Finite difference schemes satisfying an optimality condition for the unsteady heat equation

In this paper we present a formulation of a finite difference Crank–Nicolson scheme for the numerical solution of the unsteady heat equation in 2 + 1 dimensions, a problem which has not been extensively studied when the spatial domain has an irregular shape. It is based on a second order difference scheme defined by an optimality condition, which has been developed to solve Poisson-like equations whose domains are approximated by structured convex grids over very irregular regions generated by the direct variational method. Numerical examples are presented and the results are very satisfactory.