I have a question about the precision of the GMRES algorith and its variation a s a function of the size of the Krylov subspace. I want to solve a Poisson equation using a spectral method. My problem looks like this $$\nabla^2f+g\nabla f=h$$ So in my three dimensional case I apply the GMRES algorithm to the problem $$Af=h$$ where $A=\nabla^2 +g\nabla$. In this case I do not construct the matrix directly (I work in three dimesnion and the size of the problems are very important), but I compute only the effect of $A$, using preconditioning.
When I chose the dimension of the Krylov subspace too large I start to have the norm of the vector very small until I get a $NaN$.
The problem is that it is not an indication of the convergence of the method cause the residuals are still very large.
Moreover in my case small problem (100^3 points)"accept" a bigger Krylov base then bigger case (500^3 -1000^3 points).
Precision: if I use a "good " base (one that do not give $NaN$) and I restart the algorithm, the residuals decrease, even if very slowly.
So the question is, it this something that should happen ? or do you think simply that there is a bug somewhere ?