Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
4. Preconditioners for Elliptic Linear Systems of Equations
 The linear systems that arise in the solution of e.g. the Poisson equation have the following properties
  • sparse,  non-symmetric coefficient matrix
  • condition numbers grows  like 4^L with the maximal level of refinement L
  • the transpose of the coefficient matrix is not easily available
Therefore, preconditioned BiCGStab,BiCGStab(l) or GMRES solvers are used.  In [4,6,7] two preconditioners were analysed: 
simple diagonal scaling and a combination of diagonal scaling and a basis transform to so-called Lifting-Wavelets. 
Simple diagonal scaling works fine for 4th and higher order  wavelets; while the Lifting-preconditioner yields almost level-independent condition numbers for all wavelets. The figures below show convergence histories for a Poisson problem on a regular sparse grids of different size with the low order Hierarchical basis. Left: the results for the diagonal scaling. Right: the Lifting-preconditioner.