I have a large sparse linear system generated as a part of PDE solution for flows in the form Ax=b. The condition number of matrix A is very bad - of the order 3000!. But I get expected solutions with direct solvers. So, now I want to precondition the matrix so that I can use iterative solvers and use the sparseness. I have tried Jacobi preconditioner, but it does not work well as the matrix is not diagonally dominant. I need some help in proceeding further:
1) Imagine I get an approximate solution for x (generated by one run of biconjugate gradient solver). Now can I get "inverse of A" (for preconditioning) from this, seems like it must be possible but I am unable to figure out how! i.e knowing x and b can I calculate the A inverse (which may be used as preconditioner!).
2) Any other way of preconditioning which you feel would be worth a try?
3) Any way to circumvent pre-conditioning for iterative schemes for bad condition number systems?
Thanks a lot in advance for any help. Any comments are welcome.