Parallelization of the Mesh Refinement Algorithm of the FDEM Program Package

摘要

The Finite Difference Element Method (FDEM) program package is a robust and efficient black-box solver. It solves by a Finite Difference Method arbitrary non-linear systems of elliptic and parabolic partial differential equations under arbitrary non-linear boundary conditions on arbitrary domains in 2-D and 3-D, given by a FEM mesh. From formulas of different order, we get an easy access to the discretization error. By the knowledge of this error, the mesh may be refined locally to reduce the error to a prescribed relative tolerance. For the refinement of the elements, we first determine the refinement nodes because of either error or data organization reasons. The refinement of an element is based on halving its edges. We explain the difficulties in parallelizing the mesh refinement algorithm on distributed memory parallel computers where the processors have only local data and the refinement must be synchronized by the message passing paradigm.