MiS Preprint Repository

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore they cannot be represented by standard techniques. In this paper, we prove that these matrices can be approximated by $\mathcal{H}$- and ${\mathcal{H}}^2$-matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for $\mathcal{H}$- and ${\mathcal{H}}^2$-matrix approximations of the entire matrices.