Abstract
We discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from
Funding statement: The authors were supported by the German Research Foundation (DFG) under the project “High-order discontinuous Galerkin for the exa-scale” (ExaDG) within the priority program “Software for Exascale Computing” (SPPEXA) and acknowledge support by the state of Baden-Württemberg through bwHPC.
Acknowledgements
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The implementation is based on the deal.II library [1].
References
[1] D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin and D. Wells, The deal.II library, version 8.5, J. Numer. Math. 25 (2017), no. 3, 137–145. 10.1515/jnma-2017-0058Search in Google Scholar
[2] D. Arndt, N. Fehn, G. Kanschat, K. Kormann, M. Kronbichler, P. Munch, W. A. Wall and J. Witte, ExaDG – High-order discontinuous Galerkin for the exa-scale, Software for Exascale Computing—SPPEXA 2016–2019, Springer, Cham (2020), 189–224. 10.1007/978-3-030-47956-5_8Search in Google Scholar
[3] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. 10.1137/0719052Search in Google Scholar
[4] 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 (2001/02), no. 5, 1749–1779. 10.1137/S0036142901384162Search in Google Scholar
[5]
D. N. Arnold, R. S. Falk and R. Winther,
Preconditioning in
[6]
D. N. Arnold, R. S. Falk and R. Winther,
Multigrid in
[7] P. Bastian, E. H. Müller, S. Müthing and M. Piatkowski, Matrix-free multigrid block-preconditioners for higher order discontinuous Galerkin discretisations, J. Comput. Phys. 394 (2019), 417–439. 10.1016/j.jcp.2019.06.001Search in Google Scholar
[8] S. Beuchler, AMLI preconditioner for the p-version of the FEM, Numer. Linear Algebra Appl. 10 (2003), no. 8, 721–732. 10.1002/nla.329Search in Google Scholar
[9] J. H. Bramble and J. E. Pasciak, The analysis of smoothers for multigrid algorithms, Math. Comp. 58 (1992), no. 198, 467–488. 10.1090/S0025-5718-1992-1122058-0Search in Google Scholar
[10] D. Brélaz, New methods to color the vertices of a graph, Comm. ACM 22 (1979), no. 4, 251–256. 10.1145/359094.359101Search in Google Scholar
[11] W. Couzy, Spectral element discretization of the unsteady Navier–Stokes equations and its iterative solution on parallel computers, PhD thesis, EPFL, 1995. Search in Google Scholar
[12] M. Dryja and P. Krzyżanowski, A massively parallel nonoverlapping additive Schwarz method for discontinuous Galerkin discretization of elliptic problems, Numer. Math. 132 (2016), no. 2, 347–367. 10.1007/s00211-015-0718-5Search in Google Scholar PubMed PubMed Central
[13] P. F. Fischer, N. I. Miller and H. M. Tufo, An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, Parallel Solution of Partial Differential Equations (Minneapolis 1997), IMA Vol. Math. Appl. 120, Springer, New York (2000), 159–180. 10.1007/978-1-4612-1176-1_7Search in Google Scholar
[14] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer. Math. 95 (2003), no. 3, 527–550. 10.1007/s002110200392Search in Google Scholar
[15] W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Springer Ser. Comput. Math. 42, Springer, Heidelberg, 2012. 10.1007/978-3-642-28027-6Search in Google Scholar
[16] G. Kanschat, Discontinuous Galerkin finite element methods for advection-diffusion problems, Habilitationsschrift, Universität Heidelberg, 2003. Search in Google Scholar
[17] G. Kanschat, Robust smoothers for high-order discontinuous Galerkin discretizations of advection-diffusion problems, J. Comput. Appl. Math. 218 (2008), no. 1, 53–60. 10.1016/j.cam.2007.04.032Search in Google Scholar
[18] G. Kanschat, R. Lazarov and Y. Mao, Geometric multigrid for Darcy and Brinkman models of flows in highly heterogeneous porous media: A numerical study, J. Comput. Appl. Math. 310 (2017), 174–185. 10.1016/j.cam.2016.05.016Search in Google Scholar
[19]
G. Kanschat and Y. Mao,
Multigrid methods for
[20] G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd ed., Numer. Math. Sci. Comput., Oxford University, New York, 2005. 10.1093/acprof:oso/9780198528692.001.0001Search in Google Scholar
