# A coercive heterogeneous media Helmholtz model: formulation, wavenumber-explicit analysis, and preconditioned high-order FEM

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## Abstract

We consider a frequency-domain heterogeneous wave propagation model governed by the Helmholtz partial differential equation (PDE) and an impedance boundary condition. The celebrated standard (*H*^{1}) variational formulation of the model is non-coercive. It is an open problem to establish a coercive variational formulation of the heterogeneous model. The main focus of this article is on solving this continuous model formulation and analysis problem, and hence establishing an efficient preconditioned numerical algorithm for simulating our novel coercive variational formulation. We develop the variational formulation for the heterogeneous model (in a Hilbert space *V* equipped with a stronger norm than the *H*^{1}-norm) and prove that the associated sesquilinear form is coercive, with a wavenumber-independent coercivity constant. We use this result to derive a wavenumber-independent bound for solutions of the heterogeneous media wave propagation model in the *V* -norm. Additionally, we prove continuity of the sesquilinear form, with a wavenumber-explicit continuity constant. Using our analysis-supported coercive formulation, we develop a high-order frequency robust-preconditioned finite element method (FEM)-based heterogeneous media discrete wave model. For demonstrating efficiency and convergence of the coercive high-order FEM model, we use non-convex media comprising curved and non-smooth boundaries and low- to high-frequency input incident waves. For the heterogeneous media, with size varying from tens to hundreds of wavelengths, we demonstrate that our new preconditioned-FEM model requires a very low number of GMRES iterations, and the number of iterations is independent of the wavenumber of the model. We also use a class of additive Schwarz domain decomposition (DD) algorithms to implement the preconditioned-FEM model. The DD-based high-order preconditioned-FEM results and comparisons further demonstrate efficiency of the coercive formulation to simulate wave propagation in heterogenous media.

## Keywords

Helmholtz equation Heterogeneous Coercive Wavenumber-explicit Finite element method High-order Preconditioner Domain decomposition Additive Schwarz## Mathematics Subject Classification (2010)

35J05 35J20 65F08 65F10 65N30## Notes

### Acknowledgments

Support of the Colorado Golden Energy Computing Organization (GECO) is gratefully acknowledged. The first author would like to thank Dr. Stuart Hawkins and Dr. Euan Spence for their constructive comments on an earlier version of this article.

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