Numerical Analysis Seminar

Date: November 17, 2021

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: William Pazner, LLNL

Title: Low-order methods for high-order finite element discretizations and solvers

Abstract: In this talk, I will discuss two applications of low-order methods to increase the efficiency and robustness of high-order finite element discretizations and solvers. In the first part of the talk, I will discuss matrix-free linear solvers for high-order discontinuous Galerkin discretizations of elliptic problems. These solvers are based on the spectral equivalence of the high-order discretization with a low-order refined discretization (often known as the "FEM-SEM equivalence"). A novel extension of this equivalence to the case of nonconforming meshes and variable polynomial degrees will be presented. Using the subspace correction (additive Schwarz) framework, robust preconditioners for DG discretizations with (nonconforming) hp-refinement will be constructed that result in uniform convergence with respect to mesh size, polynomial degree, and DG penalty parameter. This method is amenable for use on adaptively refined meshes with any degree of irregularity. Examples are shown using the interior penalty and BR2 methods. In the second part of the talk, I will discuss the construction of invariant domain preserving discontinuous Galerkin methods using subcell convex limiting (cf. Guermond and Popov, 2016). The high-order DG method is augmented with a sparse low-order Lax-Friedrichs discretization constructed on a refined mesh. A key feature is that the low-order method does not become more dissipative as the polynomial degree of the high-order method is increased, in contrast with other graph viscosity techniques. The high-order and low-order methods are blended using an efficient dimension-by-dimension convex limiting procedure that can be used to guarantee the preservation of any number of user-specified convex invariants while retaining subcell resolution. Several numerical examples for the Euler equations will be shown, for which this method preserves the positivity of density, pressure and internal energy, and satisfies a minimum principle for the specific entropy.