Finite difference methods and spatial a posteriori error estimates for solving parabolic equations in three space dimensions on grids with irregular nodes

Adaptive methods for solving systems of partial differential equations have become widespread. Much of the effort has focused on finite element method s. In this paper modified finite difference approximations are obtained for grids with irregular nodes. The modifications are required to ensure consi stency and stability. Asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method. These est imates are derived from interpolation estimates and are computed using cent ral difference approximations of second derivatives of the solution at grid nodes. The interpolation error estimates are shown to converge for irregul ar grids while the a posteriori error estimates are shown to converge for u niform grids. Computational results demonstrate the convergence of the fini te difference method and a posteriori error estimates for cases not covered by the theory.