# Finite elements for biharmonic equation

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:

$$\Delta ^{2}\varphi =f$$

• The problem with $H^2$ conforming finite element spaces is that you need to ensure continuity of not only the function across interelement boundaries but also of the first derivatives. I suggest you read more, e.g., in Brezzi Boffi Fortin Mixed Finite Element Methods and Applications, section 2.2 – VorKir Feb 23 at 5:12