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Institute
In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy–Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands.
FEM–BEM coupling for the thermoelastic wave equation with transparent boundary conditions in 3D
(2022)
We consider the thermoelastic wave equation in three dimensions with transparent boundary conditions on a bounded, not necessarily convex domain. In order to solve this problem numerically, we introduce a coupling of the thermoelastic wave equation in the interior domain with time-dependent boundary integral equations. Here, we want to highlight that this type of problem differs from other wave-type problems that dealt with FEM–BEM coupling so far, e.g., the acoustic as well as the elastic wave equation, since our problem consists of coupled partial differential equations involving a vector-valued displacement field and a scalar-valued temperature field. This constitutes a nontrivial challenge which is solved in this paper. Our main focus is on a coercivity property of a Calderón operator for the thermoelastic wave equation in the Laplace domain, which is valid for all complex frequencies in a half-plane. Combining Laplace transform and energy techniques, this coercivity in the frequency domain is used to prove the stability of a fully discrete numerical method in the time domain. The considered numerical method couples finite elements and the leapfrog time-stepping in the interior with boundary elements and convolution quadrature on the boundary. Finally, we present error estimates for the semi- and full discretization.