# The panel-clustering method for the wave equation in two spatial dimensions

Falletta, Silvia; Sauter, Stefan A (2016). The panel-clustering method for the wave equation in two spatial dimensions. Journal of Computational Physics, 305:217-243.

## Abstract

We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks. The resulting method has a complexity of $\mathcal{O}(\mathit{N}(\mathit{N}+\mathit{M})\mathit{q}^{4+\mathit{s}})$, $\mathit{s} \in$ {0,1}, where $\mathit{N}$ is the number of time points, $\mathit{M}$ denotes the dimension of the boundary element space, and $\mathit{q}=\mathcal{O}$ (log $\mathit{N}$ + log $\mathit{M})$ is the order of the panel-clustering expansion. Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

## Abstract

We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks. The resulting method has a complexity of $\mathcal{O}(\mathit{N}(\mathit{N}+\mathit{M})\mathit{q}^{4+\mathit{s}})$, $\mathit{s} \in$ {0,1}, where $\mathit{N}$ is the number of time points, $\mathit{M}$ denotes the dimension of the boundary element space, and $\mathit{q}=\mathcal{O}$ (log $\mathit{N}$ + log $\mathit{M})$ is the order of the panel-clustering expansion. Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physics and Astronomy (miscellaneous), Computer Science Applications English January 2016 10 Aug 2016 07:32 19 Aug 2018 03:41 Elsevier 0021-9991 Green https://doi.org/10.1016/j.jcp.2015.10.033

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