We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete kexplicit stability (including k-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size h and the approximation order p are selected such that kh/p is sufficiently small and p = O(log k), and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation.

Abstract-Recently, it has been argued that natural, intact stalagmites in caves give important constraints on seismic hazard since they have survived all earthquakes over their (rather long) life span. This suggests that the pattern of oscillation should be fully understood, including the splitting of eigenfrequencies that has occurred in recent cave observations. In the present study, we simulate the oscillation of a given stalagmite by setting up four simplified models of the stalagmite. The dimensions of the intact stalagmite were measured in situ, and the geo-mechanical and elastic parameters of broken stalagmite samples, determined in geo-mechanical laboratory, have been taken into account. The eigenfrequencies of the stalagmite are then calculated numerically, by the finite element method, and compared with the measured in situ values. The latter have shown splitting of eigenfrequencies, which we were able to reproduce by the numerical model calculations taking into account the asymmetric shape of the stalagmite.

We study the interaction of a seismic wavefield with a spherical acoustic gas‐ or fluid‐filled cavity. The intention of this study is to clarify whether seismic resonances can be expected, a characteristic feature that may help in detecting cavities in the subsurface. This is important for many applications, in particular the detection of underground nuclear explosions, which are to be prohibited by the Comprehensive Test Ban Treaty. To calculate the full seismic wavefield from an incident plane wave that interacts with the cavity, we considered an analytic formulation of the problem. The wavefield interaction consists of elastic scattering and the wavefield interaction between the acoustic and elastic media. Acoustic resonant modes caused by internal reflections in the acoustic cavity show up as spectral peaks in the frequency domain. The resonant peaks coincide with the eigenfrequencies of the un‐damped system described by the particular acoustic medium bounded in a sphere with stiff walls. The filling of the cavity could thus be determined by the observation of spectral peaks from acoustic resonances. By energy transmission from the internal oscillations back into the elastic domain, the oscillations experience damping, resulting in a frequency shift and a limitation of the resonance amplitudes. In case of a gas‐filled cavity, the impedance contrast is still high, which means low damping of the internal oscillations resulting in very narrow resonances of high amplitude. In synthetic seismograms calculated in the surrounding elastic domain, the acoustic resonances of gas‐filled cavities show up as persisting oscillations. However, due to the weak acoustic–elastic coupling in this case, the amplitudes of the oscillations are very low. Due to a lower impedance contrast, a fluid‐filled cavity has a stronger acoustic–elastic coupling, which results in wide spectral peaks of lower amplitudes. In the synthetic seismograms derived in the surrounding medium of fluid‐filled cavities, acoustic resonances show up as strong but fast decaying reverberations.

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We have studied the scattering of P-waves from an acoustic inclusion in a 2D half-space with a free surface. The motivation for our study comes from detecting a cavity that might be caused by an underground nuclear explosion. This is relevant to on-site inspections, an element of the Comprehensive Nuclear-Test-Ban Treaty (CTBT). The waveform modeling we address is implemented in the frequency domain; i.e., we consider the wavefield as well as the source to be time harmonic. We numerically investigate the cases in which the source of the scattered field is either a plane wave from the bottom or the side as from passive sources, such as teleseismic waves or ambient noise, or a spherical wave from the surface as from an active point source, such as a vibroseis or an explosion. To this end, we split the total field in an incident and an unknown scattered field to understand the effects more explicitly. Modeling the response of a void in a medium is not trivial, and many numerical algorithms commonly used for seismic propagation modeling will fail. Therefore, we want to highlight the advantage of high-order methods for this type of application in general and reveal the benefit of using the finite-element method code Ngsolve. This is in particular the case for the situation we have at hand, in which the ratio between the size and the depth of the cavity is notably high. We have addressed this scenario numerically for the first time because there are few field observations of the effects and the number of papers addressing the theoretical basis is sparse. Finally, we found that our splitting strategy together with the numerical scheme that we apply give rise to a constructive approach for studying this specific issue.