Hierarchical matrix approximation to Green's function via boundary concentrated FEM

Abstract

In the preceding paper [24], a method is described for an explicit hierarchical (ℋ-matrix) approximation to the inverse of an elliptic differential operator with piecewise constant/smooth coefficients in ℝ d . In the present paper, we proceed with the ℋ-matrix approximation to the Green function. Here, it is represented by a sum of an ℋ-matrix and certain correction term including the product of data-sparse matrices of hierarchical formats based on the so-called boundary concentrated FEM [26]. In the case of jumping coefficients with respect to non-overlapping 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 corresponding to the boundary concentrated FEM. Our Schur complement matrix provides the cheap spectrally equivalent preconditioner to the conventional interface operator arising in the iterative substructuring methods by piecewise linear finite elements.