The complete discretization of the dual mixed method for the heat diffusion equation in a polygonal domain

Abstract

The purpose of this paper is to prove a priori error estimates for the completely discretized problem of the dual mixed method for the non-stationary heat diffusion equation in a polygonal domain of R2. Due to the geometric singularities of the domain, the exact solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces in our analysis. To obtain optimal convergence rates of the discrete solutions, it is necessary to refine adequately the considered meshings near the reentrant corners of the polygonal domain. In a previous work, using the Raviart–Thomas vectorfields of degree 0 for the discretization of the heat flux density vector, we have obtained a priori error estimates of order 1 for the semi-discrete solutions of this problem. In our actual paper, we complete the discretization of the problem in time by using Euler’s implicit scheme, and we obtain optimal error estimates of order 1, in time and space. Numerical results are given to illustrate this improvement of the convergence orders.