High-Order Non-Conforming Discontinuous Galerkin Methods for the Acoustic Conservation Equations

TL;DR

This paper compares two Nitsche-type approaches for high-order DG methods on non-conforming meshes in acoustic equations, demonstrating that exact integration ensures stability and optimal convergence, unlike point-to-point interpolation.

Contribution

It introduces and analyzes the necessity of exact integration in Nitsche-type DG methods for acoustic equations on non-conforming meshes, highlighting stability and convergence benefits.

Findings

Point-to-point interpolation causes instabilities in DG acoustic discretizations.

Exact integration maintains stability and achieves optimal convergence.

Non-conforming discretizations effectively handle overlaps and different fluids.

Abstract

This work compares two Nitsche-type approaches to treat non-conforming triangulations for a high-order discontinuous Galerkin (DG) solver for the acoustic conservation equations. The first approach (point-to-point interpolation) uses inexact integration with quadrature points prescribed by a primary element. The second approach uses exact integration (mortaring) by choosing quadratures depending on the intersection between non-conforming elements. In literature, some excellent properties regarding performance and ease of implementation are reported for point-to-point interpolation. However, we show that this approach can not safely be used for DG discretizations of the acoustic conservation equations since, in our setting, it yields spurious oscillations that lead to instabilities. This work presents a test case in that we can observe the instabilities and shows that exact integration is required to maintain a stable method. Additionally, we provide a detailed analysis of the method with exact integration. We show optimal spatial convergence rates globally and in each mesh region separately. The method is constructed such that it can natively treat overlaps between elements. Finally, we highlight the benefits of non-conforming discretizations in acoustic computations by a numerical test case with different fluids.