Iterative solution of sparse linear systems

• Such systems arise from discretization of partial differential equations with the finite element method.
• The size of the linear system is large, thus the storage requirements for direct solution are huge and the solution time might also be too long.
• There are lot of unsolved problems with the iterative linear solvers like:
  • The iteration process can be parameter dependent.
    As an example the pinched cylindrical shell is analysed. The behaviour of the IC(0)-preconditioned conjugate gradient method depends strongly on the relative thickness of the shell. One octant of the shell is discretized by using quadrilateral stabilized MITC4 elements with drilling degrees of freedom in a uniform 30x30 mesh. The mesh and the convergence behaviour are shown below for three different values of the relative thickness: t/R = 0.1, 0.01, 0.001. The problem gets harder when the thickness gets smaller. This small problem has 5489 unknowns.
    The crucial point is the preconditioning technique, see a survey by Michele Benzi.
    

Collaborators

Main publications

• M. Benzi, R. Kouhia and M. Tůma; Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics, Computer Methods in Applied Mechanics and Engineering, 190, 2001, No 49-50, pp. 6533-6554. URL:http://dx.doi.org/10.1016/S0045-7825(01)00235-3
• M. Benzi, R. Kouhia and M. Tůma; An assesment of some preconditioning techniques in shell problems, Communications in Numerical Methods in Engineering, 14, 897--906, No 10, 1998. URL:http://dx.doi.org/10.1002/(SICI)1099-0887(1998100)14:10<897::AID-CNM196>3.0.CO;2-L

Some links

• Color plot of matrices
• MatrixMarket
• Tim Davis: University of Florida Sparse Matrix collection
  -- Just have a look on the beautifull pictures even though you are not interested in sparse matrices