Dynamic domain decomposition and grid modification for parabolic problems

A dynamic grid modification and domain decomposition method is given and analyzed for parabolic problems. This method allows one to apply different domain decompositions, and different grids and interpolation polynomials on the subdomains at different time levels when necessary. The procedure relies on an implicit Galerkin method in the subdomains and explicit flux calculation on the inter-domain boundaries. In addition, a dynamic finite element scheme is proposed and analyzed, which is applicable to general parabolic problems. These methods are well suited to large-scale time-dependent problems involving localized phenomena, such as sharp fronts or layers, which also change with time. Convergence and stability analyses in the L2 norm are given. Numerical experiments are provided to check the performance of the methods and make comparison with other methods.