### How do i evaluate the solution of an (elliptic) PDE on a sub-manifold in 3D?

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I would like to use FEniCS to solve a PDE on a 2D manifold embedded in a 3D domain (where i solve a different PDE).

The solution from the (elliptic) PDE in 3D should be used as an input for the computations on the manifold.

For the moment it would be sufficient to model the 2D manifold as planar.

I saw Dr. Rognes presentation (and the accompanying paper) about using FEniCS 1.2 to solve PDEs on a manifold (https://www.researchgate.net/publication/279398104_Automating_the_solution_of_PDEs_on_the_sphere_and_other_manifolds_in_FEniCS_12 ).

I would be really grateful if somebody could point me to some example where a solution from 3D is evaluated on an embedded 2D-manifold.

The solution from the (elliptic) PDE in 3D should be used as an input for the computations on the manifold.

For the moment it would be sufficient to model the 2D manifold as planar.

I saw Dr. Rognes presentation (and the accompanying paper) about using FEniCS 1.2 to solve PDEs on a manifold (https://www.researchgate.net/publication/279398104_Automating_the_solution_of_PDEs_on_the_sphere_and_other_manifolds_in_FEniCS_12 ).

I would be really grateful if somebody could point me to some example where a solution from 3D is evaluated on an embedded 2D-manifold.

Community: FEniCS Project

### 1 Answer

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I had experimented with this idea at one point... My solution involved extracting the "BoundaryMesh" submesh of the exterior, then using "LagrangeInterpolator" to put the 3D variables on the 2D mesh. After this, the DEq's can be solved via standard 2D Galerkin methods.

https://fenicsproject.org/olddocs/dolfin/1.6.0/python/programmers-reference/cpp/function/LagrangeInterpolator.html

https://fenicsproject.org/olddocs/dolfin/1.5.0/python/programmers-reference/cpp/mesh/BoundaryMesh.html

https://fenicsproject.org/olddocs/dolfin/1.6.0/python/programmers-reference/cpp/function/LagrangeInterpolator.html

https://fenicsproject.org/olddocs/dolfin/1.5.0/python/programmers-reference/cpp/mesh/BoundaryMesh.html

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