Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$
H
2
-matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.
In this article, we propose new gradient recovery schemes for the BEM-based Finite Element Method (BEM-based FEM) and Virtual Element Method (VEM). Supporting general polytopal meshes, the BEM-based FEM and VEM are highly flexible and efficient tools for the numerical solution of boundary value problems in two and three dimensions. We construct the recovered gradient from the gradient of the finite element approximation via local averaging. For the BEM-based FEM, we show that, under certain requirements on the mesh, superconvergence of the recovered gradient is achieved, which means that it converges to the true gradient at a higher rate than the untreated gradient. Moreover, we propose a simple and very efficient a posteriori error estimator, which measures the difference between the unprocessed and recovered gradient as an error indicator. Since the BEM-based FEM and VEM are specifically suited for adaptive refinement, the resulting adaptive algorithms perform very well in numerical examples.
In this work, semi-analytical formulae for the numerical evaluation of surface integrals occurring in Galerkin boundary element methods (BEM) in 3D are derived. The integrals appear as the entries of BEM matrices and are formed over pairs of surface triangles. Since the integrands become singular if the triangles have non-empty intersection, the transformation presented in [1] is used to remove the singularities. It is shown that the resulting integrals admit analytical formulae if the triangles are identical or share a common edge. Moreover, the four-dimensional integrals are reduced to oneor two-dimensional integrals for triangle pairs with common vertices or disjoint triangles respectively. The efficiency and accuracy of the formulae is demonstrated in numerical experiments.
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