Searching for a good preconditioner: FEM-based sparse approximations of matrix blocks

The task to solve sparse linear systems of algebraic equations using a fast, computer-resources-efficient and robust computational method arises within various application settings and remains central to many numerical simulations in Scientific Computing. The ever increasing scale of these linear systems entails the necessity to use iterative solution methods and, thus, the need for efficient preconditioners.

Constructing good preconditioners has been a topic of intensive research and implementation over the last thirty years and has still not lost its importance. Despite the significant success in development of general algebraic preconditioning techniques, it still remains true that most efficient preconditioners are those, where in the construction we incorporate more information than what is contained in the system matrix only, i.e. if the preconditioners are more 'problem-aware'.

In the talk, we discuss possibilities to construct efficient two-by-two block factorized preconditioners for general matrices, arising from finite element discretizations of scalar or vector partial differential equations. The finite element method offers the possibility to work on a local, element level, where the corresponding matrices and vectors have small dimensions and even exact matrix computations are feasible to perform. Utilizing the latter we are able to obtain approximations of matrix blocks by assembly of matrices of small dimension, obtained by manipulation of local finite element matrices. In particular, within the construction phase we compute simultaneously sparse approximations of a matrix inverse, a Schur complement matrix and a matrix product.

We show some theoretical estimates for the quality of these approximations. We also illustrate the efficiency of the preconditioner in standard finite element and in adaptive finite element setting on matrices arising from various applications, among which for problems with discontinuous coefficients, convection-diffusion, parabolic problems and phase-field models, such as binary alloy solidification and wetting effects.