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This work describes the development of matrix-free GPU-accelerated solversfor high-order finite element problems in $H(\mathrm{div})$. The solvers areapplicable to grad-div and Darcy problems in saddle-point formulation, and haveapplications in radiation diffusion and porous media flow problems, amongothers. Using the interpolation-histopolation basis (cf. SIAM J. Sci. Comput.,45 (2023), A675-A702, arXiv:2203.02465), efficient matrix-free preconditionerscan be constructed for the $(1,1)$-block and Schur complement of the blocksystem. With these approximations, block-preconditioned MINRES converges in anumber of iterations that is independent of the mesh size and polynomialdegree. The approximate Schur complement takes the form of an M-matrix graphLaplacian, and therefore can be well-preconditioned by highly scalablealgebraic multigrid methods. High-performance GPU-accelerated algorithms forall components of the solution algorithm are developed, discussed, andbenchmarked. Numerical results are presented on a number of challenging testcases, including the "crooked pipe" grad-div problem, the SPE10 reservoirmodeling benchmark problem, and a nonlinear radiation diffusion test case.

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