Immersed Discontinuous Galerkin Methods for Acoustic Wave Propagation in Inhomogeneous Media

Abstract

We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curl-free and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity.

The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation.

Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods.