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B. N. Khoromskij

Data-sparse elliptic operator inverse based on explicit approximation to the Green function

Keywords: Elliptic equations, BEM, FEM, data-sparse approximate inverse, hierarchical matrices

A class of data-sparse hierarchical matrices (?-matrices) allows to approximate nonlocal (integral) operators with almost linear complexity [22 – 31]. In the present paper, a method is described for an explicit?-matrix approximation to the inverse of an elliptic differential operator with piecewise constant coefficients in ?d . Our approach is based on the additive splitting to the corresponding Green function, which leads to the sum of an ?-matrix and certain correction term including the product of data-sparse matrices of different hierarchical formats. In the case of jumping coefficients with respect to conformal domain decomposition, the approximate inverse operator is obtained as a direct sum of local inverses over subdomains and the Schur complement inverse on the interface. As a by-product, we obtain an explicit approximate inverse preconditioner with the data sparsity inherited from the ?-matrix format.

Journal of Numerical Mathematics, Walter de Gruyter

Print ISSN: 1570-2820
Volume: 11, 06/2003
Pages: 135 - 162

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