Scalable low-order finite element preconditioners for high-order spectral element Poisson solvers

Low-order finite element (FE) systems are considered as preconditioners for spectral element (SE) discretizations of the Poisson problem in canonical and complex domains. The FE matrices are based on the same mapped set of Gauss-Lobatto-Legendre (GLL) points as the SE discretization. Three different versions of the preconditioner based on combinations of the low-order stiffness and mass matrices are tested for 2D and 3D geometries. When building the preconditioning operators, a new meshing approach that allows elements to overlap without needing to fill out the volume of the mesh is explored and shown to be better than traditional schemes. These preconditioners are robust with respect to cell aspect ratio, demonstrate bounded iteration counts with h- and p-refinement, and have lower iteration counts than scalable hybrid-Schwarz multigrid schemes currently used in production-level SE simulations. Overall costs for large-scale parallel applications are dependent on fast and robust solvers for the sparse FE systems. Algebraic multigrid is shown to offer a pathway to realizing a robust and fast preconditioning strategy in this context.