MATRIX-FREE HIGH-PERFORMANCE SADDLE-POINT SOLVERS FOR HIGH-ORDER PROBLEMS IN H(div)

Abstract

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 (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 (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.