Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators

TL;DR

This paper analyzes various simultaneous approximation terms (SATs) for diffusion problems discretized with multidimensional summation-by-parts (SBP) operators, establishing conditions for stability, consistency, and accuracy, and demonstrating equivalences among different numerical schemes.

Contribution

It provides a unified framework for analyzing SATs in SBP discretizations, identifies conditions for stability and accuracy, and shows equivalences among several popular numerical methods.

Findings

Error in output functionals is of order h^{2p} for degree p SBP operators.

Different numerical schemes like Bassi-Rebay and interior penalty are equivalent with SBP diagonal-E operators.

Numerical experiments confirm theoretical error estimates and scheme equivalences.

Abstract

Several types of simultaneous approximation term (SAT) for diffusion problems discretized with diagonal-norm multidimensional summation-by-parts (SBP) operators are analyzed based on a common framework. Conditions under which the SBP-SAT discretizations are consistent, conservative, adjoint consistent, and energy stable are presented. For SATs leading to primal and adjoint consistent discretizations, the error in output functionals is shown to be of order $h^{2p}$ when a degree $p$ multidimensional SBP operator is used to discretize the spatial derivatives. SAT penalty coefficients corresponding to various discontinuous Galerkin fluxes developed for elliptic partial differential equations are identified. We demonstrate that the original method of Bassi and Rebay, the modified method of Bassi and Rebay, and the symmetric interior penalty method are equivalent when implemented with SBP diagonal-E operators that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator in one space dimension. Similarly, the local discontinuous Galerkin and the compact discontinuous Galerkin schemes are equivalent for this family of operators. The analysis remains valid on curvilinear grids if a degree $\le p+1$ bijective polynomial mapping from the reference to physical elements is used. Numerical experiments with the two-dimensional Poisson problem support the theoretical results.