Using the System of Graph Grammar for generation of quasi optimal element partition trees in two dimensions

Abstract

The paper presents a graph grammar based approach for h-adaptive finite element method and multi-frontal solver algorithm. The multi-frontal solver is used for solving systems of linear equations created by finite element method. The multi-frontal solver is controlled by so-called ordering. The quality of ordering influences hardly the solver effectiveness. In our approach, the finite element mesh is represented by means of a hypergraph and corresponding element partition tree. The finite element operations like mesh generation or h-adaptation are modelled by graph grammar production. Additionally graph grammar productions have corresponding productions for the construction of the element partition tree. The element partition trees are transformed into the ordering that controlls execution of the solver algorithm. We show that the ordering resulting from our element partititon tree results in better performance of the parallel solver than the state of the art nested-dissection ordering available through MUMPS interface on the class of grids refined towards singularities.