[21] M. Kronbichler and K. Kormann, A generic interface for parallel cell-based finite element operator application, Comput. & Fluids 63 (2012), 135–147. 10.1016/j.compfluid.2012.04.012Search in Google Scholar
[22] M. Kronbichler and K. Kormann, Fast matrix-free evaluation of discontinuous Galerkin finite element operators, ACM Trans. Math. Software 45 (2019), no. 3, Article ID 29. 10.1145/3325864Search in Google Scholar
[23] M. Kronbichler, K. Kormann, N. Fehn, P. Munch and J. Witte, A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators, preprint (2019), https://arxiv.org/abs/1907.08492. 10.1007/978-3-319-96415-7_53Search in Google Scholar
[24] M. Kronbichler and K. Ljungkvist, Multigrid for matrix-free high-order finite element computations on graphics processors, ACM Trans. Parallel Comput. 6 (2019), 10.1145/3322813. 10.1145/3322813Search in Google Scholar
[25] J. W. Lottes and P. F. Fischer, Hybrid multigrid/Schwarz algorithms for the spectral element method, J. Sci. Comput. 24 (2005), no. 1, 45–78. 10.1007/s10915-004-4787-3Search in Google Scholar
[26] J. P. Lucero Lorca, Multilevel Schwarz methods for multigroup radiation transport problems, PhD thesis, Heidelberg University, 2018. Search in Google Scholar
[27] J. P. Lucero Lorca and G. Kanschat, Multilevel schwarz preconditioners for singularly perturbed symmetric reaction-diffusion systems, preprint (2018), https://arxiv.org/abs/1811.03839v2. 10.1553/etna_vol54s89Search in Google Scholar
[28] R. E. Lynch, J. R. Rice and D. H. Thomas, Direct solution of partial difference equations by tensor product methods, Numer. Math. 6 (1964), 185–199. 10.1007/BF01386067Search in Google Scholar
[29] S. Müthing, M. Piatkowski and P. Bastian, High-performance implementation of matrix-free high-order discontinuous Galerkin methods, preprint (2017), https://arxiv.org/abs/1711.10885. Search in Google Scholar
[30] S. A. Orszag, Spectral methods for problems in complex geometries, J. Comput. Phys. 37 (1980), no. 1, 70–92. 10.1016/0021-9991(80)90005-4Search in Google Scholar
[31] W. Pazner and P.-O. Persson, Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods, J. Comput. Phys. 354 (2018), 344–369. 10.1016/j.jcp.2017.10.030Search in Google Scholar
[32] J. Stiller, Robust multigrid for high-order discontinuous Galerkin methods: a fast Poisson solver suitable for high-aspect ratio Cartesian grids, J. Comput. Phys. 327 (2016), 317–336. 10.1016/j.jcp.2016.09.041Search in Google Scholar
[33] J. Stiller, Nonuniformly weighted Schwarz smoothers for spectral element multigrid, J. Sci. Comput. 72 (2017), no. 1, 81–96. 10.1007/s10915-016-0345-zSearch in Google Scholar
[34] A. Toselli and O. Widlund, Domain Decomposition Methods—Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer, Berlin, 2005. 10.1007/b137868Search in Google Scholar
[35] J. Treibig, G. Hager and G. Wellein, Likwid: A lightweight performance-oriented tool suite for x86 multicore environments, International Conference on Parallel Processing Workshops, IEEE Press, Piscataway (2010), 207–2016. 10.1109/ICPPW.2010.38Search in Google Scholar
[36] B. Turcksin, M. Kronbichler and W. Bangerth, WorkStream—A design pattern for multicore-enabled finite element computations, ACM Trans. Math. Software 43 (2016), no. 1, Article ID 2. 10.1145/2851488Search in Google Scholar
[37] C. F. Van Loan, The ubiquitous Kronecker product, J. Comput. Appl. Math. 123 (2000), no. 1–2, 85–100. 10.1016/S0377-0427(00)00393-9Search in Google Scholar
[38] C. F. Van Loan and N. Pitsianis, Approximation with Kronecker products, Linear Algebra for Large Scale and Real-Time Applications (Leuven 1992), NATO Adv. Sci. Inst. Ser. E Appl. Sci. 232, Kluwer Academic, Dordrecht (1993), 293–314. 10.1007/978-94-015-8196-7_17Search in Google Scholar
[39] R. S. Varga, Matrix Iterative Analysis, Springer Ser. Comput. Math. 27, Springer, Berlin, 2000. 10.1007/978-3-642-05156-2Search in Google Scholar
[40] P. E. J. Vos, S. J. Sherwin and R. M. Kirby, From h to p efficiently: implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations, J. Comput. Phys. 229 (2010), no. 13, 5161–5181. 10.1016/j.jcp.2010.03.031Search in Google Scholar
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