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

Abstract

In this work we analyse a method to construct numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorized preconditioners for matrices with a two-bytwo block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar elliptic problems, however the approach is appropriate also for other types of problems such as parabolic problems or systems of equations. The technique is applicable for both selfadjoint and non-selfadjoint problems, in two as well as in three space dimensions. We analyse in detail the two-dimensional case and provide extensive numerical evidence for the efficiency of the proposed matrix approximations, both serial and parallel. Two-and three-dimensional tests are included.Keywords Sparse approximate inverses · Finite element method · Element-by-element technique · Block preconditioners Mathematics Subject Classifications (2010) 65F08 · 65F10 · 65F50 · 65M60 Communicated by

Preconditioning the Incompressible Navier-Stokes Equations with Variable Viscosity

“…This method is referred to as the element-by-element sparse approximate inverse (EBE-SPAI) technique and can be applied to any finite element matrix, provided that the element matrices are invertible. For more details about the EBE-SPAI method applied to approximate various FEM matrices and the properties of the obtained approximations, see for instance, [24,5,27,21,26]. One can use the diagonal entries of M …”