Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in $\boldsymbol{H}(\operatorname{\textbf{div}})$

Pazner, Will; Kolev, Tzanio; Vassilevski, Panayot S.

doi:10.1137/23m1568806

Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in $\boldsymbol{H}(\operatorname{\textbf{div}})$

Journal Article · Thu May 02 00:00:00 EDT 2024 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/23m1568806· OSTI ID:2440339

^[1]; ^[2]; Vassilevski, Panayot S. ^[1]

Portland State Univ., OR (United States)
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Here, this work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in H(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation–histopolation basis, efficient matrix-free preconditioners can be constructed for the (1, 1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the “crooked pipe” grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 2440339

Report Number(s):: LLNL--JRNL-848035; 1073091

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 3 Vol. 46; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (42)

Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows: High-performance discontinuous Galerkin for turbulent flows Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin International Journal for Numerical Methods in Fluids, Vol. 88, Issue 1 https://doi.org/10.1002/fld.4511	journal	May 2018
Preconditioning discretizations of systems of partial differential equations Mardal, Kent-Andre; Winther, Ragnar Numerical Linear Algebra with Applications, Vol. 18, Issue 1 https://doi.org/10.1002/nla.716	journal	April 2010
Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation: IMPLICIT VERSUS EXPLICIT HDG Kronbichler, M.; Schoeder, S.; Müller, C. International Journal for Numerical Methods in Engineering, Vol. 106, Issue 9 https://doi.org/10.1002/nme.5137	journal	October 2015
Performance Characteristics of HYDRA – A Multi-physics Simulation Code from LLNL Langer, Steven H.; Karlin, Ian; Marinak, Michael M. Lecture Notes in Computer Science https://doi.org/10.1007/978-3-319-17353-5_15	book	January 2015
Performance, Design, and Autotuning of Batched GEMM for GPUs Abdelfattah, Ahmad; Haidar, Azzam; Tomov, Stanimire Lecture Notes in Computer Science, p. 21-38 https://doi.org/10.1007/978-3-319-41321-1_2	book	June 2016
Mixed Finite Element Methods and Applications Boffi, Daniele; Brezzi, Franco; Fortin, Michel Springer Series in Computational Mathematics https://doi.org/10.1007/978-3-642-36519-5	book	January 2013
Spectral element multigrid. I. Formulation and numerical results R�nquist, Einar M.; Patera, Anthony T. Journal of Scientific Computing, Vol. 2, Issue 4 https://doi.org/10.1007/BF01061297	journal	December 1987
Multigrid in H (div) and H (curl) Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar Numerische Mathematik, Vol. 85, Issue 2 https://doi.org/10.1007/PL00005386	journal	April 2000
On the accuracy of high-order discretizations for underresolved turbulence simulations Gassner, Gregor J.; Beck, Andrea D. Theoretical and Computational Fluid Dynamics, Vol. 27, Issue 3-4 https://doi.org/10.1007/s00162-011-0253-7	journal	January 2012
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method Lottes, James W.; Fischer, Paul F. Journal of Scientific Computing, Vol. 24, Issue 1 https://doi.org/10.1007/s10915-004-4787-3	journal	July 2005
Spectral methods for problems in complex geometries Orszag, Steven A. Journal of Computational Physics, Vol. 37, Issue 1 https://doi.org/10.1016/0021-9991(80)90005-4	journal	August 1980
Differencing of the diffusion equation in Lagrangian hydrodynamic codes Kershaw, David S. Journal of Computational Physics, Vol. 39, Issue 2 https://doi.org/10.1016/0021-9991(81)90158-3	journal	February 1981
BoomerAMG: A parallel algebraic multigrid solver and preconditioner Henson, Van Emden; Yang, Ulrike Meier Applied Numerical Mathematics, Vol. 41, Issue 1 https://doi.org/10.1016/S0168-9274(01)00115-5	journal	April 2002
On condition numbers in hp-FEM with Gauss–Lobatto-based shape functions Melenk, J. M. Journal of Computational and Applied Mathematics, Vol. 139, Issue 1 https://doi.org/10.1016/S0377-0427(01)00391-0	journal	February 2002
Geometric Multigrid for Darcy and Brinkman models of flows in highly heterogeneous porous media: A numerical study Kanschat, Guido; Lazarov, Raytcho; Mao, Youli Journal of Computational and Applied Mathematics, Vol. 310 https://doi.org/10.1016/j.cam.2016.05.016	journal	January 2017
MFEM: A modular finite element methods library Anderson, Robert; Andrej, Julian; Barker, Andrew Computers & Mathematics with Applications, Vol. 81 https://doi.org/10.1016/j.camwa.2020.06.009	journal	January 2021
Short note on the mass matrix for Gauss–Lobatto grid points Teukolsky, Saul A. Journal of Computational Physics, Vol. 283 https://doi.org/10.1016/j.jcp.2014.12.012	journal	February 2015
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods Pazner, Will; Persson, Per-Olof Journal of Computational Physics, Vol. 354 https://doi.org/10.1016/j.jcp.2017.10.030	journal	February 2018
Porting hypre to heterogeneous computer architectures: Strategies and experiences Falgout, Robert D.; Li, Ruipeng; Sjögreen, Björn Parallel Computing, Vol. 108 https://doi.org/10.1016/j.parco.2021.102840	journal	December 2021
