Numerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster maya

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dc.contributor.authorStonko, David
dc.contributor.authorKhuvis, Samuel
dc.contributor.authorGobbert, Matthias K.
dc.date.accessioned2018-09-25T19:42:36Z
dc.date.available2018-09-25T19:42:36Z
dc.date.issued2014
dc.description.abstractDiscretizing the elliptic Poisson equation with homogeneous Dirichlet boundary conditions by the finite difference method results in a system of linear equations with a large, sparse, highly structured system matrix. It is a classical test problem for comparing the performance of direct and iterative linear solvers. We compare in this report Gaussian elimination applied to a dense system matrix, Gaussian elimination applied to a sparse system matrix, the classical iterative methods of Jacobi, Gauss-Seidel, and SOR, and finally, the conjugate gradient method without preconditioning, and the conjugate gradient method with SSOR preconditioning. The key conclusions are: (i) The comparison of dense and sparse storage shows the crucial importance of sparse storage mode to solve problems even of intermediate size. (ii) The conjugate gradient method outperforms the classical iterative methods in all cases. (iii) Preconditioning can speed up the conjugate gradient method by an order of magnitude. (iv) We find that in two dimensions Gaussian elimination of a sparse system matrix is the fastest method, but runs out of memory eventually, where iterative methods can still solve the problem, but at the price of possibly extremely long run times. (v) However, in three dimensions, the iterative methods can be significantly faster than Gaussian elimination and can solve significantly larger problems. This explains the importance of iterative methods for three-dimensional problems.en_US
dc.description.sponsorshipThe hardware used in the computational studies is part of the UMBC High Performance Computing Facility (HPCF). The facility is supported by the U.S. National Science Foundation through the MRI program (grant nos. CNS–0821258 and CNS–1228778) and the SCREMS program (grant no. DMS–0821311), with additional substantial support from the University of Maryland, Baltimore County (UMBC). See www.umbc.edu/hpcf for more information on HPCF and the projects using its resources. This project began as the class project of the first author in Math 630 Numerical Linear Algebra in Spring 2014 at UMBC. It is a heavily modified version of that report [5], which considered the same test problem on a commodity laptop. The second author acknowledges financial support as HPCF RA.en_US
dc.description.urihttps://userpages.umbc.edu/~gobbert/papers/StonkoEtAl_HPCF2014.pdfen_US
dc.format.extent14 pagesen_US
dc.genretechnical reporten_US
dc.identifierdoi:10.13016/M2H98ZH4P
dc.identifier.urihttp://hdl.handle.net/11603/11382
dc.language.isoen_USen_US
dc.relation.isAvailableAtThe University of Maryland, Baltimore County (UMBC)
dc.relation.ispartofUMBC Mathematics Department Collection
dc.relation.ispartofUMBC Faculty Collection
dc.relation.ispartofseriesHPCF Technical Report;HPCF-2014-9
dc.rightsThis item is likely protected under Title 17 of the U.S. Copyright Law. Unless on a Creative Commons license, for uses protected by Copyright Law, contact the copyright holder or the author.
dc.subjectUMBC High Performance Computing Facility (HPCF)en_US
dc.subjectconjugate gradient method without preconditioning
dc.subjectconjugate gradient method with SSOR preconditioning
dc.subjectPoisson Equation
dc.subjectFinite Difference Method
dc.subjectIterative Methods
dc.subjectMatlab
dc.titleNumerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster mayaen_US
dc.typeTexten_US

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