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Comparative study of finite element methods using the time-accuracy-size (TAS) spectrum analysis. (English) Zbl 1417.65224
For comparison of different finite element methods using continuous and discontinuous Galerkin approaches, a performance analysis metrics is introduced. An extended performance spectrum model is presented, based on the work of J. Chang et al. [“A performance spectrum for parallel computational frameworks that solve PDEs”, Concurrency Comput. Pract. Exp. 30, No. 11, e4401 (2017; doi:10\.1002/cpe.4401)], which takes into account time-to-solution, accuracy of the solution and the problem size.
Thus hardware and algorithmic trade-offs can be interpreted. The proposed metrics are illustrated for the Poisson equation using various meshes on a \(2d\) unit square and a unit cube, for the latter using parallel computations.

MSC:
65Y05 Parallel numerical computation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] M. F. Adams, Evaluation of three unstructured multigrid methods on \(3\)D finite element problems in solid mechanics, Int. J. Numer. Methods Engrg., 55 (2002), pp. 519–534. · Zbl 1076.74547
[2] M. F. Adams, H. Bayraktar, T. Keaveny, and P. Papadopoulos, Ultrascalable implicit finite element analyses in solid mechanics with over a half a billion degrees of freedom, in Proceedings of the 2004 ACM/IEEE Conference on Supercomputing (SC ’04), Pittsburgh, PA, 2004, 34.
[3] R. Agelek, M. Anderson, W. Bangerth, and W. L. Barth, On orienting edges of unstructured two-and three-dimensional meshes, ACM Trans. Math. Software (TOMS), 44 (2017), 5. · Zbl 06920067
[4] M. Aln\aes, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes, and G. N. Wells, The FEniCS project version 1.5, Arch. Numer. Software, 3 (2015), pp. 9–23.
[5] G. M. Amdahl, Validity of the single processor approach to achieving large scale computing capabilities, in Proceedings of the April 18-20, 1967, Spring Joint Computer Conference, AFIPS ’67 (Spring), ACM, New York, 1967, pp. 483–485, .
[6] D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, Unified analysis of discontinuous galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), pp. 1749–1779, . · Zbl 1008.65080
[7] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Tech. report ANL-95/11—Revision 3.9, Argonne National Laboratory, Lemont, IL, 2018.
[8] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, R. T. Mills, T. Munson, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Web Page, , 2018.
[9] W. Bangerth, D. Davydov, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B. Turcksin, and D. Wells, The deal.II library, version 8.4, J. Numer. Math., 24 (2016), pp. 135–141. · Zbl 1348.65187
[10] G.-T. Bercea, A. T. T. McRae, D. A. Ham, L. Mitchell, F. Rathgeber, L. Nardi, F. Luporini, and P. H. J. Kelly, A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake, Geosci. Model Dev., 9 (2016), pp. 3803–3815, .
[11] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, New York, 2002. · Zbl 1012.65115
[12] J. Brown, Threading tradeoffs in domain decomposition, presented at the SIAM Conference on Parallel Processing for Scientific Computing as part of the Minisymposia “To Thread or Not to Thread,” Paris, 2016, .
[13] J. Brown, B. Smith, and A. Ahmadia, Achieving textbook multigrid efficiency for hydrostatic ice sheet flow, SIAM J. Sci. Comput., 35 (2013), pp. B359–B375, . · Zbl 1266.86001
[14] J. Chang, S. Karra, and K. B. Nakshatrala, Large-scale optimization-based non-negative computational framework for diffusion equations: Parallel implementation and performance studies, J. Sci. Comput., 70 (2017), pp. 243–271. · Zbl 1359.65250
[15] J. Chang and K. B. Nakshatrala, Variational inequality approach to enforce the non-negative constraint for advection-diffusion equations, Comput. Methods Appl. Mech. Engrg., 320 (2017), pp. 287–334.
[16] J. Chang, K. B. Nakshatrala, M. G. Knepley, and L. Johnsson, A performance spectrum for parallel computational frameworks that solve PDEs, Concurrency and Computation Practice and Experience, 30 (2017), e4401, .
[17] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365, . · Zbl 1205.65312
[18] V. Eijkhout, Introduction to High Performance Scientific Computing, Texas Advanced Computing Center (TACC), The University of Texas at Austin, 2014, .
[19] M. S. Fabien, M. G. Knepley, and B. M. Rivière, A hybridizable discontinuous Galerkin method for two-phase flow in heterogeneous porous media, Int. J. Numer. Methods Engrg., 116 (2018), pp. 161–177, .
[20] R. D. Falgout and U. M. Yang, HYPRE: A library of high performance preconditioners, in Proceedings of the International Conference on Computational Science, Springer, New York, 2002, pp. 632–641. · Zbl 1056.65046
[21] D. Gaston, C. Newman, G. Hansen, and D. Lebrun-Grandie, MOOSE: A parallel computational framework for coupled systems of nonlinear equations, Nuclear Engrg. Des., 239 (2009), pp. 1768–1778.
