I am working on the flow stability problem. In this work the main complication is solving generalized eigenvalue problem for a large scale Non-Hermitian matrix. I need only one eigenvalue (most left on real axis), so I used shift-invert transformation. I used SLEPc library to solve this problem, especially Krylov-Shur method with MUMPS LU decomposition. It works fine for small problem (matrix size up to 500k). But real cases give me matrix size about 15M, so I just do not have such amount of memory to solve these cases with MUMPS. I tried to find an iterative method for my problem (method with small memory requirement), but they do not work for me. Generalized Davidson and Jacobi-Davidson are not converged. Also I tried iterative linear solver instead of direct MUMPS for Krylov-Shur method, but they also are not converged. I concentrated on one relatively small problem (about 70k matrix size) that solved fine using MUMPS and cannot find any iterative method to solve it.
Is it possible that iterative eigenvalue solver (or solver with small memory requirement) does not exist for my matrix? I will very appreciate if someone gives me an advice what I can do more to solve my problem.