Skip to main content
SpringerLink
Log in

High-Performance Implementation of Discontinuous Galerkin Methods with Application in Fluid Flow

  • Chapter
  • First Online:
Efficient High-Order Discretizations for Computational Fluid Dynamics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 602))

  • 1066 Accesses

Abstract

In this book chapter, the high-performance implementation of discontinuous Galerkin methods is reviewed, with the main focus on sum factorization algorithms. The main computational properties of the algorithms are compared to capabilities of modern computer hardware, highlighting the opportunities and limitations of discontinuous Galerkin discretizations. The chapter closes with a presentation of how to apply these algorithms to the compressible Euler equations, the acoustic wave equation, and the incompressible Navier–Stokes equations.

The author acknowledges joint algorithm and code development with Niklas Fehn, Katharina Kormann, Benjamin Krank, Karl Ljungkvist, Peter Munch, and Svenja Schoeder, as well as collaboration with the deal.II community.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    As of June 2020, the fastest machine listed on the Top-500 list, https://top500.org, consists of 152,064 nodes with 48-core A64FX CPUs.

  2. 2.

    Commit 385c588, retrieved on August 21, 2020.

  3. 3.

    SuperMUC-NG supercomputer, https://doku.lrz.de/display/PUBLIC/SuperMUC-NG, retrieved on July 27, 2020.

  4. 4.

    Retrieved on July 27, 2020.

References

  • Adams, M., Brezina, M., Hu, J., & Tuminaro, R. (2003). Parallel multigrid smoothing: Polynomial versus Gauss-Seidel. Journal of Computational Physics, 188, 593–610. https://doi.org/10.1016/S0021-9991(03)00194-3.

  • Alnæs, M. S., Logg, A., Ølgaard, K. B., Rognes, M. E., & Wells, G. N. (2014). Unified form language. ACM Transactions on Mathematical Software, 40(2), 1–37. https://doi.org/10.1145/2566630.

  • Alnæs, M. S., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., et al. (2015). The FEniCS project version 1.5. Archive of Numerical Software, 3(100). https://doi.org/10.11588/ans.2015.100.20553.

  • Alzetta, G., Arndt, D., Bangerth, W., Boddu, V., Brands, B., Davydov, D., et al. (2018). The deal.II library, version 9.0. Journal of Numerical Mathematics, 26(4), 173–184. https://doi.org/10.1515/jnma-2018-0054.

  • Amdahl, G. M. (1967). Validity of the single processor approach to achieving large scale computing capabilities. In AFIPS Conference Proceedings (Vol. 30, pp. 483–485). https://doi.org/10.1145/1465482.1465560.

  • Anderson, R., Andrej, J., Barker, A., Bramwell, J., Camier, J.-S., Cerveny, J., et al. (2020). MFEM: A modular finite element methods library. Computers and Mathematics with Applications, in press. https://doi.org/10.1016/j.camwa.2020.06.009.

  • Arndt, D., Bangerth, W., Blais, B., Clevenger, T. C., Fehling, M., Grayver, A. V., et al. (2020a). The deal.II library, version 9.2. Journal of Numerical Mathematics, in press. https://doi.org/10.1515/jnma-2020-0043.

  • Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., et al. (2020b). The deal.II finite element library: Design, features, and insights. Computers and Mathematics with Applications, in press. https://doi.org/10.1016/j.camwa.2020.02.022.

  • Arndt, D., Fehn, N., Kanschat, G., Kormann, K., Kronbichler, M., Munch, P., et al. (2020c). ExaDG – high-order discontinuous Galerkin for the exa-scale. In H.-J. Bungartz, S. Reiz, B. Uekermann, P. Neumann, & W. E. Nagel (Eds.), Software for exascale computing – SPPEXA 2016–2019. Lecture notes in computational science and engineering (Vol. 136, pp. 189–224). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-47956-5_8.

  • Bangerth, W., Burstedde, C., Heister, T., & Kronbichler, M. (2011). Algorithms and data structures for massively parallel generic finite element codes. ACM Transactions on Mathematical Software, 38(2), 14:1–14:28. https://doi.org/10.1145/2049673.2049678.

