Directional $\mathcal{H}^2$ Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation (Communications in Computational Physics, 2022)
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 hypersingularequation 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-frequencyproblems. We introduced the necessary special optimisations to make this algorithmmore efficient in the case of a DG stiffness matrix. Moreover, an automatic parametertuning 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 suitablefor 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.