An overlapping decomposition framework for wave propagation in heterogeneous and unbounded media: Formulation, analysis, algorithm, and simulation

Abstract

A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium wave propagation model, and avoid an approximation of the RC. In this work we first present and analyze an overlapping decomposition framework that is equivalent to the full-space heterogeneous-homogenous continuous model, governed by the Helmholtz equation with a spatially dependent refractive index and the RC. Our novel overlapping framework allows the user to choose two free boundaries, and gain the advantage of applying established high-order finite and boundary element methods (FEM and BEM) to simulate an equivalent coupled model.The coupled model comprises auxiliary interior bounded heterogeneous and exterior unbounded homogeneous Helmholtz problems. A smooth boundary can be chosen for simulating the exterior problem using a spectrally accurate BEM, and a simple boundary can be used to setup a high-order FEM for the interior problem. Thanks to the spectral accuracy of the exterior computational model, the resulting coupled system in the overlapping region is relatively very small. Using the decomposed equivalent framework, we develop a novel overlapping FEM-BEM algorithm for simulating the acoustic or electromagnetic wave propagation in two dimensions. Our FEM-BEM algorithm for the full-space model incorporates the RC exactly. Numerical experiments demonstrate the efficiency of the FEM-BEM approach for simulating smooth and non-smooth wave fields, with the latter induced by a complex heterogeneous medium and a discontinuous refractive index.

Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media

“…Then I − K : Proof. This result is proven in [10]. We will give here a sketch of the proof for the sake of completeness.…”

“…For the given wave propagation problem (2.1), next we recall an equivalent decomposition framework introduced and analyzed in [10]. The framework introduce two artificial curves Γ and Σ with interior Ω 1 and Ω 2 , respectively, satisfying…”

“…In [10], the authors proved that the unique solution u of (2.1) can be constructed in two steps as follows:…”

“…Using two artificial interfaces a mathematical framework was developed over four decades ago in [25] and was subsequently used in [8,22]. A domain-overlapped framework [illustrated in Figure 1 with overlapped region (R 2 \ Ω 1 ) ∩ Ω 2 ] was developed recently by the authors in [10] that takes advantage of a polygonal boundary (Σ) required for a wide-class FEM, and a smooth boundary (Γ) for highorder spectral BEM. Mathematical analysis of the continuous framework in [10] establishes the equivalence and regularity of the decomposed model.…”

“…In this work we present rigorous numerical analysis of the FEM-BEM adaptive coupling framework introduced in [10] for solving the Helmholtz acoustic/electromagnetic wave propagation problem on the plane with a bounded heterogeneous region, and demonstrate efficiency of the numerical algorithm for complex Janus-type heterogeneous configurations with computational experiments. The algorithm works on a suitable partition of the plane defined from a polygonal domain containing a smooth curve that surround all heterogeneity.…”