On a numerical method for a class of parabolic problems in composite media

This paper deals with a numerical method, viz. a Rothe Galerkin finite element method, for a class of parabolic problems in composite media with jump conditions for the unknown at the interfaces of the subdomains, retaining the continuity of its conormal derivative. Crucial to our approach is a nonstandard variational formulation in a product Sobolev space setting. We begin with the discretization in time by the Rothe method, which, in essence, involves a backward finite difference scheme and which provides a constructive method for proving existence of a (unique) variational exact solution under weak conditions for the data. We are left with a recurrent system of elliptic problems at each subsequent time point which are solved numerically by a Galerkin finite element method. The resulting algorithm may be implemented by adapting standard codes for parabolic problems in one-component domains, when suitably taking into account the specific transmission conditions at the internal boundaries. The emphasis of the paper is on convergence results and error estimates for both the fully discrete and the semi-discrete approximation of the exact solution. The effectiveness of the approach is illustrated by means of a 1D and a 2D example, the analytical solution of which is known. © 1993 John Wiley & Sons, Inc.