I am a graduate student majoring scientific computing. The numeric model I made caused a very ugly-looking saddle-point linear system. It is not symmetric at all and I will attach the sparsity pattern below. So far, I've only dealt with matrices with non-zero diagonal entries, so incomplete LU-type preconditioner with GMRES worked fine. But GMRES tends to converge slowly on this one and even worse, I cannot use any LU type preconditioners.
Could you recommend an iterative linear solver for this problem?