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Hi.<br>
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As Christian said this doesn't look like a too difficult example (except the corner case g(u)=|grad(u)| which will cause problems with the non linear solver).<br>
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So, I am wondering which problems you encountered with FreeFem since I would except any standard PDE software package should at least manage the case g(u)=u^2. Are there any additional details about your problem that caused the problems? Is it the a-posteriori
adaptivity that made things difficult or the DG formulation of the problem?</div>
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Perhaps have a look at<br>
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<a href="https://dune-project.org/sphinx/content/sphinx/dune-fem/discontinuousgalerkin_nb.html">https://dune-project.org/sphinx/content/sphinx/dune-fem/discontinuousgalerkin_nb.html</a></div>
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That's DG for Laplace but adding nonlinearities should be in general straightforward like in</div>
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<a href="https://dune-project.org/sphinx/content/sphinx/dune-fem/dune-fempy_nb.html">https://dune-project.org/sphinx/content/sphinx/dune-fem/dune-fempy_nb.html</a></div>
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Best</div>
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Andreas<br>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Dune <dune-bounces@lists.dune-project.org> on behalf of Christian Engwer <christian.engwer@uni-muenster.de><br>
<b>Sent:</b> 26 June 2021 14:39<br>
<b>To:</b> Hyun-Geun Shin <shgmath@gmail.com><br>
<b>Cc:</b> dune@lists.dune-project.org <dune@lists.dune-project.org><br>
<b>Subject:</b> Re: [Dune] A question about solving my problem</font>
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<div class="PlainText">Dear Hyun-Geun,<br>
<br>
> My problem is to solve a PDE like -div(g(u) * grad(u)) = f, discretized by<br>
> discontinuous Galerkin methods. Here, u is the solution, g(u) is a nonlinear<br>
> function, and f is a load function. So, g(u) can be “abs(grad(u))” or u^2.<br>
<br>
So basically you are trying to solve a non-linear diffusion<br>
problem. This is perfectly possible.<br>
<br>
Although it is not part of the usual examples the modifications<br>
regarding the typical poisson problem are not too big. You should be<br>
able to implement this relatively easily with a discretization module<br>
like dune-pdelab, which I'm using, but also dune-fem should provide<br>
the necessary flexibility. The third option would be to do thinkgs a<br>
bit more by hand and use the functions spaces from dune-functions.<br>
<br>
Regarding the non-linear solver, a standard Newton method, together<br>
with a Jacobian computed by numerical differentiation should work in<br>
most cases. Still, depending on the type on non-linearity, you might<br>
encounter convergecne issues (which you can still ignore if you don't<br>
care about efficiency).<br>
<br>
Ciao<br>
Christian<br>
<br>
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