A Nitsche-type Method for Helmholtz Equation with an Embedded Acoustically Permeable Interface
This work provides a numerical method for wave propagation problems with permeable interfaces, addressing a gap in handling vanishing impedance, which is important for computational acoustics.
The paper proposes a new finite element method for the Helmholtz equation with an embedded acoustically permeable interface, using a variant of Nitsche's method that handles complex-valued impedance functions, including vanishing impedance. Stability and a priori error estimates are proved, and numerical experiments in 2D and 3D demonstrate performance.
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 Gårding 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.