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.

翻译：本文描述了针对$H(\mathrm{div})$中高阶有限元问题的无矩阵GPU加速求解器的开发。所提出的求解器适用于鞍点形式中的梯度-散度和达西问题，并在辐射扩散和多孔介质流动等问题中具有应用前景。利用插值-历史化基（参见SIAM J. Sci. Comput., 45 (2023), A675-A702, arXiv:2203.02465），可以为块系统的$(1,1)$-块和舒尔补构造高效的无矩阵预条件子。借助这些近似，块预条件MINRES的收敛迭代次数与网格尺寸和多项式阶数无关。近似舒尔补采用M矩阵图拉普拉斯形式，因此可通过高度可扩展的代数多重网格方法进行良好预条件。文中开发、讨论并测试了求解算法各组成部分的高性能GPU加速算法。在多个具有挑战性的算例上给出了数值结果，包括“弯曲管道”梯度-散度问题、SPE10油藏建模基准问题以及一个非线性辐射扩散测试案例。