I'm trying to solve semi-discrete equation on unstructured grids by implicit method.

Is there any good way to efficiently inverse matrix produced by implicit time integration?

I'm wonder if it is possible to use a factorisation method used in structured grids.

To solve your systems of equations, I strongly suggest using iterative methods such as the conjugate gradient or GMRES. If the system is well conditionned and the main diagonal is dominant you can even use Gauss-Siedel or SOR. The directs methods are very unefficient (time and memory cost), even for discrete equations obtained from structured grids.

The direct methods are actually not that bad in two-dimensions. In three dimensions, the fill-in is much too high in (LU) factorization.

In 2-dimensions, direct methods are helpful if the (structured) grid has very few points along one direction compared to the other.

I agree that for very-very ill conditioned system in 2D that a direct method can more efficient than the iterative solver for time cost.

Dear friend,

If you problem is well posed & formulated u will get a good coefficient matrix. My experience shows Gauss-Siedel/SOR works ok. But sometin like CG/GMRES will give better convergence. Also something not often stated in paper/books. U will need to reorder your mesh ( I mean renumber cells) before u can implement Implicit schemes. If re-ordering is not done your formulation may not be Implicit.You will then face stability problems.

bye

Abhijit Tilak

The main problem with all iterative techniques is that they are particularily bad at smoothing long wavelength errors. To help circumvent this, many people use multigrid acceleration. Additionally, direct solutions are definitely not a good idea on a 3D unstructured mesh (even 2D sounds pretty bad to me).

In the end though, it depends what you want to do, if you are doing transient calculations, maybe something like GMRES is better. I'm not sure though as I have never bothered to figure out the plan with it. Certainly multigrid is a proven technology at this point though, and it just makes so much sense.

Dan.