Radial perfectly matched layers and infinite elements for the anisotropic wave equation

Abstract

We consider the scalar anisotropic wave equation. Recently a convergence analysis for radial perfectly matched layers (PML) in the frequency domain was reported and in the present article we continue this approach into the time domain. First we explain why there is a good hope that radial complex scalings can overcome the instabilities of PML methods caused by anisotropic materials. Next we discuss some sensitive details, which seem like a paradox at the first glance: If the absorbing layer and the inhomogeneities are sufficiently separated, then the solution is indeed stable. However, for more general data the problem becomes unstable. In numerical computations we observe instabilities regardless of the position of the inhomogeneities, although the instabilities arise only for fine enough discretizations. As a remedy we propose a complex frequency shifted scaling and discretizations by Hardy space infinite elements or truncation-free PMLs. We show numerical experiments which confirm the stability and convergence of these methods.

Figures (12)

• Figure 1: An example slowness curve for an anisotropic wave equation. With blue we mark the vectors of the group and phase velocity that have projections of different signs on the direction $\mathbf{e}_x$, and with violet those that have projections of different signs on the direction $\mathbf{e}_y$. Remark that the phase velocity is directed in the radial direction, and the group velocity is directed along the exterior normal to the slowness curve. Clearly, the projection of the group velocity on the phase velocity is always positive.
• Figure 2: (In-)stability of standard PMLs.
• Figure 3: Unstable PML solution with $h=0.075$ at different times $t$ (cf. also Figure \ref{['fig:coarse_stability']}).
• Figure 4: An illustration to the notation \ref{['eq:notation1']}. The PML medium is marked in gray and the physical medium in white.
• Figure 5: Left: the quantity $\operatorname{sign}\gamma_{12}(\hat{\mathbf{x}},\mathbf{y})$ as defined in \ref{['eq:notation2']} depending on $\mathbf{y}\in B(0,1)$, for a fixed $\mathbf{x}=(\cos0.8\pi,\sin0.8\pi)$, $R_{\mathrm{pml}}=1$. In dark blue we mark $\mathbf{y}$, s.t. $\operatorname{sign}\gamma_{12}(\hat{\mathbf{x}},\mathbf{y})=-1$, while in yellow $\mathbf{y}$ with $\operatorname{sign}\gamma_{12}(\hat{\mathbf{x}},\mathbf{y})=1$. The point $\mathbf{x}$ is marked in red. Right: the sets $\Omega_{\operatorname{st}}$ (in yellow) and $\Omega_{\operatorname{inst}}$ (in blue). In all the experiments, the matrix $\mathbf{B}=\operatorname{diag}(1,1/4)$.

Theorems & Definitions (111)

• Remark 3.1
• Remark 3.2
• Theorem 3.4
• proof
• Theorem 3.5
• Lemma 3.6
• proof
• Corollary 3.7
• Proposition 3.8: Fundamental solution of the PML problem
• proof