08:30
MS11: Numerical methods for wave propagation with applications in electromagnetics and geophysics (Part 1)
Chair: Barbara Verfürth
08:30
25 mins
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A Trefftz discontinuous Galerkin method for acoustic scattering by small circular obstacles
Monique Dauge, Ilaria Perugia, Alexander Pichler
Abstract: In this talk, following the concept of the ultra weak variational formulation of Cessenat and Després, we present a Trefftz discontinuous Galerkin (DG) finite element method for the numerical approximation of solutions to acoustic scattering problems by small circular obstacles. This method is characterized by the following two features: firstly, it is applicable to all kind of polygonal meshes resolving the position of the obstacles, but not necessarily their size; secondly, it directly takes into account the Sommerfeld radiation condition. Both these features are realized by the use of specifically tailored basis functions that automatically satisfy the boundary conditions and are additionally Trefftz, i.e. local solution to the homogeneous Helmholtz equation. Due to the Trefftz property, the employed basis functions provide high-order approximation, and a large reduction in the required number of degrees of freedom for a given accuracy, as compared to standard finite element methods, can be obtained. Theoretical results, as well as a variety of numerical experiments, will be discussed.
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08:55
25 mins
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Asymptotically constant-free, P-robust and guaranteed a posteriori error estimates for the Helmholtz equation
Théophile Chaumont-Frelet, Alexandre Ern, Martin Vohralik
Abstract: We propose a novel a posteriori error estimator that relies on a flux equilibration technique. We provide guaranteed and fully computable upper-bounds on the finite element error. We demonstrate that the proposed estimator is p-robust, which means that its efficiency does not deteriorates when the polynomial degree is large. We also show that asymptotically, when the finite element mesh is sufficiently refined, the upper-bound can be simplified into a constant-free estimate. We present numerical experiments that illustrate our analysis and highlight the robustness of the proposed error estimates.
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09:20
25 mins
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On energy preserving high-order discretizations for nonlinear acoustics
Herbert Egger, Vsevolod Shashkov
Abstract: This paper addresses the numerical solution of the Westervelt equation, which arises as one of the model equations in nonlinear acoustics. The problem is rewritten in a canonical form that allows the systematic discretization by Galerkin approximation in space and time. Exact energy preserving methods of formally arbitrary order are obtained and their efficient realization as well as the relation to other frequently used methods is discussed.
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09:45
25 mins
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Data-driven modeling for wave-propagation
Tristan van Leeuwen, Sergiy Zhuk, Peter Jan van Leeuwen
Abstract: Many imaging modalities, such as ultrasound and radar, rely heavily on the ability to accurately model wave propagation. In most applications, the response of an object to an incident wave is recorded and the goal is to characterize the object in terms of its physical parameters (e.g., density or sound speed). The resulting data-fitting problem to recover the parameters involves repeatedly solving the wave-equation, for which standard numerical methods suffice. Recently, a different approach was proposed, where one replaces the wave-equation by a data-assimilation problem. A benefit of this modified procedure is that it makes it easier to recover the parameters. In this paper, I give an overview of numerical methods that can be used to solve the data-assimilation problem.
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