  • Barra, V., Beams, N., Brown, J., Camier, J.-S., Dobrev, V., Dudouit, Y., et al. (2020). libCEED development site. https://github.com/ceed/libceed.

  • Bassi, F., Botti, L., Colombo, A., Crivellini, A., Franciolini, M., Ghidoni, A., et al. (2020). A p-adaptive matrix-free discontinuous Galerkin method for the implicit LES of incompressible transitional flows. Flow, Turbulence and Combustion, 105(2), 437–470. https://doi.org/10.1007/s10494-020-00178-2.

  • Bastian, P., Müller, E. H., Müthing, S., & Piatkowski, M. (2019). Matrix-free multigrid block-preconditioners for higher order discontinuous Galerkin discretisations. Journal of Computational Physics, 394, 417–439. https://doi.org/10.1016/j.jcp.2019.06.001.

  • Bastian, P., Blatt, M., Dedner, A., Dreier, N.-A., Engwer, C., Fritze, R., et al. (2020). The DUNE framework: Basic concepts and recent developments. Computers and Mathematics with Applications, in press. https://doi.org/10.1016/j.camwa.2020.06.007.

  • Bramble, J. H., Pasciak, J. E., & Xu, J. (1991). The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Mathematics of Computation, 56(193), 1–1. https://doi.org/10.1090/s0025-5718-1991-1052086-4.

  • Brightwell, R., Riesen, R., & Underwood, K. D. (2005). Analyzing the impact of overlap, offload, and independent progress for message passing interface applications. International Journal of High Performance Computing Applications, 19(2), 103–117. https://doi.org/10.1177/1094342005054257.

  • Brown, J. (2010). Efficient nonlinear solvers for nodal high-order finite elements in 3D. Journal of Scientific Computing, 45(1–3), 48–63.

    Article  MathSciNet  Google Scholar 

  • Buis, P. E., & Dyksen, W. R. (1996). Efficient vector and parallel manipulation of tensor products. ACM Transactions on Mathematical Software, 22(1), 18–23.

    Article  MathSciNet  Google Scholar 

  • Burstedde, C., Wilcox, L. C., & Ghattas, O. (2011). p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journal on Scientific Computing, 33(3), 1103–1133. https://doi.org/10.1137/10079163, http://p4est.org.

  • Cantwell, C. D., Sherwin, S. J., Kirby, R. M., & Kelly, P. H. J. (2011). Form h to p efficiently: Selecting the optimal spectral/\(hp\) discretisation in three dimensions. Mathematical Modelling of Natural Phenomena, 6.

    Google Scholar 

  • Cantwell, C. D., Moxey, D., Comerford, A., Bolis, A., Rocco, G., Mengaldo, G., et al. (2015). Nektar++: An open-source spectral/hp element framework. Computer Physics Communications, 192, 205–219. https://doi.org/10.1016/j.cpc.2015.02.008.

  • Chang, J., Fabien, M. S., Knepley, M. G., & Mills, R. T. (2018). Comparative study of finite element methods using the time-accuracy-size (TAS) spectrum analysis. SIAM Journal on Scientific Computing, 40(6), C779–C802. https://doi.org/10.1137/18m1172260.

  • Davis, T. A. (2004). Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Transactions on Mathematical Software, 30, 196–199. https://doi.org/10.1145/992200.992206.

  • Deville, M. O., Fischer, P. F., & Mund, E. H. (2002). High-order methods for incompressible fluid flow (Vol. 9). Cambridge: Cambridge University Press.

    Google Scholar 

  • Diosady, L. T., & Murman, S. M. (2019). Scalable tensor-product preconditioners for high-order finite-element methods: Scalar equations. Journal of Computational Physics, 394, 759–776. https://doi.org/10.1016/j.jcp.2019.04.047.

  • Elman, H., Silvester, D., & Wathen, A. (2005). Finite elements and fast iterative solvers with applications in incompressible fluid dynamics. Oxford: Oxford Science Publications.

    Google Scholar 

  • Fehn, N., Wall, W. A., & Kronbichler, M. (2017). On the stability of projection methods for the incompressible Navier–Stokes equations based on high-order discontinuous Galerkin discretizations. Journal of Computational Physics, 351, 392–421. https://doi.org/10.1016/j.jcp.2017.09.031.

