publication . Article . Other literature type . Preprint . 2016

High level implementation of geometric multigrid solvers for finite element problems: Applications in atmospheric modelling

Lawrence Mitchell; Eike Hermann Müller;
Open Access
  • Published: 01 Dec 2016
  • Publisher: Elsevier
  • Country: United Kingdom
Abstract
Comment: 22 pages, 5 figures, 9 tables. Submitted to JCP
Subjects
free text keywords: Applied Mathematics, Mathematical Sciences, Physical Sciences, Engineering, Computer Science - Mathematical Software, Mathematics - Numerical Analysis, Physics - Fluid Dynamics, 65F08, 65N55, 76M10, 86A10, D.2.2, G.1.3, G.1.8, G.4, J.2, Physics and Astronomy (miscellaneous), Computer Science Applications, Abstraction layer, Multigrid method, Preconditioner, Supercomputer, Finite element method, Computational science, Mathematical theory, Computer science, Source code, media_common.quotation_subject, media_common, Weak formulation
Related Organizations
39 references, page 1 of 3

[1] F. Brezzi, J. Douglas Jr, M. Fortin, L. D. Marini, E cient rectangular mixed nite elements in two and three space variables, ESAIM: Mathematical Modelling and Numerical Analysis - Modelisation Mathematique et Analyse Numerique 21 (4) (1987) 581{604.

[2] C. J. Cotter, J. Shipton, Mixed nite elements for numerical weather prediction, Journal of Computational Physics 231 (2012) 7076{7091. arXiv:1103.2440. [OpenAIRE]

[3] A. T. T. McRae, C. J. Cotter, Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic nite elements, Quarterly Journal of the Royal Meteorological Society 140 (684) (2014) 2223{2234. arXiv:1305.4477.

[4] A. Staniforth, J. Thuburn, Horizontal grids for global weather and climate prediction models: a review, Quarterly Journal of the Royal Meteorological Society 138 (662) (2012) 1{26.

[5] S. Borm, R. Hiptmair, Analysis Of Tensor Product Multigrid, Numerical Algorithms 26 (2001) 219{234.

[6] E. H. Muller, R. Scheichl, Massively parallel solvers for elliptic partial di erential equations in numerical weather and climate prediction, Quarterly Journal of the Royal Meteorological Society 140 (685) (2014) 2608{2624. arXiv: 1307.2036. [OpenAIRE]

[7] E. H. Muller, R. Scheichl, E. Vainikko, Petascale elliptic solvers for anisotropic PDEs on GPU clusters, Parallel Computing 50 (2015) 53{69. arXiv:1402.3545.

[8] A. Dedner, E. Muller, R. Scheichl, E cient multigrid preconditioners for atmospheric ow simulations at high aspect ratio, International Journal for Numerical Methods in Fluids 80 (1) (2016) 76{102. arXiv:1408.2981.

[9] F. Rathgeber, G. R. Markall, L. Mitchell, N. Loriant, D. A. Ham, C. Bertolli, P. H. J. Kelly, PyOP2: A High-Level Framework for Performance-Portable Simulations on Unstructured Meshes, in: High Performance Computing, Networking Storage and Analysis, SC Companion:, IEEE Computer Society, 2012, pp. 1116{1123. [OpenAIRE]

[10] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G.-T. Bercea, G. R. Markall, P. H. J. Kelly, Firedrake: automating the nite element method by composing abstractions, submitted (2015). arXiv:1501.01809.

[11] W. Bangerth, R. Hartmann, G. Kanschat, deal.II { a General Purpose Object Oriented Finite Element Library, ACM Transactions on Mathematical Software 33 (4) (2007) 24/1{24/27.

[12] P. Bastian, F. Heimann, S. Marnach, Generic implementation of nite element methods in the distributed and uni ed numerics environment (DUNE), Kybernetika 46 (2) (2010) 294{315.

[13] A. Dedner, R. Klofkorn, M. Nolte, M. Ohlberger, A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module, Computing 90 (3) (2010) 165{196. [OpenAIRE]

[14] P. Bastian, M. Blatt, R. Scheichl, Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems, Numerical Linear Algebra with Applications 19 (2) (2012) 367{388.

[15] G. Kanschat, Y. Mao, Multigrid methods for H(div)-conforming discontinuous Galerkin methods for the Stokes equations, Journal of Numerical Mathematics 23 (1) (2015) 51{66. arXiv:1501.06021.

