Stability and Convergence of a Generalized Crank-Nicolson Scheme on a Variable Mesh for the Heat Equation

Abstract

In a previous paper devoted to the numerical solution of the Stefan problem, the author proposed a numerical scheme to solve the heat equation on a variable mesh; this scheme is a generalization of the classical Crank-Nicolson scheme since it is identical to the Crank-Nicolson scheme in the particular case of a fixed mesh. Numerical experiments have been performed in one and two space dimensions, but no mathematical results had been proved. In the present paper, the stability and convergence of the scheme are established together with an error estimate.