A Nitsche-type Method for Helmholtz Equation with an Embedded Acoustically Permeable Interface

TL;DR

This paper introduces a novel finite element method based on a modified Nitsche's approach for solving the Helmholtz equation with embedded acoustically permeable interfaces, capable of handling complex impedance functions including zero impedance cases.

Contribution

The paper develops a new Nitsche-type finite element method that weakly enforces impedance conditions and handles complex, possibly vanishing, impedance functions in Helmholtz problems.

Findings

Method is stable under a discrete G{ a}rding inequality.

Provides an a priori error estimate for impedance bounded away from zero.

Numerical experiments confirm effectiveness in 2D and 3D cases.

Abstract

We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche's method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function $Z$ that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete G{\aa}rding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.