# Iterative solution of ill-conditioned matrix systems

I want to solve a matrix system of the form $$Ax=b$$ where $$A$$ is ill-conditioned. The matrix system comes from a structural simulation problem which was discretized using finite elements. I do not have access to the finite element model. I know that proper boundary conditions improve the condition number of the matrix but this should not be the point here.

The matrix system should be solved using an iterative method like CG. What can I do numerically to improve the convergence speed and the accuracy of the solution or even get a useful solution?

• any particular reason why only iterative techniques are considered? – Anton Menshov Aug 12 '19 at 22:31
• The matrix systems are quite large. They can have billions of unknowns which makes it hard for direct methods to get a solution in a reasonable time. – vydesaster Aug 12 '19 at 22:37
• In this post you can find references to multigrid technique and polynomial precondition. – Mauro Vanzetto Aug 15 '19 at 14:28
• I already tried multigrid methods but they do not work very well for matrices which arise from FE structural problems and are ill conditioned (condition number about 1e12). The null space seems to play a big role here but I have not yet found an algorithm which shows good results. – vydesaster Aug 16 '19 at 20:28