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I am a beginner in adaptive mesh refinement (AMR). After I am done with the first two papers by Dr. Marsha Berger, I was trying to write my own code for a problem which has a parabolic partial differential equation. In this regard, I wish to know if I could use the same trick as described by Dr. Berger to this problem? Because she dealt with hyperbolic PDEs.

I am trying to simulate a two dimensional heat conduction problem of which governing equation is just a Laplace equation and temperature at a nodal point average of neighboring four nodal points.

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    $\begingroup$ What papers? There are thousands of AMR papers and probably dozens by Dr. Berger. There also also hundreds if not thousands of AMR papers for parabolic and elliptic problems, so why not go look for them? $\endgroup$ – Bill Barth May 19 '15 at 12:07
  • $\begingroup$ Indeed. Specifically, what "trick described by Dr. Berger" do you refer to? $\endgroup$ – Wolfgang Bangerth May 20 '15 at 5:36
  • $\begingroup$ @BillBarth I was referring to the papers published in 1984 and 1989 which are relatively more famous and lay the basic groundwork of AMR. I was just wondering if I could use exactly the same "ideology" for parabolic PDE. I know there would be a lot of papers on parabolic PDE particularly but I mistakenly wrote my code using the Berger-Oliger algorithm and it did not seem to run, so I just wanted to know if the same algorithm is suitable for parabolic PDE or not. $\endgroup$ – Tanmay Agrawal May 20 '15 at 13:12
  • $\begingroup$ Maybe you can write down what you tried and what went wrong? It's easier for us if we don't have to go read two full papers. $\endgroup$ – Bill Barth May 20 '15 at 13:13
  • $\begingroup$ @WolfgangBangerth Professor, I meant the idea of adaptive mesh. Trick should not be a good word. I am estimating error using the procedure mentioned in the paper of 1989 (Local adaptive mesh refinement for shock hydrodynamics) and I am using a very simple method to generate the new grids and using the same difference equation on fine grid and putting the solution back to coarser grid. Somehow my code did not seem to work well, that is why I asked if I could use her algorithm for parabolic PDE. $\endgroup$ – Tanmay Agrawal May 20 '15 at 13:15
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Yes, adaptivity is usable for parabolic problems. We put together an example here: https://www.dealii.org/8.2.1/doxygen/deal.II/step_26.html . Take a look at the results section in particular.

I discuss this program in lectures 29 and 30 here: http://www.math.tamu.edu/~bangerth/videos.html .

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  • $\begingroup$ 1/2 Thank you Professor for this help. I am using finite differences for my AMR and therefore I at least can get some idea from the FEM based formulation. One more thing I would like to know: Suppose if I just have the coarsest mesh and I have just begun the code. After say 4 steps, I made a finer mesh on top of it. For this finer mesh, I calculated initial values from the coarse mesh and solved my PDE on this (say twice if refinement is twice). Then I get the updated value of coarse grid points using this fine mesh. $\endgroup$ – Tanmay Agrawal May 21 '15 at 12:06
  • $\begingroup$ 2/2 As per me, I should have both coarse and fine mesh in my solution. If I want to fetch this solution, how can I do so? I have different arrays for the coarse and fine mesh and their size is different as well. May you please let me know if anything is wrong in what I have said? $\endgroup$ – Tanmay Agrawal May 21 '15 at 12:08
  • $\begingroup$ What you say seems reasonable. You need to interpolate the solution from the coarse mesh to the finer mesh (sometimes this operation is also called "prolongation"). $\endgroup$ – Wolfgang Bangerth May 21 '15 at 16:06

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