39 references, page 1 of 3
Abstract
Comment: 22 pages, 5 figures, 9 tables. Submitted to JCP
Subjects
free text keywords: Applied Mathematics, Mathematical Sciences, Physical Sciences, Engineering, Computer Science - Mathematical Software, Mathematics - Numerical Analysis, Physics - Fluid Dynamics, 65F08, 65N55, 76M10, 86A10, D.2.2, G.1.3, G.1.8, G.4, J.2, Physics and Astronomy (miscellaneous), Computer Science Applications, Abstraction layer, Multigrid method, Preconditioner, Supercomputer, Finite element method, Computational science, Mathematical theory, Computer science, Source code, media_common.quotation_subject, media_common, Weak formulation
Related Organizations
39 references, page 1 of 3

[1] F. Brezzi, J. Douglas Jr, M. Fortin, L. D. Marini, E cient rectangular mixed nite elements in two and three space variables, ESAIM: Mathematical Modelling and Numerical Analysis - Modelisation Mathematique et Analyse Numerique 21 (4) (1987) 581{604.

[2] C. J. Cotter, J. Shipton, Mixed nite elements for numerical weather prediction, Journal of Computational Physics 231 (2012) 7076{7091. arXiv:1103.2440. [OpenAIRE]

[3] A. T. T. McRae, C. J. Cotter, Energy- and enstrophy-conserving schemes for the shallow-water equations, based on mimetic nite elements, Quarterly Journal of the Royal Meteorological Society 140 (684) (2014) 2223{2234. arXiv:1305.4477.

[4] A. Staniforth, J. Thuburn, Horizontal grids for global weather and climate prediction models: a review, Quarterly Journal of the Royal Meteorological Society 138 (662) (2012) 1{26.

[5] S. Borm, R. Hiptmair, Analysis Of Tensor Product Multigrid, Numerical Algorithms 26 (2001) 219{234.

[6] E. H. Muller, R. Scheichl, Massively parallel solvers for elliptic partial di erential equations in numerical weather and climate prediction, Quarterly Journal of the Royal Meteorological Society 140 (685) (2014) 2608{2624. arXiv: 1307.2036. [OpenAIRE]

[7] E. H. Muller, R. Scheichl, E. Vainikko, Petascale elliptic solvers for anisotropic PDEs on GPU clusters, Parallel Computing 50 (2015) 53{69. arXiv:1402.3545.

[8] A. Dedner, E. Muller, R. Scheichl, E cient multigrid preconditioners for atmospheric ow simulations at high aspect ratio, International Journal for Numerical Methods in Fluids 80 (1) (2016) 76{102. arXiv:1408.2981.

[9] F. Rathgeber, G. R. Markall, L. Mitchell, N. Loriant, D. A. Ham, C. Bertolli, P. H. J. Kelly, PyOP2: A High-Level Framework for Performance-Portable Simulations on Unstructured Meshes, in: High Performance Computing, Networking Storage and Analysis, SC Companion:, IEEE Computer Society, 2012, pp. 1116{1123. [OpenAIRE]

[10] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. T. McRae, G.-T. Bercea, G. R. Markall, P. H. J. Kelly, Firedrake: automating the nite element method by composing abstractions, submitted (2015). arXiv:1501.01809.

[11] W. Bangerth, R. Hartmann, G. Kanschat, deal.II { a General Purpose Object Oriented Finite Element Library, ACM Transactions on Mathematical Software 33 (4) (2007) 24/1{24/27.

[12] P. Bastian, F. Heimann, S. Marnach, Generic implementation of nite element methods in the distributed and uni ed numerics environment (DUNE), Kybernetika 46 (2) (2010) 294{315.

[13] A. Dedner, R. Klofkorn, M. Nolte, M. Ohlberger, A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module, Computing 90 (3) (2010) 165{196. [OpenAIRE]

[14] P. Bastian, M. Blatt, R. Scheichl, Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems, Numerical Linear Algebra with Applications 19 (2) (2012) 367{388.

[15] G. Kanschat, Y. Mao, Multigrid methods for H(div)-conforming discontinuous Galerkin methods for the Stokes equations, Journal of Numerical Mathematics 23 (1) (2015) 51{66. arXiv:1501.06021.

39 references, page 1 of 3
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