Approximation of solution operators of elliptic partial differential equations by ℋ- and ℋ2-matrices

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 Hand H2-matrices. The key results are existence proofs for local lowrank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for Hand H2-matrix approximations of the entire matrices.