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M. Faustmann, J. Melenk, D. Praetorius:
"A new proof for existence of $\mathcal{H}$-matrix approximants to the inverse of FEM matrices: the Dirichlet problem for the Laplacian";
in: "ASC Report 51/2012", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2012, ISBN: 978-3-902627-05-6, 1 - 10.

We study the question of approximability of the inverse of the FEM stiffness matrix for the Laplace problem with Dirichlet boundary conditions 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. Unlike prior works [BH03, B¨or10a], our analysis avoids any a priori coupling r = O(| log h|) of r and the mesh width h. Moreover, the techniques
developed can be used to analyze other boundary conditions as well.

http://www.asc.tuwien.ac.at/preprint/2012/asc51x2012.pdf