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A. Rieder:
"Convolution Quadrature and Boundary Element Methods in wave propagation: a time domain point of view";
Supervisor, Reviewer: J. Melenk, L. Banjai, F. Sayas; Institut für Analysis und Scientific Computing, 2021; oral examination: 2017-06-13.

In this thesis, we consider different classes of time dependent wave propagation problems, and investigate whether they can be efficiently approximated using boundary integral methods. The idea of these methods is to replace partial differential equations with an integral equation on the boundary of the domain of interest. One of the main advantages of this approach is that problems posed on unbounded domains can be handled without further difficulties. For stationary problems boundary integral methods are well established as an alternative to more classical finite element based methods. In order to treat time dependent problems, one possibility is to apply Lubich's method of Convolution Quadrature. This approach has many favorable properties, including an equivalence principle, which relates the CQ approximation to the approximation of the underlying semigroup with an appropriate time-stepping scheme. In this work, we exploit this equivalence to analyze the discretization schemes under consideration. Our approach differs from the more standard way of treating time domain boundary integral equations, which relies on estimate in the Laplace domain in order to infer convergence results. The pure time-domain approach has the benefit of yielding stronger estimates with fewer regularity assumptions than the Laplace domain counterpart. In this thesis, we consider three different model problems, namely the time dependent Schrödinger equation posed in R^d, treated by a coupling of Finite- and Boundary Element Methods, a nonlinear scattering problem in the exterior domain consisting of the linear wave equation augmented by a nonlinear impedance boundary condition, and a scattering problem by a composite material characterized by a non-constant wave number. For all of these model problems we answer questions regarding convergence and stability of the discretization scheme. Numerical simulations support the theoretical findings.

convolution quadrature / boundary element method / wave propgation; semigroup theory; Runge Kutta method; wave equation; Schrodinger equation