  • Fehn, N., Wall, W. A., & Kronbichler, M. (2018a). Robust and efficient discontinuous Galerkin methods for under-resolved turbulent incompressible flows. Journal of Computational Physics, 372, 667–693. https://doi.org/10.1016/j.jcp.2018.06.037.

  • Fehn, N., Wall, W. A., & Kronbichler, M. (2018b). Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 88(1), 32–54. https://doi.org/10.1002/fld.4511.

  • Fehn, N., Kronbichler, M., Lehrenfeld, C., Lube, G., & Schroeder, P. W. (2019a). High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of \(L^{2}\) and \(H\)(div) methods. International Journal for Numerical Methods in Fluids, 91(11), 533–556. https://doi.org/10.1002/fld.4763.

  • Fehn, N., Wall, W. A., & Kronbichler, M. (2019b). A matrix-free high-order discontinuous Galerkin compressible Navier–Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows. International Journal for Numerical Methods in Fluids, 89(3), 71–102. https://doi.org/10.1002/fld.4683.

  • Fehn, N., Heinz, J., Wall, W. A., & Kronbichler, M. (2020a). High-order arbitrary Lagrangian-Eulerian discontinuous Galerkin methods for the incompressible Navier–Stokes equations. Technical report. arXiv:2003.07166.

  • Fehn, N., Munch, P., Wall, W. A., & Kronbichler, M. (2020b). Hybrid multigrid methods for high-order discontinuous Galerkin discretizations. Journal of Computational Physics, 415, 109538. https://doi.org/10.1016/j.jcp.2020.109538.

  • Fischer, P., Min, M., Rathnayake, T., Dutta, S., Kolev, T., Dobrev, V., et al. (2020). Scalability of high-performance PDE solvers. International Journal of High Performance Computing Applications, 34(5), 562–586. https://doi.org/10.1177/1094342020915762.

  • Fischer, P. F. (1997). An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. Journal of Computational Physics, 133(1), 84–101. https://doi.org/10.1006/jcph.1997.5651.

  • Fischer, P. F., & Patera, A. T. (1991). Parallel spectral element solution of the Stokes problem. Journal of Computational Physics, 92(2), 380–421. https://doi.org/10.1016/0021-9991(91)90216-8.

  • Fischer, P. F., Kerkemeier, S., et al. (2020). Nek5000 Web page. https://nek5000.mcs.anl.gov.

  • Franco, M., Camier, J.-S., Andrej, J., & Pazner, W. (2020). High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners. Computers and Fluids, 203, 104541. https://doi.org/10.1016/j.compfluid.2020.104541.

  • Gholami, A., Malhotra, D., Sundar, H., & Biros, G. (2016). FFT, FMM, or multigrid? A comparative study of state-of-the-art Poisson solvers for uniform and nonuniform grids in the unit cube. SIAM Journal on Scientific Computing, 38(3), C280–C306. https://doi.org/10.1137/15M1010798.

  • Gmeiner, B., Rüde, U., Stengel, H., Waluga, C., & Wohlmuth, B. (2015). Towards textbook efficiency for parallel multigrid. Numerical Mathematics-Theory, Methods and Applications, 8(1), 22–46.

    Google Scholar 

  • Göddeke, D., Strzodka, R., & Turek, S. (2007). Performance and accuracy of hardware-oriented native-, emulated-and mixed-precision solvers in FEM simulations. International Journal of Parallel, Emergent and Distributed Systems, 22(4), 221–256. https://doi.org/10.1080/17445760601122076.

  • Grote, M. J., & Huckle, T. (1997). Parallel preconditioning with sparse approximate inverses. SIAM Journal on Scientific Computing, 18(3), 838–853. https://doi.org/10.1137/s1064827594276552.

  • Guermond, J.-L., & Minev, P. (2019). High-order adaptive time stepping for the incompressible Navier–Stokes equations. SIAM Journal on Scientific Computing, 41(2), A770–A788. https://doi.org/10.1137/18m1209301.

