Sponsor
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 (LLNL-JRNL-848035). The research of the first and second authors was partially supported by the LLNL-LDRD Program under project 20-ERD-002. The research of the first author was also partially supported the ORAU Ralph E. Powe Junior Enhancement Award, 2023.
Published In
Siam Journal on Scientific Computing
Document Type
Pre-Print
Publication Date
8-25-2024
Subjects
Mathematics, Polynomials -- Mathematical models
Abstract
This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in . 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 (cf. [W. Pazner, T. Kolev, and C. R. Dohrmann, SIAM J. Sci. Comput., 45 (2023), pp. A675–A702]), efficient matrix-free preconditioners can be constructed for the -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.
Rights
© 2024 the Authors
Locate the Document
DOI
10.1137/23M1568806
Persistent Identifier
https://archives.pdx.edu/ds/psu/42471
Citation Details
Pazner, W., Kolev, T., & Vassilevski, P. S. (2024). Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in \(\boldsymbol{H}(\operatorname{\textbf{div}})\). SIAM Journal on Scientific Computing, 46(3), B179–B204. https://doi.org/10.1137/23m1568806
Description
Pre-print
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: Pazner, W., Kolev, T., & Vassilevski, P. S. (2024). Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in. SIAM Journal on Scientific Computing, 46(3), B179-B204.