It seems that 'P3dc' element in FreeFem++ does not support third order derivatives in the code. The third order derivative is very necessary for the weak formulation.
Can someone please help me how to overcome these shortcomings?
I don't see why you need a third-order derivative for the biharmonic problem, since by integrating by part twice, you only have second-order derivatives.
Anyway, you can construct a third order derivative by creating first a matrix from P3dc to P1dc which differentiates twice, and then a matrix form P1dc to P0 which differentiate once, and then take the product of these two matrices (you can use the command interpolate for that).
http://www.um.es/freefemv3/ff++/pmwiki.php?n=FFDoc.InterpolationMatrix
When we apply integration by parts to the biharmonic problem with test function from the discontinuous space, it will leave third order edge terms. This third order derivatives is essential for dG formulation (see http://www.ams.org/mathscinet-getitem?mr=2048235). This is why dG formulation with P3 element demands third order derivative in the formulation.
Ok, I understand.
I see two ways of bypassing this problem:
_you can create interpolation matrices and variational matrices as mentioned, but using these matrices are a bit technical to handle.
_introduce an auxiliary variable U in P1dc and its dual V and modify your weak formulation in a way such that Laplacian(U)=u and Laplacian(V)=v and use U,V when you need a third-order derivative. Not sure that it works but worth trying.
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