# Spectral/hp-finite elements for 4th order PDEs

Does anyone know of references discussing the solution of 4th order PDEs by way of spectral/hp-finite element methods? Specifically, I'm interested in the extension of the spectral element method, which uses high-order Lagrange polynomial interpolants (at a suitable distribution of nodes) as shape functions. In this case, the resulting finite elements are C0 continuous, which is sufficient for 2nd order PDEs.

For fourth order PDEs, C1 continuous finite elements are required. Hence, cubic (or quintic) Hermite interpolation is typically used to construct the shape functions, as they produce finite elements with this property. I'm interested in how one might best construct higher order "spectral" elements that are C1 continuous and have the fast convergence properties associated with the spectral element method.

I'm aware that an alternative approach would be to split the 4th order PDE into two 2nd order PDEs. However, I think this would require solving for the 2nd derivative of the unknown, which may become infinite at points on the boundary in my case. Thus, I'm interested in methods where I need only solve for the unknown and its first derivative to avoid this trouble.

• Why do you think a C1 continuous finite element is required. C0 is used for first order as well as second order. The only requirement would be for your basis polynomials to have sufficient order to solve for the 4th order PDE. Of course the stability or accuracy properties would have to be worked out. – Vikram Jul 7 '16 at 12:15
• Good question. This comes from the smoothness requirements on the trial and test functions that appear in the weak variational form of the PDE. For 4th order equations, you end up with integrals that involve 2nd derivatives of these functions (at minimum), and hence C1 continuity is required across elements so that these integrals may be evaluated. – Nick C. Jul 8 '16 at 14:05