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 - <p style="text-align: justify;">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.</p>