  • Guermond, J. L., Minev, P., & Shen, J. (2006). An overview of projection methods for incompressible flows. Computer Methods in Applied Mechanics and Engineering, 195(44–47), 6011–6045. https://doi.org/10.1016/j.cma.2005.10.010.

  • Gustafson, J. L. (1988). Reevaluating Amdahl’s law. Communications of the ACM, 31(5), 532–533. https://doi.org/10.1145/42411.42415.

  • Hager, G., & Wellein, G. (2011). Introduction to high performance computing for scientists and engineers. Boca Raton: CRC Press.

    Google Scholar 

  • Hager, G., Treibig, J., Habich, J., & Wellein, G. (2016). Exploring performance and power properties of modern multi-core chips via simple machine models. Concurrency and Computation, 28(2), 189–210. https://doi.org/10.1002/cpe.3180.

  • Hesthaven, J. S., & Warburton, T. (2008). Nodal discontinuous Galerkin methods: Algorithms, analysis, and applications. Berlin: Springer. https://doi.org/10.1007/978-0-387-72067-8.

  • Hindenlang, F., Gassner, G., Altmann, C., Beck, A., Staudenmaier, M., & Munz, C.-D. (2012). Explicit discontinuous Galerkin methods for unsteady problems. Computers and Fluids, 61, 86–93.

    Article  MathSciNet  Google Scholar 

  • Hoefler, T., & Belli, R. (2015). Scientific benchmarking of parallel computing systems. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC’15. ACM Press. https://doi.org/10.1145/2807591.2807644.

  • Huismann, I., Stiller, J., & Fröhlich, J. (2017). Factorizing the factorization - a spectral-element solver for elliptic equations with linear operation count. Journal of Computational Physics, 346, 437–448. https://doi.org/10.1016/j.jcp.2017.06.012.

  • Huismann, I., Stiller, J., & Fröhlich, J. (2019). Scaling to the stars – a linearly scaling elliptic solver for p-multigrid. Journal of Computational Physics, 398, 108868. https://doi.org/10.1016/j.jcp.2019.108868.

  • Huismann, I., Stiller, J., & Fröhlich, J. (2020). Efficient high-order spectral element discretizations for building block operators of CFD. Computers and Fluids, 197, 104386. https://doi.org/10.1016/j.compfluid.2019.104386.

  • Ibeid, H., Olson, L., & Gropp, W. (2019). FFT, FMM, and multigrid on the road to exascale: Performance challenges and opportunities. Journal of Parallel and Distributed Computing, 136, 63–74. https://doi.org/10.1016/j.jpdc.2019.09.014.

  • Karakus, A., Chalmers, N., Świrydowicz, K., & Warburton, T. (2019). A GPU accelerated discontinuous Galerkin incompressible flow solver. Journal of Computational Physics, 390, 380–404. https://doi.org/10.1016/j.jcp.2019.04.010.

  • Karniadakis, G., & Sherwin, S. J. (2005). Spectral/hp element methods for computational fluid dynamics (2nd ed.). Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198528692.001.0001.

  • Karniadakis, G. E., Israeli, M., & Orszag, S. A. (1991). High-order splitting methods for the incompressible Navier–Stokes equations. Journal of Computational Physics, 97(2), 414–443. https://doi.org/10.1016/0021-9991(91)90007-8.

  • Karypis, G., & Kumar, V. (1998). A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1), 359–392. https://doi.org/10.1137/S1064827595287997.

  • Kempf, D., Hess, R., Müthing, S., & Bastian, P. (2018). Automatic code generation for high-performance discontinuous Galerkin methods on modern architectures. Technical report. arXiv:1812.08075.

  • Kennedy, C. A., Carpenter, M. H., & Lewis, R. M. (2000). Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Applied Numerical Mathematics, 35(3), 177–219. https://doi.org/10.1016/s0168-9274(99)00141-5.

  • Keyes, D. E., McInnes, L. C., Woodward, C., Gropp, W., Myra, E., Pernice, M., et al. (2013). Multiphysics simulations: Challenges and opportunities. International Journal of High Performance Computing Applications, 27(1), 4–83. https://doi.org/10.1177/1094342012468181.

