Graph Grammar based Multi-thread Multi-frontal Direct Solver with Galois Scheduler

In this paper, we present a multi-frontal solver algorithm for the adaptive finite element method expressed by graph grammar productions. The graph grammar productions construct first the binary elimination tree, and then process frontal matrices stored in distributed manner in nodes of the elimination tree. The solver is specialized for a class of one, two and three dimensional h refined meshes whose elimination tree has a regular structure. In particular, this class contains all one dimensional grids, two and three dimensional grids refined towards point singularities, two dimensional grids refined in an anisotropic way towards edge singularity as well as three dimensional grids refined in an anisotropic way towards edge or face singularities. In all these cases, the structure of the elimination tree and the structure of the frontal matrices are similar. The solver is implemented within the Galois environment, which allows parallel execution of graph grammar productions. We also compare the performance of the Galois implementation of our graph grammar based solver with the MUMPS solver.