[22] J. L. Gustafson, Reevaluating Amdahl’s law, Comm. ACM, 31 (1988), pp. 532–533, .
[23] M. Homolya and D. A. Ham, A parallel edge orientation algorithm for quadrilateral meshes, SIAM Journal on Scientific Computing, 38 (2016), pp. S48–S61, . · Zbl 1404.65266
[24] R. M. Kirby, S. J. Sherwin, and B. Cockburn, To CG or to HDG: A comparative study, J. Sci. Comput., 51 (2012), pp. 183–212. · Zbl 1244.65174
[25] B. S. Kirk, J. W. Peterson, R. H. Stogner, and G. F. Carey, libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations, Engrg. Comput., 22 (2006), pp. 237–254, .
[26] M. G. Knepley, Computational Science I, Lecture Notes for CAAM 519, Department of Computational and Applied Mathematics, William Marsh Rice University, Houston, TX, 2017, .
[27] M. G. Knepley and D. A. Karpeev, Mesh algorithms for PDE with Sieve I: Mesh distribution, Sci. Programming, 17 (2009), pp. 215–230, .
[28] M. Lange, M. G. Knepley, and G. J. Gorman, Flexible, scalable mesh and data management using PETSc DMPlex, in Proceedings of the 3rd International Conference on Exascale Applications and Software Conference, Edinburgh, 2015, pp. 71–76, .
[29] M. Lange, L. Mitchell, M. G. Knepley, and G. J. Gorman, Efficient mesh management in Firedrake using PETSc DMPlex, SIAM J. Sci. Comput., 38 (2016), pp. S143–S155, . · Zbl 1352.65613
[30] A. Logg, Efficient representation of computational meshes, Int. J. Comput. Sci. Engrg., 4 (2009), pp. 283–295.
[31] N. K. Mapakshi, J. Chang, and K. B. Nakshatrala, A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity, J. Comput. Phys., 359 (2018), pp. 137–163. · Zbl 1383.76342
[32] D. A. May, J. Brown, and L. L. Laetitia, pTatin3D: High-performance methods for long-term lithospheric dynamics, in Proceedings of the International Conference for High Performance Computing, Network, Storage and Analysis (SC ’14), IEEE Press, Piscataway, NJ, 2014, pp. 274–284.
[33] A. T. T. McRae, G.-T. Bercea, L. Mitchell, D. A. Ham, and C. J. Cotter, Automated generation and symbolic manipulation of tensor product finite elements, SIAM J. Sci. Comput., 38 (2016), pp. S25–S47, . · Zbl 1352.65615
[34] H. Morgan, M. G. Knepley, P. Sanan, and L. R. Scott, A stochastic performance model for pipelined Krylov methods, Concurrency and Computation Practice and Experience, 28 (2016), pp. 4532–4542, .
[35] F. Rathgeber, D. A. Ham, L. Mitchell, M. Lange, F. Luporini, A. T. McRae, G.-T. Bercea, G. R. Markall, and P. H. Kelly, Firedrake: Automating the finite element method by composing abstractions, ACM Trans. Math. Software (TOMS), 43 (2016), 24. · Zbl 1396.65144
[36] P. A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Springer, New York, 1977, pp. 292–315. · Zbl 0362.65089
[37] M. Sala, J. J. Hu, and R. S. Tuminaro, ML3.1 Smoothed Aggregation User’s Guide, Tech. report SAND2004-4821, Sandia National Laboratories, Albuquerque, NM, 2004.
[38] M. Shabouei and K. Nakshatrala, Mechanics-based solution verification for porous media models, Comm. Comput. Phys., 20 (2016), pp. 1127–1162. · Zbl 1373.76315
[39] K. Shahbazi, An explicit expression for the penalty parameter of the interior penalty method, J. Comput. Phys., 205 (2005), pp. 401–407. · Zbl 1072.65149
[40] S. Williams, A. Waterman, and D. Patterson, Roofline: An insightful visual performance model for multicore architectures, Comm. ACM, 54 (2009), pp. 65–76.
[41] Zenodo/coneoproject/Coffee, coneoproject/COFFEE: A Compiler for Fast Expression Evaluation, 2017, .
[42] Zenodo/FIAT, FIAT: The Finite Element Automated Tabulator, 2018, .
[43] Zenodo/FInAT, FInAT: A Smarter Library of Finite Elements, 2017, .
[44] Zenodo/Firedrake, Firedrake: An Automated Finite Element System, 2018, .
[45] Zenodo/PETSc, PETSc: Portable, Extensible Toolkit for Scientific Computation, 2017, .
[46] Zenodo/Petsc4py, Petsc4py: The Python Interface to PETSc, 2017, .
[47] Zenodo/PyOP2, OP2/PyOP2: Framework for Performance-Portable Parallel Computations on Unstructured Meshes, 2018, .
[48] Zenodo/TSFC, TSFC: The Two Stage Form Compiler, 2018, .
[49] Zenodo/UFL, UFL: The Unified Form Language, 2018, .
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