Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in \ (H (div) \)

. 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. KeywordsH (div) matrix-free solversGPU-accelerated solvershigh-order finite elementsDarcyradiation diffusionMSC codes65F0865N3065Y0565Y1065Y20