I have a question regarding the conforming finite elements for biharmonic equation: if we want to discretize the weak formulation for this problem using a conforming finite element space $V^h$, is it possible to do it using Lagrange elements? If this is possible, what is the minimal required order of the Lagrange elements?
When I obtained the weak formulation, I noticed the function space should be $H^2$ with compact support if we are given homogeneous Dirichlet boundary conditions. By the embedding Sobolev theorem, this means the solution should be in $C^1$ but I don't really get if we can use the Lagrangian polynomials for the discretised problem. The biharmonic equation is:
$$\Delta ^{2}\varphi =f $$
Thanks in advance
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