Approximation of solution operators of elliptic partial differential equations by $${\mathcal{H}}$$ - and $${\mathcal{H}^2}$$ -matrices

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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 representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse $${\mathcal{H}}$$- 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.