I have a sparse $5\,656\,236 * 5\,656\,236$ matrix $A$ with $166\,526\,888$ non-zero elements. The matrix comes from using the finite element method on a linear elasticity problem and is positive semi-definite. I'm trying to use the preconditioned conjugate gradient method to solve it, particularly, the pcg() function in MATLAB.
Can I expect this to converge at all? I tried
L = ichol(A); [u,flag,res,iter,resvec] = pcg(A,F,1e-6,max_ter,L,L');
But it does not seem to converge at all. The residual starts by increasing a lot and after 2000 iterations (which takes some hours) the relative residual is up at $10^4$.
So I guess my question is, how would you attack a system of this size? I'm sure there are a lot of ways to lure the method into converging faster. Any hints and tips are welcome!