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 methods.  In this paper modified finite difference approximations are obtained for grids with irregular nodes.  The modifications are required to ensure consistency and stability.  Asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method.  These estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives at grid nodes.  The interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids.  Computational results demonstrate the convergence of the finite difference method and a posteriori error estimates for cases not covered by the theory.