Subcontract B635525 (Final Report)

Abstract

During my visits to LLNL July 8–12 and July 22–26, 2019, I worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear systems arising from $\textit{hp}$ adaptive finite element calculations, and is implemented in the $\textit{PLTMG}$ software package, version 13. This preconditioner typically requires 3–20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive definite elliptic equations, convection dominated convection-diffusion equations, and highly indefinite Helmholtz equations, among others). The convergence rate is not independent of the polynomial degree $\textit{p}$ as $\textit{p} → ∞$, but remains strong for $\textit{p}$ ≤ 9, which is the highest polynomial degree allowed in $\textit{PLTMG}$, due to limitations of the numerical quadrature rules implemented in the soft- ware package. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in. Like traditional geometric multilevel methods, this scheme relies on knowledge of the underlying finite element space in order to construct the smoother and the coarse grid correction.