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 ?

  • $\begingroup$ Your question is unclear. Have you incorporated boundary conditions in your discretization of the Poisson equation? Are you using restarted GMRES? How big is your Krylov subspace? What preconditioner are you using? Please add details so that someone can help you. $\endgroup$ – smh Jan 28 at 16:09

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