Maya Neytcheva
Division of Scientific Computing
Department of Information Technology
Uppsala University
In this 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 work to be presented is based on two milestones.
The first one is the elegancy of the finite element method (FEM) which is not only a powerful discretization tool but also offers the possibility to perform much analysis on a local, element level.
There, since the corresponding matrices and vectors have small dimensions, even exact and symbolic computations are
feasible to perform. Furthermore, adaptive finite element methods (AFEM) have undergone a substantial development in theory and implementation, offering the advantage to reduce the required degrees of freedom by 'zooming' only in those parts of the problem domain where it turns out to be necessary.
The second milestone is the ever increasing scale of linear systems to be solved within various application settings, entailing the use of 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-dependent'.
We will consider a class of of potentially very powerful algebraically constructed preconditioners, based on a presumed block two-by-two structure of the matrix of the linear system of equations we want to solve. We will show techniques to construct sparse approximations of matrix blocks needed in that context and the behaviour of the resulting preconditioner will be illustrated on a number of test problems, such as elliptic problems with discontinuous coefficients, parabolic problems and systems of equations as arising from discretizing the so-called Cahn-Hilliard equation.