[Dune] A question about solving my problem

[Christian Engwer]

Dear Hyun-Geun,

> My problem is to solve a PDE like -div(g(u) * grad(u)) = f, discretized by
> discontinuous Galerkin methods. Here, u is the solution, g(u) is a nonlinear
> function, and f is a load function. So, g(u) can be “abs(grad(u))” or u^2.

So basically you are trying to solve a non-linear diffusion
problem. This is perfectly possible.

Although it is not part of the usual examples the modifications
regarding the typical poisson problem are not too big. You should be
able to implement this relatively easily with a discretization module
like dune-pdelab, which I'm using, but also dune-fem should provide
the necessary flexibility. The third option would be to do thinkgs a
bit more by hand and use the functions spaces from dune-functions.

Regarding the non-linear solver, a standard Newton method, together
with a Jacobian computed by numerical differentiation should work in
most cases. Still, depending on the type on non-linearity, you might
encounter convergecne issues (which you can still ignore if you don't
care about efficiency).

Ciao
Christian