We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format introduced in [Hac99]. We show that exponential convergence in the local block rank r can be achieved. We also show that exponentially accurate LU -decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

Abstract. We consider the question of approximating the inverse W = V −1 of the Galerkin stiffness matrix V obtained by discretizing the simple-layer operator V with piecewise constant functions. The block partitioning of W is assumed to satisfy any one of several standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator V. We show that W can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of V.

We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-space format of H-matrices. We show the approximability result for two types of discretizations. The first one is a saddle point formulation, which incorporates the constraint of vanishing mean of the solution. The second discretization is based on a stabilized hyper-singular operator, which leads to symmetric positive definite matrices. In this latter setting, we also show that the hierarchical Cholesky factorization can be approximated at an exponential rate in the block rank. Main Result Notation and settingThroughout this paper, we assume that Ω ⊂ R d , d ∈ {2, 3} is a bounded Lipschitz domain such that Γ := ∂Ω is polygonal (for d = 2) or polyhedral (for d = 3). We assume that Γ is connected.

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We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $${\mathcal{H}}$$ H -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 $${\mathcal{H}}$$ 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.