A Robust Helmholtz-Decomposition-Based Real Compressed Layer Method for Time-Harmonic Elastic Wave Scattering

Abstract

Time-harmonic elastic wave scattering involves both compressional (P-) and shear (S-) waves, which propagate with different wavenumbers and polarization characteristics. The naive construction of perfectly matched layer (PML)-type methods based on complex coordinate stretching may lack robustness, or even fail, particularly when the wavenumbers are highly contrasted. The recently developed real compressed layer (RCL) technique build upon real compression transformations and explicit extraction of resulting oscillatory patterns for time-harmonic Helmholtz problems may not work, since the oscillations cannot be explicitly extracted by a single change of variables. This paper intends to bridge this gap by developing a robust RCL method for two-dimensional time-harmonic elastic wave scattering in unbounded domains with compactly supported inhomogeneities. A key observation is that, through the Helmholtz decomposition, the displacement field in the exterior homogeneous region decoupled into P-wave and S-wave and each has a distinctive separation of its oscillatory pattern and decaying behaviours in polar coordinates. We then apply the real compression coordinate transformation in the radial direction to each component. We further propose a coupled displacement-potential RCL formulation that seamlessly integrates the Helmholtz-decomposed wave components with the interior displacement field. We show that, under this framework, the essential oscillations in the layer can be effectively removed. We prove the well-posedness of the resulting coupled problem and establish the exponential convergence of the RCL solution to the original scattering solution in the truncated domain of interest. We discretize the RCL-system using high-order spectral element method and demonstrate the effectiveness and robustness of the proposed method through ample numerical results.