Startseite Mathematik Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
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Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods

  • Julius Witte ORCID logo , Daniel Arndt ORCID logo und Guido Kanschat ORCID logo EMAIL logo
Veröffentlicht/Copyright: 11. November 2020
Computational Methods in Applied Mathematics
Aus der Zeitschrift Computational Methods in Applied Mathematics Band 21 Heft 3

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 đ’Ș ⁹ ( k 3 ⁹ d ) to đ’Ș ⁹ ( d ⁹ k d + 1 ) by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.

Keywords: Geometric Multigrid; Domain Decomposition; Fast Diagonalization; Discontinuous Galerkin Finite Element
MSC 2010: 65N55; 65Y20

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-WĂŒrttemberg 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-2017-0058Suche 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_8Suche 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. 10.1137/0719052Suche 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/S0036142901384162Suche in Google Scholar

[5] D. N. Arnold, R. S. Falk and R. Winther, Preconditioning in H ⁱ ( div ) and applications, Math. Comp. 66 (1997), no. 219, 957–984. 10.1090/S0025-5718-97-00826-0Suche in Google Scholar

[6] D. N. Arnold, R. S. Falk and R. Winther, Multigrid in H ⁱ ( div ) and H ⁱ ( curl ) , Numer. Math. 85 (2000), no. 2, 197–217. 10.1007/PL00005386Suche in Google Scholar

[7] P. Bastian, E. H. MĂŒller, S. MĂŒthing 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.2019.06.001Suche 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.329Suche 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/S0025-5718-1992-1122058-0Suche in Google Scholar

[10] D. BrĂ©laz, New methods to color the vertices of a graph, Comm. ACM 22 (1979), no. 4, 251–256. 10.1145/359094.359101Suche 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. Suche in Google Scholar

[12] M. Dryja and P. KrzyĆŒanowski, A massively parallel nonoverlapping additive Schwarz method for discontinuous Galerkin discretization of elliptic problems, Numer. Math. 132 (2016), no. 2, 347–367. 10.1007/s00211-015-0718-5Suche 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_7Suche in Google Scholar

[14] J. Gopalakrishnan and G. Kanschat, A multilevel discontinuous Galerkin method, Numer. Math. 95 (2003), no. 3, 527–550. 10.1007/s002110200392Suche in Google Scholar

[15] W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Springer Ser. Comput. Math. 42, Springer, Heidelberg, 2012. 10.1007/978-3-642-28027-6Suche in Google Scholar

[16] G. Kanschat, Discontinuous Galerkin finite element methods for advection-diffusion problems, Habilitationsschrift, UniversitÀt Heidelberg, 2003. Suche 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.2007.04.032Suche 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.2016.05.016Suche in Google Scholar

[19] G. Kanschat and Y. Mao, Multigrid methods for H div -conforming discontinuous Galerkin methods for the Stokes equations, J. Numer. Math. 23 (2015), no. 1, 51–66. 10.1515/jnma-2015-0005Suche in Google Scholar

[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/9780198528692.001.0001Suche 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.2012.04.012Suche 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. 10.1145/3325864Suche 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_53Suche 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. 10.1145/3322813Suche 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/s10915-004-4787-3Suche in Google Scholar

[26] J. P. Lucero Lorca, Multilevel Schwarz methods for multigroup radiation transport problems, PhD thesis, Heidelberg University, 2018. Suche 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_vol54s89Suche 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/BF01386067Suche in Google Scholar

[29] S. MĂŒthing, M. Piatkowski and P. Bastian, High-performance implementation of matrix-free high-order discontinuous Galerkin methods, preprint (2017), https://arxiv.org/abs/1711.10885. Suche 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)90005-4Suche 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.2017.10.030Suche 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.2016.09.041Suche 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-zSuche 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/b137868Suche 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.2010.38Suche 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. 10.1145/2851488Suche 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)00393-9Suche 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_17Suche in Google Scholar

[39] R. S. Varga, Matrix Iterative Analysis, Springer Ser. Comput. Math. 27, Springer, Berlin, 2000. 10.1007/978-3-642-05156-2Suche 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.2010.03.031Suche in Google Scholar
Received: 2020-05-12
Revised: 2020-10-07
Accepted: 2020-10-29
Published Online: 2020-11-11
Published in Print: 2021-07-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Sino–German Computational and Applied Mathematics
  3. An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
  4. Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
  5. Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
  6. On the Threshold Condition for Dörfler Marking
  7. An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
  8. Dimensionally Consistent Preconditioning for Saddle-Point Problems
  9. An Optimal Multilevel Method with One Smoothing Step for the Morley Element
  10. Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
  11. 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
  12. Defects in Active Nematics – Algorithms for Identification and Tracking
  13. Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
  14. Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
  15. Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
https://doi.org/10.1515/cmam-2020-0078
Schlagwörter fĂŒr diesen Artikel
Geometric Multigrid; Domain Decomposition; Fast Diagonalization; Discontinuous Galerkin Finite Element

Artikel in diesem Heft

  1. Frontmatter
  2. Sino–German Computational and Applied Mathematics
  3. An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
  4. Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
  5. Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
  6. On the Threshold Condition for Dörfler Marking
  7. An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
  8. Dimensionally Consistent Preconditioning for Saddle-Point Problems
  9. An Optimal Multilevel Method with One Smoothing Step for the Morley Element
  10. Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
  11. 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
  12. Defects in Active Nematics – Algorithms for Identification and Tracking
  13. Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
  14. Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
  15. Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
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