A parallel direct solver for the self-adaptive hp Finite Element Method

  • Authors:

  • Maciej Paszyński;David Pardo;Carlos Torres-Verdín;Leszek Demkowicz;Victor Calo

  • Affiliations:

  • Department of Computer Science, AGH University of Science and Technology, Cracow, Poland;IKERBASQUE (Basque Foundation for Sciences) and BCAM (Basque Center for Applied Mathematics) Bilbao, Spain;Department of Petroleum and Geosystems Engineering, The University of Texas in Austin, USA;Institute for Computational Engineering and Sciences, The University of Texas in Austin, USA;Earth Science and Engineering, Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), Saudi Arabia

  • Venue:
  • Journal of Parallel and Distributed Computing
  • Year:
  • 2010

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Abstract

In this paper we present a new parallel multi-frontal direct solver, dedicated for the hp Finite Element Method (hp-FEM). The self-adaptive hp-FEM generates in a fully automatic mode, a sequence of hp-meshes delivering exponential convergence of the error with respect to the number of degrees of freedom (d.o.f.) as well as the CPU time, by performing a sequence of hp refinements starting from an arbitrary initial mesh. The solver constructs an initial elimination tree for an arbitrary initial mesh, and expands the elimination tree each time the mesh is refined. This allows us to keep track of the order of elimination for the solver. The solver also minimizes the memory usage, by de-allocating partial LU factorizations computed during the elimination stage of the solver, and recomputes them for the backward substitution stage, by utilizing only about 10% of the computational time necessary for the original computations. The solver has been tested on 3D Direct Current (DC) borehole resistivity measurement simulations problems. We measure the execution time and memory usage of the solver over a large regular mesh with 1.5 million degrees of freedom as well as on the highly non-regular mesh, generated by the self-adaptive hp-FEM, with finite elements of various sizes and polynomial orders of approximation varying from p=1 to p=9. From the presented experiments it follows that the parallel solver scales well up to the maximum number of utilized processors. The limit for the solver scalability is the maximum sequential part of the algorithm: the computations of the partial LU factorizations over the longest path, coming from the root of the elimination tree down to the deepest leaf.