FINITE-DIFFERENCES SCHEMES ON GRIDS WITH LOCAL REFINEMENT IN TIME AND SPACE FOR PARABOLIC PROBLEMS .1. DERIVATION, STABILITY, AND ERROR ANALYSIS

Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat). The approximation at the grid points near the fine and coarse grid interface is based on the approach proposed by the authors in a previous paper for selfadjoint elliptic equations. The proposed schemes are implicit of backward Euler type and are shown to be unconditionally stable. Error analysis is also presented. © 1990 Springer-Verlag.