  • Klöckner, A. (2014). Loo.py: Transformation-based code generation for GPUs and CPUs. In Proceedings of ARRAY ‘14: ACM SIGPLAN Workshop on Libraries, Languages, and Compilers for Array Programming, Edinburgh, Scotland, 2014. Association for Computing Machinery. https://doi.org/10.1145/2627373.2627387.

  • Knepley, M. G., Brown, J., Rupp, K., Smith, B. F. (2013). Achieving high performance with unified residual evaluation. Technical report. arXiv:1309.1204.

  • Knoll, D. A., & Keyes, D. E. (2004). Jacobian-free Newton–Krylov methods: A survey of approaches and applications. Journal of Computational Physics, 193(2), 357–397. https://doi.org/10.1016/j.jcp.2003.08.010.

  • Komatitsch, D., et al. (2015). SPECFEM 3D cartesian user manual. Technical report, Computational Infrastructure for Geodynamics, Princeton University, CNRS and University of Marseille, and ETH Zürich.

    Google Scholar 

  • Kopriva, D. A. (2006). Metric identities and the discontinuous spectral element method on curvilinear meshes. Journal of Scientific Computing, 26(3), 301–327. https://doi.org/10.1007/s10915-005-9070-8.

  • Kopriva, D. A. (2009). Implementing spectral methods for partial differential equations. Berlin: Springer.

    Google Scholar 

  • Kopriva, D. A., & Gassner, G. J. (2016). Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable. Applied Mathematics and Computation, 272, 274–290. https://doi.org/10.1016/j.amc.2015.08.047.

  • Krais, N., Beck, A., Bolemann, T., Frank, H., Flad, D., Gassner, G., et al. (2020). FLEXI: A high order discontinuous Galerkin framework for hyperbolic–parabolic conservation laws. Computers and Mathematics with Applications. https://doi.org/10.1016/j.camwa.2020.05.004.

  • Krank, B., Fehn, N., Wall, W. A., & Kronbichler, M. (2017). A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow. Journal of Computational Physics, 348, 634–659. https://doi.org/10.1016/j.jcp.2017.07.039.

  • Kronbichler, M., & Allalen, M. (2018). Efficient high-order discontinuous Galerkin finite elements with matrix-free implementations. In H.-J. Bungartz, D. Kranzlmüller, V. Weinberg, J. Weismüller, & V. Wohlgemuth (Eds.), Advances and new trends in environmental informatics (pp. 89–110). Berlin: Springer. https://doi.org/10.1007/978-3-319-99654-7_7.

  • Kronbichler, M., & Kormann, K. (2012). A generic interface for parallel cell-based finite element operator application. Computers and Fluids, 63, 135–147.

    Article  MathSciNet  Google Scholar 

  • Kronbichler, M., & Kormann, K. (2019). Fast matrix-free evaluation of discontinuous Galerkin finite element operators. ACM Transactions on Mathematical Software, 45(3), 29:1–29:40. https://doi.org/10.1145/3325864.

  • Kronbichler, M., & Ljungkvist, K. (2019). Multigrid for matrix-free high-order finite element computations on graphics processors. ACM Transactions on Parallel Computing, 6(1), 2:1–2:32. https://doi.org/10.1145/3322813.

  • Kronbichler, M., & Wall, W. A. (2018). A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers. SIAM Journal on Scientific Computing, 40(5), A3423–A3448. https://doi.org/10.1137/16M110455X.

  • Kronbichler, M., Schoeder, S., Müller, C., & Wall, W. A. (2016). Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. International Journal for Numerical Methods in Engineering, 106(9), 712–739. https://doi.org/10.1002/nme.5137.

  • Kronbichler, M., Kormann, K., Pasichnyk, I., & Allalen, M. (2017). Fast matrix-free discontinuous Galerkin kernels on modern computer architectures. In J. M. Kunkel, R. Yokota, P. Balaji, & D. E. Keyes (Eds.), ISC high performance 2017. LNCS (Vol. 10266, pp. 237–255). https://doi.org/10.1007/978-3-319-58667-0_13.

  • Kronbichler, M., Diagne, A., & Holmgren, H. (2018). A fast massively parallel two-phase flow solver for microfluidic chip simulation. International Journal of High Performance Computing Applications, 32(2), 266–287.

