Second derivative boundary conditions

variational formulation
1

Hello,
I want to solve the biharmonic equation where the boundary conditions are the second derivetives (d^2/dx^2 for example). Does anyone know how to do that or gve me a reference?

Thank you!

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See: https://fenicsproject.org/olddocs/dolfin/latest/python/demos/biharmonic/demo_biharmonic.py.html?highlight=biharmonic

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Thank you for your response. From what I understand, a(u,v) is composed of two elements: the first is equivalent to the biharmonic equation, and the second is error control. The boundary condition on the Laplacian of the function is used in omitting another boundary term.
I have two questions:

  1. I don’t want the Laplacian of the function to vanish on the boundary, but I want a specific second derivative to have a particular value (d^2/dx^2=c, d^2/dy^2 unknown) on one boundary (while the others are free). Do you know how can I do that?
  2. I haven’t seen the error control term in other examples. Why is it needed here?

Thank you

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The standard finite element formulation of the biharmonic problem seeks a solution which (roughly speaking) has to be C^1 continuous. There are some standard finite elements which conform with this requirement, e.g., DefElement: Argyris and DefElement: Hermite. However, these elements can be a pain to implement. In fact these elements are not supported by legacy DOLFIN nor (as far as I’m aware as of Sept 2023, but I believe there’s some ambition to get them supported in the future) DOLFINx.

Another technique is to exploit discontinuous Galerkin (DG) methods. A subset of DG methods employ interior penalty Galerkin (IPG) formulations. Employing an IPG method you can use a discontinuous basis to weakly enforce C^1 continuity of the FE solution. Although the resulting formulation is verbose and tricky to implement, you don’t have to worry about complicated finite element bases.

Nonconforming FE methods like DG methods; however, are typically more expensive than their conforming, i.e., continuous Galerkin (CG), counterparts. We can exploit CG methods with a DG formulation for the biharmonic problem since we can strongly enforce the C^0 continuity with a CG basis, and weakly enforce C^1 continuity with the DG FE formulation terms which do not vanish as a consequence. The result is the formulation in the demo referenced.

The imposition of \nabla^2 u = g_{1} where g_{1} is the boundary data in this case follows naturally from the DG FE formulation.

See, for example, Engel (2002), Georgoulis (2009).

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