Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

TL;DR

This paper analyzes multidimensional SBP discretizations with interior penalties for linear elliptic equations, generalizing existing methods and verifying stability and consistency through numerical experiments on unstructured grids.

Contribution

It introduces conditions for SAT penalties in SBP discretizations, generalizes BR2 and SIPG methods within this framework, and verifies theoretical results numerically.

Findings

Conditions for SAT penalties ensuring stability and consistency

Generalization of BR2 and SIPG methods to SBP-SAT discretizations

Numerical verification on unstructured triangular grids

Abstract

This work focuses on multidimensional summation-by-parts (SBP) discretizations of linear elliptic operators with variable coefficients. We consider a general SBP discretization with dense simultaneous approximation terms (SATs), which serve as interior penalties to enforce boundary conditions and inter-element coupling in a weak sense. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties. Based on these conditions, we generalize the modified scheme of Bassi and Rebay (BR2) and the symmetric interior penalty Galerkin (SIPG) method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.