Adaptive and high order methods for time domain boundary elements
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This thesis concerns eﬃcient and accurate time domain boundary element methods for the wave equation, with an emphasis on wave scattering in singular geometries and also towards higher frequencies in 3d. The work focuses on three approaches: Mesh reﬁnements, as obtained in adaptive boundary elements or with algebraically graded meshes; a p-version of the boundary element method; and generalized boundary elements based on enriching the approximation spaces with either singular functions or plane waves. An application to traﬃc noise addresses computations for the sound ampliﬁcation in the singular horn geometry between tyre and road. The thesis ﬁrst shows that algebraically graded meshes recover quasi-optimal convergence rates in polyhedral domains. For more general singularities of the solution, adaptive mesh reﬁnements based on residual and Zienkiewicz-Zhu a posteriori error indicators lead to improved convergence rates. In benchmark examples for wave scattering on screens the convergence rates recover the rates known for time-independent problems. Then a p-version of the time domain boundary element method is presented, based on increasing the polynomial degree on a ﬁxed coarse space-time mesh. Compared to the standard h-version the p-methods are shown to double the convergence rate on screens. Generalized boundary elements are studied for both screen and plane-wave scattering. For screens the ansatz and test functions are enriched with the singular functions known from the asymptotic expansion of the exact solution near edges and corners. For planewave scattering enrichments based on plane waves are presented. In examples we obtain a rapid convergence to engineering accuracy even on coarse meshes. The work is supplemented with the study of an eﬃcient preconditioner for the linear systems arising in the time domain method. It leads to eﬃcient computation times for a wide range of discretisations.