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

Handle URI:
http://hdl.handle.net/10754/561441
Title:
A parallel direct solver for the self-adaptive hp Finite Element Method
Authors:
Paszyński, Maciej R.; Pardo, David; Torres-Verdín, Carlos; Demkowicz, Leszek F.; Calo, Victor M. ( 0000-0002-1805-4045 )
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 h p-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. © 2009 Elsevier Inc. All rights reserved.
KAUST Department:
Applied Mathematics and Computational Science Program; Earth Science and Engineering Program; Physical Sciences and Engineering (PSE) Division; Environmental Science and Engineering Program; Numerical Porous Media SRI Center (NumPor)
Publisher:
Elsevier BV
Journal:
Journal of Parallel and Distributed Computing
Issue Date:
Mar-2010
DOI:
10.1016/j.jpdc.2009.09.007
Type:
Article
ISSN:
07437315
Appears in Collections:
Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program

Full metadata record

DC FieldValue Language
dc.contributor.authorPaszyński, Maciej R.en
dc.contributor.authorPardo, Daviden
dc.contributor.authorTorres-Verdín, Carlosen
dc.contributor.authorDemkowicz, Leszek F.en
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2015-08-02T09:11:24Zen
dc.date.available2015-08-02T09:11:24Zen
dc.date.issued2010-03en
dc.identifier.issn07437315en
dc.identifier.doi10.1016/j.jpdc.2009.09.007en
dc.identifier.urihttp://hdl.handle.net/10754/561441en
dc.description.abstractIn 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 h p-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. © 2009 Elsevier Inc. All rights reserved.en
dc.publisherElsevier BVen
dc.subject3D borehole resistivityen
dc.subjectFinite Element Methoden
dc.subjecthp adaptivityen
dc.subjectParallel direct solversen
dc.titleA parallel direct solver for the self-adaptive hp Finite Element Methoden
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentEarth Science and Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEnvironmental Science and Engineering Programen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalJournal of Parallel and Distributed Computingen
dc.contributor.institutionDepartment of Computer Science, AGH University of Science and Technology, Cracow, Polanden
dc.contributor.institutionIKERBASQUE (Basque Foundation for Sciences) and BCAM (Basque Center for Applied Mathematics) Bilbao, Spainen
dc.contributor.institutionDepartment of Petroleum and Geosystems Engineering, The University of Texas in Austin, United Statesen
dc.contributor.institutionInstitute for Computational Engineering and Sciences, The University of Texas in Austin, United Statesen
kaust.authorCalo, Victor M.en
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