e( \cdot ,t)$ is proportional to $t^k $ or t times the maximal truncation error, where the factor depends on the method under consideration. In some multistep methods, $t^k $ can be replaced by the dimension of the subspace S showing thereby that the error does not increase with time t. For the Crank–Nicholson method and the backward differentiation method of order two in time-dependent parabolic problems, even better results are obtained by simple estimations. However, these latter results are difficult to deduce in higher order methods.