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

Will Pazner, Tzanio Kolev, Panayot Vassilevski

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.

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 . 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 -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.
Paper Structure (17 sections, 5 theorems, 48 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 17 sections, 5 theorems, 48 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretization and linear system
  3. Interpolation-histopolation basis
  4. Block preconditioners for saddle-point systems
  5. Application to the grad-div system
  6. Application to the Darcy system
  7. L2 mass inverse
  8. Notes on the untransformed system
  9. Algorithmic details and GPU acceleration
  10. GPU acceleration of the DG mass inverse
  11. Schur complement preconditioning
  12. Numerical results
  13. Grad-div problem: crooked pipe
  14. Darcy flow: SPE10 benchmark
  15. Nonlinear radiation diffusion
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\mathcal{B}$ be the block-diagonal preconditioner where $\mathsf{S} = \mathsf{C} + \mathsf{D} \mathsf{A}^{-1} \mathsf{D}^\mathsf{T}$ is the (negative) Schur complement of $\mathcal{A}$ with respect to the $(1,1)$-block. This preconditioner is optimal in the sense that

Figures (10)

  • Figure 1: Left: high-order ($p=9$) hexahedral element with 10 Gauss--Lobatto nodes in each dimension. Right: subelement mesh with vertices at the Gauss--Lobatto points.
  • Figure 2: Condition number of the $L^2$ mass matrix preconditioned by diagonal scaling for different choices of basis. Left: 2D case on a skewed quadrilateral. Right: 3D case on a skewed hexahedron.
  • Figure 3: Computed condition numbers of preconditioner Schur complement. The transformed system is given by $\widetilde{\mathsf{S}}^{-1}\mathsf{S}$, and the untransformed system is given by $\widetilde{\mathsf{S}}'^{-1} \mathsf{S}'$.
  • Figure 4: Solver diagram for block preconditioners applied to the transformed saddle-point system \ref{['eq:transformed-saddle-point']}.
  • Figure 5: GPU DG mass inverse setup phase.
  • ...and 5 more figures

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • ...and 2 more