    Article  Google Scholar 

  • Kronbichler, M., Kormann, K., Fehn, N., Munch, P., Witte, J. (2019). A Hermite-like basis for faster matrix-free evaluation of interior penalty discontinuous Galerkin operators. Technical report. arXiv:1907.08492.

  • LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge.

    Google Scholar 

  • Loppi, N. A., Witherden, F. D., Jameson, A., & Vincent, P. E. (2018). A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet. Computer Physics Communications, 233, 193–205. https://doi.org/10.1016/j.cpc.2018.06.016.

  • Lottes, J. W., & Fischer, P. F. (2005). Hybrid multigrid-Schwarz algorithms for the spectral element method. Journal of Scientific Computing, 24, 613–646. https://doi.org/10.1007/s10915-004-4787-3.

  • Lynch, R. E., Rice, J. R., & Thomas, D. H. (1964). Direct solution of partial difference equations by tensor product methods. Numerische Mathematik, 6, 185–199. https://doi.org/10.1007/BF01386067.

  • Maday, Y., Patera, A. T., & Rønquist, E. M. (1990). An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow. Journal of Scientific Computing, 5(4), 263–292. https://doi.org/10.1007/bf01063118.

  • Manzanero, J., Rubio, G., Kopriva, D. A., Ferrer, E., & Valero, E. (2020). An entropy–stable discontinuous Galerkin approximation for the incompressible Navier–Stokes equations with variable density and artificial compressibility. Journal of Computational Physics, 408, 109241. https://doi.org/10.1016/j.jcp.2020.109241.

  • May, D. A., Brown, J., & Le Pourhiet, L. (2014). pTatin3D: High-performance methods for long-term lithospheric dynamics. In J. M. Kunkel, T. Ludwig, & H. W. Meuer (Eds.), Supercomputing (SC14), New Orleans (pp. 1–11). https://doi.org/10.1109/SC.2014.28.

  • May, D. A., Brown, J., & Le Pourhiet, L. (2015). A scalable, matrix-free multigrid preconditioner for finite element discretizations of heterogeneous Stokes flow. Computer Methods in Applied Mechanics and Engineering, 290, 496–523.

    Article  MathSciNet  Google Scholar 

  • Moxey, D., Amici, R., & Kirby, M. (2020a). Efficient matrix-free high-order finite element evaluation for simplicial elements. SIAM Journal on Scientific Computing, 42(3), C97–C123. https://doi.org/10.1137/19m1246523.

  • Moxey, D., Cantwell, C. D., Bao, Y., Cassinelli, A., Castiglioni, G., Chun, S., et al. (2020b). Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods. Computer Physics Communications, 249, 107110. https://doi.org/10.1016/j.cpc.2019.107110.

  • Müthing, S., Piatkowski, M., & Bastian, P. (2017). High-performance implementation of matrix-free high-order discontinuous Galerkin methods. Technical report. arXiv:1711.10885.

  • Nguyen, N. C., Peraire, J., & Cockburn, B. (2011). High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. Journal of Computational Physics, 230, 3695–3718. https://doi.org/10.1016/j.jcp.2011.01.035.

  • Noventa, G., Massa, F., Bassi, F., Colombo, A., Franchina, N., & Ghidoni, A. (2016). A high-order discontinuous Galerkin solver for unsteady incompressible turbulent flows. Computers and Fluids, 139, 248–260. https://doi.org/10.1016/j.compfluid.2016.03.007.

  • Olson, L. (2007). Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM Journal on Scientific Computing, 29(5), 2189–2209. https://doi.org/10.1137/060663465.

  • Oo, K. L., & Vogel, A. (2020). Accelerating geometric multigrid preconditioning with half-precision arithmetic on GPUs. Technical report. arxiv:2007.07539.

  • Orszag, S. A. (1980). Spectral methods for problems in complex geometries. Journal of Computational Physics, 37, 70–92.

    Article  MathSciNet  Google Scholar 

  • Patera, A. T. (1984). A spectral element method for fluid dynamics: Laminar flow in a channel expansion. Journal of Computational Physics, 54(3), 468–488. https://doi.org/10.1016/0021-9991(84)90128-1.

