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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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  • $\begingroup$ Please give references for biharmonic equation, which is not very classical. $\endgroup$ – Jean Marie Feb 20 at 21:24
  • $\begingroup$ Thanks, I put the equation $\endgroup$ – chicamate Feb 21 at 15:53
  • $\begingroup$ 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 $\endgroup$ – VorKir Feb 23 at 5:12
  • $\begingroup$ For this reason many people prefer to use nonconforming f.e. for the biharmonic equation $\endgroup$ – VorKir Feb 23 at 5:18
  • $\begingroup$ Thanks for your answer. However, does this mean there is no possibility for using piecewise Lagrangian elements? $\endgroup$ – chicamate Feb 23 at 12:36

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