MiS Preprint Repository

This article deals with the existence of blockwise low-rank approximants --- so-called $\mathcal{H}$-matrices --- to inverses of FEM matrices in the case of uniformly elliptic operators with $L^{\mathrm{\infty }}$-coefficients. Unlike operators arising from boundary element methods for which the $\mathcal{H}$-matrix theory has been extensively developed, the inverses of these operators do not benefit from the smoothness of the kernel function. However, it will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses. Numerical examples confirm the correctness of our estimates. As a side-product we analyse the $\mathcal{H}$-matrix property of the inverse of the FE mass matrix.