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Volume 31, Issue 5
Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation

Nadir-Alexandre Messaï, Sebastien Pernet & Abdesselam Bouguerra

DOI: 10.4208/cicp.OA-2021-0241

Commun. Comput. Phys., 31 (2022), pp. 1585-1635.

Published online: 2022-05

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  • 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.

  • Keywords

Integral equation, boundary element method, Helmholtz equation, discontinuous Galerkin, directional $\mathcal{H}^2$-matrix, low-rank approximation, all frequency compression algorithm.

  • AMS Subject Headings

35J05, 65D05, 65N38, 41A10, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1585, author = {Messaï , Nadir-AlexandrePernet , Sebastien and Bouguerra , Abdesselam}, title = {Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0241}, url = {http://global-sci.org/intro/article_detail/cicp/20516.html} }
TY - JOUR T1 - Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation AU - Messaï , Nadir-Alexandre AU - Pernet , Sebastien AU - Bouguerra , Abdesselam JO - Communications in Computational Physics VL - 5 SP - 1585 EP - 1635 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0241 UR - https://global-sci.org/intro/article_detail/cicp/20516.html KW - Integral equation, boundary element method, Helmholtz equation, discontinuous Galerkin, directional $\mathcal{H}^2$-matrix, low-rank approximation, all frequency compression algorithm. AB -

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.

Messaï , Nadir-AlexandrePernet , Sebastien and Bouguerra , Abdesselam. (2022). Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation. Communications in Computational Physics. 31 (5). 1585-1635. doi:10.4208/cicp.OA-2021-0241
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