Adaptive mesh refinement and solution transfer

[Yang Liu]

Dear DEAL.II developers,

I am trying to solve a Laplace problem and refine the mesh adaptively. I am using a CG solver with a symmetric Multigrid preconditioner. 

I plan to transfer the solution on the initial mesh to the refined mesh, and then use the interpolated solution as the initial guess in the iterative solver of the fine mesh. 

But the iterative solver on the fine mesh diverges. In fact, the iterative solver always diverges unless the initial guess is small (less than 1e-7). 

I wonder if condense the hanging node constraints in the system assembly is the correct way to go. 

If needed, I can attach a smallest working example. 

Best Regards,

Yang

[Wolfgang Bangerth]

Yang:

> I am trying to solve a Laplace problem and refine the mesh adaptively. I
> am using a CG solver with a symmetric Multigrid preconditioner.
>
> I plan to transfer the solution on the initial mesh to the refined mesh,
> and then use the interpolated solution as the initial guess in the
> iterative solver of the fine mesh.

You mean as the initial guess for the linear solve on the fine mesh?
That sounds like a good idea, but experience shows that in practice it
only reduces the number of CG iterations slightly and that that does not
offset the cost of transferring the solution.

The situation is different for nonlinear problems, where having a good
initial guess is very important.

But that wasn't your real question :-)

> But the iterative solver on the fine mesh diverges. In fact, the
> iterative solver always diverges unless the initial guess is small (less
> than 1e-7).
>
> I wonder if condense the hanging node constraints in the system assembly
> is the correct way to go.

That sounds strange to me. Iterative methods such as CG are supposed to
converge from any starting point, and whether you use a zero vector or
one that was obtained by interpolation from a coarse grid shouldn't
matter. What happens if you start with a randomly chosen vector?

To test whether hanging node constraints are important, you might want
to try out your strategy when using global mesh refinement. There are no
hanging node constraints in that case.

You probably also want to output the starting vector you have via
DataOut to see whether it looks as you expect it to look like.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/