Problems with Direct Solvers
 

I am a college student, currently taking a CFD course. Today we talked about the differences between Gaussian and iterative solvers.

I understand that Gaussian methods are less efficient, but my professor claims that you cannot use them for linear algebraic equations resulting from discretization of the flow equations.

His reasoning is that even if you have a matrix that includes elements that are zero, the upper triangular matrix after the forward elimination step will have all non-zero elements.

I don't understand why this is bad? Isn't that always the case in Gaussian elimination? I tried talking to him after class, but I still couldn't understand his rationale. I feel like he explained something wrong...

Thanks for any help.

maybe the discussion is about the pressure solver? the resulting matrix of the system is singular

The matrices resulting from discretizations are usually very big (depending on your mesh). In fact, they can be so big, that even storing only the nonzero entries can become a problem (although they are sparse).

This is why, storing these matrices as a 2d-array is just not feasible for non-toy problems.