To solve Biharmonic equation

Question

As said in documentation for Biharmonic equation, two ways avialable to solve the biharmonic equation using Lagrange finite element basis function:
1) spilt to two second-order equations
2) "a variational formulation can be constructed that imposes weak continuity of normal derivatives between finite element cells. "

My question is,
For a problem like below,
$$ \nabla^{4} u = f \quad {\rm in} \ \Omega $$
$$\begin{split}u &= 0 \quad {\rm on} \ \partial\Omega, \ \nabla^{2} u &= 0 \quad {\rm on} \ \partial\Omega\end{split}$$

Why can't I just use the code below to solve the problem
$$a(u, v) = \int \nabla^{2} u \nabla^{2} v \, {\rm d}x,
\
L(v) = \int_{\Omega} fv \, {\rm d}x$$

a = inner(div(grad(u)), div(grad(v)))*dx
L = f*v*dx
solve(a == L, u, bc)

Thank you, appreciate your help.

Li

Answer 1

The variational form you've written requires finite element basis functions with continous derivatives across facets. There are several such elements in two dimensions, for example the Argyris and Bell triangles, but they are not supported in FEniCS.

Comments

Thank you Magne. For who might have the same question as me, here is what I found:
The integration weak form of biharmonic equation has second order derivative, to satisfy the compatibility, patch trial functions must be C1 continuous between interconnected elements. In this case Hermite interpolation is necessary, but it's not supported in Fenics. So it is not possible to solve this problem directly. However, the code below has given a solution for this. Navier plate fenics code.
Also, if you are dealing with plate bending problem, Mindlin theory is antoher solution.