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This work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in if(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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a nonoverlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Pad\'e approximants of the nonlocal magnetic-to-electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers. [ABSTRACT FROM AUTHOR]

In this paper we introduce general transfer operators between high-order and loworder refined finite element spaces that can be used to couple high-order and low-order simulations. Under natural restrictions on the low-order refined space we prove that both the high-to-low-order and low-to-high-order linear mappings are conservative, constant preserving, and high-order accurate. While the proof holds for affine geometries, numerical experiments indicate that the results hold for more general curved meshes. We present several numerical results confirming our analysis and demonstrate the utility of the new mappings in the context of adaptive mesh refinement and conservative multidiscretization coupling. [ABSTRACT FROM AUTHOR]

With the gap between processor clock speeds and memory bandwidth speeds continuing to increase, the use of arithmetically intense schemes, such as high-order finite element methods, continues to be of considerable interest. In particular, the use of matrix-free formulations of finite element operators for tensor-product elements of quadrilaterals in two dimensions and hexahedra in three dimensions, in combination with single-instruction multiple-data instruction sets, is a wellstudied topic at present for the efficient implicit solution of elliptic equations. However, a considerable limiting factor for this approach is the use of meshes comprising of only quadrilaterals or hexahedra, the creation of which is still an open problem within the mesh generation community. In this article, we study the efficiency of high-order finite element operators for the Helmholtz equation with a focus on extending this approach to unstructured meshes of triangles, tetrahedra, and prismatic elements using the spectral/hp element method and corresponding tensor-product bases for these element types. We show that although performance is naturally degraded when going from hexahedra to these simplicial elements, efficient implementations can still be obtained that are capable of attaining 50% through 70% floating point operations of the peak of processors with both AVX2 and AVX512 instruction sets. [ABSTRACT FROM AUTHOR]

The aim of this work is to propose a novel, fast solver for the Poisson problem discretized with high-order spectral element methods (HO-SEM) in canonical geometries (rectangle in two dimensions, rectangular parallelepiped in three dimensions). This method is based on the use of the discrete Fourier transform to reduce the problem to the inversion of the symbol of the operator in the frequency space. The solver proposed is endowed with several properties. First, it preserves the efficiency of the standard FFT algorithm; then, the matrix storage is drastically reduced (in particular, it is independent of the space dimension); a pseudoexplicit singular value decomposition is used for the inversion of the symbols; and finally, it can be extended to nonperiodic boundary conditions. Furthermore, due to the underlying HO-SEM discretization, the multidimensional symbol of the operator can be efficiently computed from the one-dimensional symbol by tensorisation. [ABSTRACT FROM AUTHOR]

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of [P. Knupp, Eng. Comput., 28 (2012), pp. 419--429]. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Lagrangian--Eulerian (ALE) application [BLAST: High-order curvilinear finite elements for shock hydrodynamics. [ABSTRACT FROM AUTHOR]

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (HDG) method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the HDG method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. It is found that high-order continuous finite elements give best time to solution for smooth solutions, closely followed by the matrix-free solvers for the other two discretizations. Their performance is up to an order of magnitude higher than that of the best matrix-based methods, even after including the superconvergence effects in the matrix-based HDG method. This difference is because of the vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A rooine performance model confirms the superiority of the matrix-free implementation. [ABSTRACT FROM AUTHOR]

Motivated by the need for efficient and accurate simulation of the dynamics of the polar ice sheets, we design high-order finite element discretizations and scalable solvers for the solution of nonlinear incompressible Stokes equations. In particular, we focus on power-law, shear thinning rheologies commonly used in modeling ice dynamics and other geophysical flows. We use nonconforming hexahedral meshes and the conforming inf-sup stable finite element velocity-pressure pairings Qk × Qk-2disc or Qk × Pk-1disc, where k ≥ 2 is the polynomial order of the velocity space. To solve the nonlinear equations, we propose a Newton-Krylov method with a block upper triangular preconditioner for the linearized Stokes systems. The diagonal blocks of this preconditioner are sparse approximations of the (1,1)-block and of its Schur complement. The (1,1)-block is approximated using linear finite elements based on the nodes of the high-order discretization, and the application of its inverse is approximated using algebraic multigrid with an incomplete factorization smoother. This preconditioner is designed to be efficient on anisotropic meshes, which are necessary to match the high aspect ratio domains typical for ice sheets. As part of this work, we develop and make available extensions to two libraries--a hybrid meshing scheme for the p4est parallel adaptive mesh refinement library and a modified smoothed aggregation scheme for PETSc--to improve their support for solving PDEs in high aspect ratio domains. In a comprehensive numerical study, we find that our solver yields fast convergence that is independent of the element aspect ratio, the occurrence of nonconforming interfaces, and the mesh refinement and that depends only weakly on the polynomial finite element order. We simulate the ice flow in a realistic description of the Antarctic ice sheet derived from field data and study the parallel scalability of our solver for problems with up to 383 million unknowns. [ABSTRACT FROM AUTHOR]