Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

Abstract

This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

Error Analysis of Symmetric Linear/Bilinear Partially Penalized Immersed Finite Element Methods for Helmholtz Interface Problems

“…Now, we present the PPIFE methods developed in [29] for the Helmholtz interface problem in order to analyze them. The description of these PPIFE methods relies on the following space defined according to the mesh T h :…”

“…In this section, we present a numerical example to validate the error estimates in Theorems 3.3 and 3.4. We note that [29] provides quite a few numerical examples to illustrate convergence features of the PPIFE methods developed there for solving the Helmholtz interface problems. However, the exact solutions in the examples presented in [29] have a regularity better than piecewise H r with r > 2.…”

“…This article is about the error analysis for the linear and bilinear partially penalized immersed finite element (PPIFE) methods developed in [29] for solving interface boundary value problems of the Helmholtz equation [5,23] that is posed in a bounded domain Ω ⊆ R 2 : find u(X) that satisfies the Helmholtz equation and the first-order absorbing boundary condition:…”

“…The PPIFE methods developed in [29] are a class of finite element methods for solving Helmholtz interface problems with interface-independent meshes. Extensive numerical experiments reported in [29] clearly demonstrate the optimal convergence of these PPIFE methods, and this observation motivates us to carry out related error analysis to theoretically confirm their optimal convergence.…”

“…In this article, we also follow the framework of Schatz's argument to carry out the error analysis for the linear and bilinear symmetric PPIFE methods developed in [29] for Helmholtz interface problems with a Robin boundary condition. We utilize the coercivity and continuity of the bilinear form corresponding to the elliptic operator [14,28] to establish Gårding's inequality and continuity of the bilinear form in these PPIFE methods which are key ingredients in Schatz's argument.…”