Matrix-free GPU-accelerated saddle-point solvers for high-order problems in $H(\mathrm{div})$

Abstract

This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in $H(\mathrm{div})$. 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 (cf. SIAM J. Sci. Comput., 45 (2023), A675-A702, arXiv:2203.02465), 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.

21 pages, 10 figures

References in corpus (5)

• Efficient Exascale Discretizations: High-Order Finite Element Methods
• Multigroup radiation hydrodynamics with flux-limited diffusion and adaptive mesh refinement
• Low-order preconditioning for the high-order finite element de Rham complex
• Short note on the mass matrix for Gauss-Lobatto grid points
• End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

Cited by in corpus (2)

• Smoothers with localized residual computations for geometric multigrid methods
• Multigrid methods for the Stokes problem on GPU systems