10:40
MS11: Numerical methods for wave propagation with applications in electromagnetics and geophysics (Part 2)
Chair: Barbara Verfürth
10:40
25 mins
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A second order finite element method with mass lumping for wave equations in H(div)
Bogdan Radu, Herbert Egger
Abstract: We consider the efficient numerical approximation of acoustic wave propagation in time domain by a finite element method with mass lumping. In the presence of internal damping, the problem can be reduced to a second order formulation in time for the velocity field alone. For the spatial approximation we consider H(div)-conforming finite elements of second order. In order to allow for an efficient time integration, we propose a mass-lumping strategy based on approximation of the L2-scalar product by inexact numerical integration which leads to a block-diagonal mass matrix. A careful error analysis allows to show that second order accuracy is not reduced by the quadrature errors which is illustrated also by numerical tests.
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11:05
25 mins
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Numerical Homogenization of the Maxwell-Debye system
Jan Philip Freese, Dietmar Gallistl, Christian Wieners
Abstract: In this talk we investigate time-dependent Maxwell's equations coupled with the Debye model for orientation polarization in a medium with highly oscillatory parameters and thus with oscillating solutions.
In our context we are only interested in the macroscopic behavior of the solution and not in the oscillations which are assumed to be considerably small. We use analytical homogenization results to derive the effective Maxwell-Debye system with the corresponding cell problems. The solution of this effective system is the macroscopic part of the oscillating solution in which we are interested.
Due to homogenization the system includes a convolution with a new time-dependent parameter whose corresponding cell corrector is also time-dependent. This time-dependence is the challenge, both in the implementation and in the analysis of the numerical scheme.
We apply the Finite Element Heterogeneous Multiscale Method (FE-HMM) to solve the effective Maxwell-Debye system since the HMM is suitable in the context of locally periodic parameters.
Finally, we present our semi-discrete error analysis and show numerical results that we have realized in the finite element library deal.II.
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11:30
25 mins
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Wavenumber-explicit error analysis of non-conforming FEM applied to the time-harmonic Maxwell equations with impedance boundary conditions in a smooth domain.
Serge Nicaise, Jérôme Tomezyk
Abstract: We present a coupled regularized variational formulation of the time-harmonic Maxwell equations with
impedance boundary conditions in smooth domains. This formulation is associated with an elliptic system, hence standard shift theorem holds. Then, we propose a non-conforming finite element method to approximate this problem and give a wavenumber explicit a priori error bound.
Numerical results illustrating our theoretical analysis will be given.
The talk is based on joint works with S. Nicaise (Valenciennes).
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11:55
25 mins
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Discontinuous Galerkin Finite Element method for Port-Hamiltonian Systems.
Nishant Kumar, J.J.W. van der Vegt, H.J. Zwart
Abstract: Since a long time, the numerical mathematics community has been trying to develop numerical methods that would give the flexibility of choosing different functional spaces for different physical variables and at the same time preserve the mathematical structure of the PDE. To this I will bring forward the idea of looking at the PDE in a geometrical way and to represent it as a port-Hamiltonian system. Using the fact that port-Hamiltonian systems are based on the idea of energy conservation and that the interconnection of port-Hamiltonian structures is again a port-Hamiltonian system, I propose to use discontinuous finite elements and then consider each finite element as a port-Hamiltonian system. In this talk I will discuss the idea and construction of this novel framework of port-Hamiltonian Discontinuous Galerkin Finite element methods. I will also discuss how this framework can be applied to the acoustic wave equation and how it can accurately solve the wave equation for long periods of time and over long distances.
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