Approximating inverse FEM matrices on non-uniform meshes with [Formula: see text] -matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse [Formula: see text] -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the [Formula: see text] -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.