GPU algorithms for Efficient Exascale Discretizations Abdelfattah, Ahmad; Barra, Valeria; Beams, Natalie Parallel Computing, Vol. 108 https://doi.org/10.1016/j.parco.2021.102841	journal	December 2021
The Design and Performance of Batched BLAS on Modern High-Performance Computing Systems Dongarra, Jack; Hammarling, Sven; Higham, Nicholas J. Procedia Computer Science, Vol. 108 https://doi.org/10.1016/j.procs.2017.05.138	journal	January 2017
Numerical solution of saddle point problems Benzi, Michele; Golub, Gene H.; Liesen, Jörg Acta Numerica, Vol. 14 https://doi.org/10.1017/S0962492904000212	journal	April 2005
Multigroup radiation hydrodynamics with flux-limited diffusion and adaptive mesh refinement González, M.; Vaytet, N.; Commerçon, B. Astronomy & Astrophysics, Vol. 578 https://doi.org/10.1051/0004-6361/201525971	journal	May 2015
A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems Bramble, James H.; Pasciak, Joseph E. Mathematics of Computation, Vol. 50, Issue 181 https://doi.org/10.1090/S0025-5718-1988-0917816-8	journal	January 1988
Interior penalty preconditioners for mixed finite element approximations of elliptic problems Rusten, Torgeir; Vassilevski, Panayot S.; Winther, Ragnar Mathematics of Computation, Vol. 65, Issue 214 https://doi.org/10.1090/S0025-5718-96-00720-X	journal	April 1996
Preconditioning in \emph{\textbf{H}}$(\operatorname{div})$ and applications Arnold, Douglas N.; Falk, Richard S.; Winther, R. Mathematics of Computation, Vol. 66, Issue 219 https://doi.org/10.1090/S0025-5718-97-00826-0	journal	July 1997
Stability of Methods for Matrix Inversion Croz, Jeremy J. Du; Higham, Nicholas J. IMA Journal of Numerical Analysis, Vol. 12, Issue 1 https://doi.org/10.1093/imanum/12.1.1	journal	January 1992
Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s Alexander, Roger SIAM Journal on Numerical Analysis, Vol. 14, Issue 6 https://doi.org/10.1137/0714068	journal	December 1977
Nonstandard Norms and Robust Estimates for Saddle Point Problems Zulehner, Walter SIAM Journal on Matrix Analysis and Applications, Vol. 32, Issue 2 https://doi.org/10.1137/100814767	journal	April 2011
Parallel Auxiliary Space AMG Solver for $H(div)$ Problems Kolev, Tzanio V.; Vassilevski, Panayot S. SIAM Journal on Scientific Computing, Vol. 34, Issue 6 https://doi.org/10.1137/110859361	journal	January 2012
Natural Preconditioning and Iterative Methods for Saddle Point Systems Pestana, Jennifer; Wathen, Andrew J. SIAM Review, Vol. 57, Issue 1 https://doi.org/10.1137/130934921	journal	January 2015
A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers Kronbichler, Martin; Wall, Wolfgang A. SIAM Journal on Scientific Computing, Vol. 40, Issue 5 https://doi.org/10.1137/16M110455X	journal	January 2018
Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning Dobrev, V.; Kolev, T.; Lee, C. S. SIAM Journal on Scientific Computing, Vol. 41, Issue 3 https://doi.org/10.1137/17M1132562	journal	January 2019
Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods Pazner, Will SIAM Journal on Scientific Computing, Vol. 42, Issue 5 https://doi.org/10.1137/19M1282052	journal	January 2020
Spectral Equivalence of Low-Order Discretizations for High-Order H(curl) and H(div) Spaces Dohrmann, Clark R. SIAM Journal on Scientific Computing, Vol. 43, Issue 6 https://doi.org/10.1137/21M1392115	journal	January 2021
Conservative and Accurate Solution Transfer Between High-Order and Low-Order Refined Finite Element Spaces Kolev, Tzanio; Pazner, Will SIAM Journal on Scientific Computing, Vol. 44, Issue 1 https://doi.org/10.1137/21M1403916	journal	January 2022
Low-Order Preconditioning for the High-Order Finite Element de Rham Complex Pazner, Will; Kolev, Tzanio; Dohrmann, Clark R. SIAM Journal on Scientific Computing, Vol. 45, Issue 2 https://doi.org/10.1137/22M1486534	journal	April 2023
Multigrid for Matrix-Free High-Order Finite Element Computations on Graphics Processors Kronbichler, Martin; Ljungkvist, Karl ACM Transactions on Parallel Computing, Vol. 6, Issue 1 https://doi.org/10.1145/3322813	journal	May 2019
Scalability of high-performance PDE solvers Fischer, Paul; Min, Misun; Rathnayake, Thilina The International Journal of High Performance Computing Applications, Vol. 34, Issue 5 https://doi.org/10.1177/1094342020915762	journal	June 2020
Efficient exascale discretizations: High-order finite element methods Kolev, Tzanio; Fischer, Paul; Min, Misun The International Journal of High Performance Computing Applications, Vol. 35, Issue 6 https://doi.org/10.1177/10943420211020803	journal	June 2021
End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations Pazner, Will; Kolev, Tzanio; Camier, Jean-Sylvain The International Journal of High Performance Computing Applications, Vol. 37, Issue 5 https://doi.org/10.1177/10943420231175462	journal	June 2023
Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques Christie, M. A.; Blunt, M. J. SPE Reservoir Evaluation & Engineering, Vol. 4, Issue 04 https://doi.org/10.2118/72469-PA	journal	August 2001

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Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in \(\boldsymbol{H}(\operatorname{\textbf{div}})\)

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