On continuous and discrete maximum principles for elliptic problems with the third boundary condition
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In this paper we present the performance of our parallel multi-frontal direct solver when applied to solve linear systems of equations resulting from discretizations of a hp Finite Element Method (hp-FEM). The hp-FEM generates a sequence of computational meshes delivering exponential convergence of the numerical error with respect to the mesh size (number of degrees of freedom). A sequence of meshes is obtained by performing several hp refinements starting from an arbitrary initial mesh. The solver constructs initial elimination tree for an arbitrary initial mesh, and expands the elimination tree each time the mesh is refined. The solver has been tested on 3D Direct Current (DC) borehole resistivity measurement simulations problems. We compare the solver with two versions of the MUMPS parallel solver: with (1) distributed entries executed over the entire problem, and (2) the direct sub-structuring method with parallel MUMPS solver utilized to solve the interface problem. We show that by providing to the solver the knowledge about the structure of the hp-FEM, the order of elimination is obtained straightforward, and leads to a better performance than by submitting the entire matrix to the solver and executing a connectivity graph based ordering algorithm.