  • Patterson, D. A., & Hennessy, J. L. (2013). Computer organization and design: The hardware/software interface (5th ed.). Burlington: Morgan Kaufmann.

    MATH  Google Scholar 

  • Pazner, W. (2019). Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods. Technical report. arXiv:1908.07071.

  • Pazner, W., & Persson, P.-O. (2018). Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods. Journal of Computational Physics, 354, 344–369. https://doi.org/10.1016/j.jcp.2017.10.030.

  • Persson, P. O. (2013). A sparse and high-order accurate line-based discontinuous Galerkin method for unstructured meshes. Journal of Computational Physics, 233, 414–429. https://doi.org/10.1016/j.jcp.2012.09.008.

  • Raffenetti, K., Amer, A., Oden, L., Archer, C., Bland, W., Fujita, H., et al. (2017). Why is MPI so slow?: Analyzing the fundamental limits in implementing MPI-3.1. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC’17, New York, NY, USA, 2017 (pp. 62:1–62:12). ACM. https://doi.org/10.1145/3126908.3126963. ISBN 978-1-4503-5114-0.

  • Rathgeber, F., Ham, D. A., Mitchell, L., Lange, M., Luporini, F., McRae, A. T. T., et al. (2017). Firedrake: Automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software, 43(3), 24:1–24:27. https://doi.org/10.1145/2998441.

  • Remacle, J.-F., Gandham, R., & Warburton, T. (2016). GPU accelerated spectral finite elements on all-hex meshes. Journal of Computational Physics, 324, 246–257. https://doi.org/10.1016/j.jcp.2016.08.005.

  • Ruge, J. W.,& Stüben, K. (1987). Algebraic multigrid (AMG). In Multigrid methods (pp. 73–130). Philadelphia: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611971057.ch4.

  • Saad, Y. (2003). Iterative methods for sparse linear systems (2nd ed.). Philadelphia: SIAM.

    Book  Google Scholar 

  • Schenk, O., & Gärtner, K. (2004). Solving unsymmetric sparse systems of linear equations with PARDISO. Future Generation Computer Systems, 20(3), 475–487. https://doi.org/10.1016/j.future.2003.07.011, https://www.pardiso-project.org/.

  • Schöberl, J. (2014). C++11 implementation of finite elements in NGSolve. Technical report ASC Report No. 30/2014, Vienna University of Technology.

    Google Scholar 

  • Schoeder, S., Kormann, K., Wall, W. A., & Kronbichler, M. (2018a). Efficient explicit time stepping of high order discontinuous Galerkin schemes for waves. SIAM Journal on Scientific Computing, 40(6), C803–C826. https://doi.org/10.1137/18M1185399.

  • Schoeder, S., Kronbichler, M., & Wall, W. A. (2018b). Arbitrary high-order explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation. Journal of Scientific Computing, 76, 969–1006. https://doi.org/10.1007/s10915-018-0649-2.

  • Schoeder, S., Wall, W. A., & Kronbichler, M. (2019). ExWave: A high performance discontinuous Galerkin solver for the acoustic wave equation. SoftwareX, 9, 49–54. https://doi.org/10.1016/j.softx.2019.01.001.

  • Solomonoff, A. (1992). A fast algorithm for spectral differentiation. Journal of Computational Physics, 98(1), 174–177. https://doi.org/10.1016/0021-9991(92)90182-X.

  • Stanglmeier, M., Nguyen, N. C., Peraire, J., & Cockburn, B. (2016). An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation. Computer Methods in Applied Mechanics and Engineering, 300, 748–769. https://doi.org/10.1016/j.cma.2015.12.003.

  • Sun, T.,  Mitchell, L.,  Kulkarni, K.,  Klöckner, A., Ham, D. A., & Kelly, P. H. J. (2020). A study of vectorization for matrix-free finite element methods. International Journal of High Performance Computing Applications, page in press. https://doi.org/10.1177/1094342020945005.

  • Świrydowicz, K., Chalmers, N., Karakus, A., & Warburton, T. (2019). Acceleration of tensor-product operations for high-order finite element methods. International Journal of High Performance Computing Applications, 33(4), 735–757. https://doi.org/10.1177/1094342018816368.

