# Approximating inverse FEM matrices on non-uniform meshes with H-matrice

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse H-matrix format. For a large class of shape regular but possibly non-uniform meshes including graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the H-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.

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