Graph-grammar greedy algorithm for reutilization of partial LU factorization over 3D tetrahedral grids

In this paper, we consider three-dimensional grids with tetrahedral finite elements, and the graph grammar model for mesh generation and adaptation as well as for the simultaneous construction of the element partition tree. The element partition tree defines the ordering to be used by the multi-frontal solver algorithm for efficient LU factorization. Additionally, we present the graph grammar productions for reutilization of the LU factorizations from the previous steps of the computations, when the mesh is h refined. We verify the ordering generated by our graph grammar through the element partition tree by comparing it against AMD, PORD and METIS orderings available through MUMPS interface. We also verify the graph grammar for reutilization by comparing the sequence of computation over 3D grids refined towards point, edge and face singularities, with and without the reutilization technique