  • Thomas, J. L., Diskin, B., & Brandt, A. (2003). Textbook multigrid efficiency for fluid simulations. Annual Reviews of Fluid Mechanics, 35, 317–340. https://doi.org/10.1146/annurev.fluid.35.101101.161209.

  • Treibig, J., & Hager, G. (2010). Introducing a performance model for bandwidth-limited loop kernels. In R. Wyrzykowski, J. Dongarra, K. Karczewski, & J. Wasniewski (Eds.), Parallel Processing and Applied Mathematics: 8th International Conference, PPAM 2009, Wroclaw, Poland, 13–16 September 2009. Revised Selected Papers, Part I (pp. 615–624). Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-14390-8_64.

  • Treibig, J.,  Hager, G.,& Wellein, G. (2020). LIKWID: A lightweight performance-oriented tool suite for x86 multicore environments. In Proceedings of PSTI2010, the First International Workshop on Parallel Software Tools and Tool Infrastructures, San Diego, CA. https://doi.org/10.1109/ICPPW.2010.38, https://github.com/RRZE-HPC/likwid. Retrieved 27 July 2020.

  • Trottenberg, U., Oosterlee, C., & Schüller, A. (2001). Multigrid. London: Elsevier/Academic.

    Google Scholar 

  • Vaněk, P., Mandel, J., & Brezina, M. (1996). Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing, 56(3), 179–196. https://doi.org/10.1007/bf02238511.

  • Wang, Z. J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., et al. (2013). High-order CFD methods: Current status and perspective. International Journal for Numerical Methods in Fluids, 72(8), 811–845. https://doi.org/10.1002/fld.3767.

  • Weinzierl, T. (2019). The Peano software—parallel, automaton-based, dynamically adaptive grid traversals. ACM Transactions on Mathematical Software, 45(2), 14:1–14:41. https://doi.org/10.1145/3319797.

  • Williams, S., Waterman, A., & Patterson, D. (2009). Roofline: An insightful visual performance model for multicore architectures. Communications of the ACM, 52(4), 65–76. https://doi.org/10.1145/1498765.1498785.

  • Winters, A. R., Moura, R. C., Mengaldo, G., Gassner, G. J., Walch, S., Peiró, J., et al. (2018). A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. Journal of Computational Physics, 372, 1–21. https://doi.org/10.1016/j.jcp.2018.06.016.

  • Witte, J.,  Arndt, D., & Kanschat, G. (2019). Fast tensor product Schwarz smoothers for high-order discontinuous Galerkin methods. Technical report. arXiv:1910.11239.

  • Yan, Z.-G.,  Pan, Y.,  Castiglioni, G.,  Hillewaert, K.,  Peiró, J.,  Moxey, D., et al. (2020). Nektar++: Design and implementation of an implicit, spectral/hp element, compressible flow solver using a Jacobian-free Newton Krylov approach. Computers and Mathematics with Applications, in press. https://doi.org/10.1016/j.camwa.2020.03.009.

Download references

Author information

Authors and Affiliations

  1. Institute for Computational Mechanics, Technical University of Munich, Boltzmannstr. 15, 85748, Garching bei München, Germany

    Martin Kronbichler

Authors
  1. Martin Kronbichler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Kronbichler .

Editor information

Editors and Affiliations

  1. Institute for Computational Mechanics, Technical University Munich, Garching bei München, Germany

    Martin Kronbichler

  2. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    Per-Olof Persson

Rights and permissions

Reprints and permissions

Copyright information

© 2021 CISM International Centre for Mechanical Sciences, Udine

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kronbichler, M. (2021). High-Performance Implementation of Discontinuous Galerkin Methods with Application in Fluid Flow. In: Kronbichler, M., Persson, PO. (eds) Efficient High-Order Discretizations for Computational Fluid Dynamics. CISM International Centre for Mechanical Sciences, vol 602. Springer, Cham. https://doi.org/10.1007/978-3-030-60610-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-60610-7_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-60609-1

  • Online ISBN: 978-3-030-60610-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation