Finite-element based sparse approximate inverses for block-factorized preconditioners

In this work we analyse a method to construct a numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorised preconditioners for matrices with a two-by-two block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar elliptic problems, however the approach is appropriate for other types of problems such as parabolic or systems of equations. The technique is applicable for both selfadjoint and non-selfadjoint problems, in two as well as in three dimensions. We analyze in detail the 2D case and provide extensive numerical evidence for the efficiency of the proposed matrix approximations, both serial and parallel.