Existence of $\mathcal{H}$-matrix approximants to the inverses of BEM matrices: the simple-layer operator

Abstract

We consider the question of approximating the inverse $\mathbf{W}={\mathbf{V}}^{-1}$ of the Galerkin stiffness matrix $\mathbf{V}$ obtained by discretizing the simple-layer operator $V$ with piecewise constant functions. The block partitioning of $\mathbf{W}$ is assumed to satisfy any of the standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator $\mathbf{V}$. We show that $\mathbf{W}$ can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of $\mathbf{V}$.

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