Timeline for Projecting Finite Element solution onto new mesh
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2021 at 2:01 | comment | added | Nick Alger | @WolfgangBangerth thanks! I will take a look at your paper, and see if I can find the paper by the petsc people | |
| Jul 29, 2021 at 0:38 | comment | added | Wolfgang Bangerth | @NickAlger I've got a paper with Rene Gassmoeller and others about particle methods that talks about these sorts of issues. I also thought that the PETSc people had written a paper on interpolation from one mesh to another, but I can't find it on Matt Knepley's web site. Maybe it was a different set of authors, I just don't recall. | |
| Jul 28, 2021 at 20:18 | comment | added | Nick Alger | @WolfgangBangerth Are you aware of a good reference that explains the computational difficulty of evaluating a finite element function at a large number of arbitrary points? Clearly this is true for the reasons you have explained, and I have observed it myself. But I would like a reference to cite in a paper I am writing. | |
| Jan 16, 2015 at 14:20 | comment | added | Wolfgang Bangerth | @BillGreene: In deal.II, we typically try very hard to avoid this kind of operation. Interpolating between unrelated meshes is very expensive because you need to evaluate $u_1$ at the quadrature points of mesh ${\cal T}_2$ and that implies searching which cell of ${\cal T}_1$ a particular quadrature point lies in. This is expensive. You also lose a significant amount of accuracy if you do it frequently. So what we try very hard is to ensure that the meshes ${\cal T}_{1,2}$ are related: e.g,. that they each result by refinement from a common mesh (i.e., they are hierarchically related). | |
| Jan 16, 2015 at 14:17 | comment | added | Wolfgang Bangerth | @Eff: No. What you are asking is whether the interpolation (which is what you get if you just take nodal values) is the same as the projection (which is what I describe and you asked about in the original post). The answer is no. They're, in general, not the same. I also have that $\|Pu_1-u_1\| \le \|Iu_1-u_1\|$ where $Pu_1$ is the projection and $Iu_1$ is the interpolation of $u_1$ onto mesh 2. So the projection's error is better. On the other hand, I can show that $\|Pu_1-Iu_1\|\le Ch^2$, i.e., they are in a sense "close". | |
| Jan 15, 2015 at 18:12 | comment | added | Bill Greene | Professor Bangerth: Is this the approach you use in deal.II? I know that deal.II supports h-adaptivity and nonlinear problems so I'm assuming this type of interpolation would be needed in that context. Thanks. | |
| Jan 15, 2015 at 17:35 | vote | accept | Eff | ||
| Jan 15, 2015 at 17:22 | comment | added | Eff | (+1) Thanks a lot for the answer, I will see if I can make this general implementation. Do you know what happens if we simplify the problem to the following: Let's say that, in the mesh $\mathcal{T}_2$ that we are to project onto, that all vertices also are vertices of the initial mesh $\mathcal{T}_1$ (However, there would be other vertices in the initial mesh that would not be in $\mathcal{T}_2$). Would the solution that minimizes the norm simply be given by the series expansion with coefficients of each vertex equal to the initial solution's coefficients at those vertices? If you understand. | |
| Jan 15, 2015 at 12:51 | history | answered | Wolfgang Bangerth | CC BY-SA 3.0 |