Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation
Year: 2022
Author: Nadir-Alexandre Messaï, Sebastien Pernet, Abdesselam Bouguerra
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1585–1635
Abstract
This study aimed to specialise a directional $\mathcal{H}^2 (\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0241
Communications in Computational Physics, Vol. 31 (2022), Iss. 5 : pp. 1585–1635
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 51
Keywords: Integral equation boundary element method Helmholtz equation discontinuous Galerkin directional $\mathcal{H}^2$-matrix low-rank approximation all frequency compression algorithm.