Evaluate FEM solution at any point in domain

[Xuefeng Li]

Hi, everyone!

This is a problem related to post-processing. But I can't find a solution from the DataPostprocessor class. Here is the description of my problem.

1. Solve a PDE on a 2D domain [0, 1]X[0, 1] in the x-y plane by FEM with a fixed mesh, say, a 16X16 mesh. The FEM solution is a Vector u=u(x, y). Information of u(x, y) is known if (x, y) is a grid point of the mesh.
2.  I need to solve for a curve x=f(y) defined by u(f(y), y)=f(y), for 0<y<1, where f(0) is known.
3. On paper, we can solve for curve x=f(y) using Newton's method so long as u(x, y) and derivatives of u(x,y) are known for all (x, y) in [0, 1]X[0, 1].
4. Is it possible to evaluate solution u and its derivatives at (x, y) which is not a grid point of the mesh?
5. Ultimately, curve x=f(y) will be approximated by f(0), f(y_1), f(y_2), ..., f(y_{15}), f(1). Here is a related problem. Is there a way to get a 1D projection of a 2D mesh in dealii? In this case, I need the projection of the 16X16 mesh onto the y-axis. 

[Xuefeng Li]

I apologize for hitting the Send button before completing my questions.

I realize that the evaluation of a FEM function is an expensive process due to the fact that we'll need to determine the cell in the mesh where a point belongs before interpolation takes place. That's why I'm just asking for the possibility of existence of such a capability in dealii. 

Your suggestions are greatly appreciated!

Best,

--Xuefeng Li, (504)865-3340(phone)

   Department of Mathematics and Computer Science
   Like floating clouds, the heart rests easy
   Like flowing water, the spirit stays free

      Loyola University New Orleans

   New Orleans, Louisiana (504)865-2051(fax)

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[Praveen C]

May be this does what you are asking for

https://dealii.org/current/doxygen/deal.II/classFunctions_1_1FEFieldFunction.html

best

praveen

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[Peter Munch]

I guess what you need is something like VectorTools::point_values() (https://www.dealii.org/developer/doxygen/deal.II/namespaceVectorTools.html#a4638b2ccdbf190a9b1f6bbbee98a2978) and VectorTools::point_gradients(https://www.dealii.org/developer/doxygen/deal.II/namespaceVectorTools.html#aa96d29e9f7275de7556d6222d8023849). If you need more control, you can take a look at the underlying classes: RemotePointEvaluation (https://www.dealii.org/developer/doxygen/deal.II/classUtilities_1_1MPI_1_1RemotePointEvaluation.html) and FEPointEvaluation (https://www.dealii.org/developer/doxygen/deal.II/classFEPointEvaluation.html). A good starting point might be step-87 and step-89: https://www.dealii.org/developer/doxygen/deal.II/step_87.html and https://www.dealii.org/developer/doxygen/deal.II/step_89.html.

Best,
Peter

[Xuefeng Li]

Thank you, Praveen and Peter. Seems like what you've suggested meets my needs. 

Have a great weekend!

Best,

--Xuefeng Li, (504)865-3340(phone)

   Chair, Department of Mathematics and Computer Science

   Like floating clouds, the heart rests easy
   Like flowing water, the spirit stays free

      Loyola University New Orleans

   New Orleans, Louisiana (504)865-2051(fax)

To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/5a348336-b8dd-4bf3-bf32-0f04b9c21854n%40googlegroups.com.