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

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

Journal Article · 02 May 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.

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

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

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

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

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