$$\mathcal {H}$$ H -matrix approximability of the inverses of FEM matrices

Abstract

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.

Iterative Solution of Large Sparse Systems of Equations

“…It is shown in Bebendorf-Hackbusch [39], Faustmann-Melenk-Praetorius [129], and Faustmann [128] that the inverse matrix can be well approximated by the format H(r, P ). It is shown in Bebendorf-Hackbusch [39], Faustmann-Melenk-Praetorius [129], and Faustmann [128] that the inverse matrix can be well approximated by the format H(r, P ).…”

“…Bebendorf-Hackbusch [39], Faustmann [128], Faustmann-Melenk-Praetorius [129], and [198, §11.3]). In the case of the inversion of a sparse finite element matrix, the inverse is implicitly connected with an integral operator using the Green function G(x, y) instead of σ.…”

Survey on the Technique of Hierarchical Matrices

“…This result is remarkable since it holds for the Green function G(x, y) of differential operators div(C(x) grad) with a nonsmooth coefficient matrix C ∈ L ∞ ( ) (cf. Bebendorf-Hackbusch [4], FaustmannMelenk-Praetorius [12], and [26, Section 11.3]). These results can be transferred to the LU decomposition, i.e., also the factors L and U are well approximated by hierarchical triangular matrices in H(r, P ) (cf.…”

${\mathscr{H}}$-matrix approximability of inverses of discretizations of the fractional Laplacian

“…• Related to H-matrices is the format of H 2 -matrices discussed in [31,8,30,7]. Using the techniques employed in [7,17,18,19], one may also show that A −1 can be approximated by H 2 -matrices at an exponential rate in the block rank.…”