Weak formulation of biharmonic equation

Question

I am trying to solve the biharmonic equation in FEniCS, specifically the bending of a Kirchhoff-Love plate. My boundary conditions are those which lead to the Navier solution, namely w=0 and d2w/dn2 = 0. I found the following dolfin demo:

http://fenicsproject.org/documentation/dolfin/1.4.0/python/demo/documented/biharmonic/python/documentation.html

However, I get completely lost at the very first step: "a weak formulation of the biharmonic reads". There are no hints on how such an equation has been developed, could anyone point some references where this process can be followed?

Besides, what would be the different between this approach and simply solving the harmonic equation twice?

Comments

This approach is an interior penalty scheme for the biharmonic equation. It allows you to solve the problem in spaces which are globally only H^1-conforming. You can derive it - like it is common for DG approaches - by element-wise integration by parts, adding consistent terms, and making some rearrangements. I think the description in the example is sufficient to get the idea. Otherwise, you can google for interior penalty schemes for biharmonic equations.

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Great, thanks for the keywords!

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Christian: could you post your 'comment' as an 'answer'?

Answer 1

As Christian Waluga pointed out, looking for "penalty methods" showed a couple of interesting papers. For example, "C0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains" http://link.springer.com/article/10.1007%2Fs10915-004-4135-7.

Note for the devs: For me, this was not obvious at all when reading the demo and I still consider some sort of citation would be a nice addition.

Comments

I agree with Juan. If you're not familiar with penalty methods, the weak formulation in the tutorial is quite confusing.