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A parallel direct solver for the selfadaptive hp
 J. PARALLEL DISTRIB COMPUT
, 2010
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A Multi Level Direct Substructuring Multifrontal Parallel Solver for the hpFinite Element Method
"... We describe a new parallel direct solver for hp refined meshes, embedded into a 3D selfadaptive hp finiteelement method. The solver utilizes a substructuring method with multifrontal processing of subdomain internal nodes over each subdomain. This method of solution includes a new approach to so ..."
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We describe a new parallel direct solver for hp refined meshes, embedded into a 3D selfadaptive hp finiteelement method. The solver utilizes a substructuring method with multifrontal processing of subdomain internal nodes over each subdomain. This method of solution includes a new approach to solve the interface problem. Specifically, the solver utilizes multifrontal processing of the top of the tree of separators associated with the subdomains on which we apply the Schur complement strategy. The relative efficiency of the solver is both analyzed theoretically and measured on a sequence of meshes generated for a 3D borehole resistivity logging problem in a deviated well. Execution time and memory usage of the solver are compared against the parallel MUMPS solver executed over the entire problem, and the MUMPSbased direct substructuring method with the sequential or parallel MUMPS solvers utilized to approach the interface problem. We show that the relative efficiency of the new solver tends to infinity as the polynomial orders of approximation utilized on the hp meshes approaches infinity. This result is achieved under the assumption that the domain decomposition is performed on the level of finiteelement faces. Such a behavior comes as a consequence of the fact that the computational cost in the interior of highorder elements is several orders of magnitude higher then the computational cost over highorder elements faces. Based on the performed experiments, it follows that our new solver is up to five times faster than the MUMPS parallel solver when implemented on on 16 processors and if executed on computational meshes with high polynomial orders of approximation. Key words: Parallel direct solvers, Finite Element Method, hp adaptivity, 3D resistivity logging simulations.
Efficient Sequential and Parallel Solvers for hp Finite Element Method
"... Abstract We present a sequential and parallel direct solver designed for hp Finite Element Method (FEM) applied to solve numerous problems, including nonstationary heat transfer problem, the Stokes problem, and the resistivity logging measurement simulations. The hp FEM incorporates a selfadaptive ..."
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Abstract We present a sequential and parallel direct solver designed for hp Finite Element Method (FEM) applied to solve numerous problems, including nonstationary heat transfer problem, the Stokes problem, and the resistivity logging measurement simulations. The hp FEM incorporates a selfadaptive strategy that generates a sequence of hp refined meshes, delivering exponential convergence of the numerical error with respect to the number of degrees of freedom (mesh size or CPU time). The hp meshes generated by the selfadaptive strategy are obtained by multiple h or p refinements of the initial mesh. The selfadaptive mesh, generated in this way, is stored as refinement trees growing down from nodes of the initial mesh. First, we eliminate degrees of freedom starting from leaves of refinement trees, and then we eliminate common degrees of freedom traveling up the refinement trees. The solver is parallelized by utilizing the domain decomposition paradigm. In other words, the solver generates Schur complements of local subsystems, from bottom of refinement trees, through initial mesh elements and subdomains. Then, the global problem reduces to relatively small one common &quot;interface &quot; problem, and finally the backward substitution must be executed to propagate the solution from the common interface, through subdomains, initial mesh elements, down to leafs of refinement trees. The LU factorizations computed at different levels of elimination trees are stored at tree nodes to be reutilized by the solver after the computational mesh is