An Adaptive Overlapping Local Grid Refinement Method for Two-Dimensional Parabolic Systems.

An adaptive local refinement element method for solving vector systems of parabolic partial differential equations in two-space dimensions and time. The algorithm uses the finite element Galerkin method in space and backward Euler temporal integration. Each time step obtains an estimate of the error on each element, group the elements whose error violates a user prescribed tolerance, form new local grids, and solve the problem again on each of the new grids. We discuss several aspects of the algorithm including the necessary data structures, the error estimation technique, and the determination of initial and boundary conditions at coarse-fine mesh interfaces. Finally, several examples demonstrate the viability of this approach. Keywords: Finite element methods, Adaptive methods, Overlapping grids, Local refinement, Parabolic systems.