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 diierential equations have become widespread. Much of the eeort has focused on nite element methods. In this paper modiied nite diierence approximations are obtained for grids with irregular nodes. The modiications are required to ensure consistency and stability. Asymptotically exact a posteriori error estimates of the spatial error are presented for the nite diierence method. These estimates are derived from interpolation estimates and are computed using central diierence approximations of second derivatives of the solution 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 nite diierence method and a posteriori error estimates for cases not covered by the theory.