I want to solve \begin{align} \nabla^4\psi+\alpha\nabla^2\psi+\beta\psi=F(x,y);\quad \nabla \psi\cdot \hat{n}=\nabla^3\psi\cdot \hat{n}=0\quad \text{on boundaries} \end{align} with a 2d FEM scheme. With the biharmonic operator, linear test functions are ruled out. I can integrate the first term by parts twice, so that the weak version of the PDE includes a term \begin{align} \int_A\nabla^2\phi_i \nabla^2\phi_j\,da \end{align} Quadratic test functions will allow me to evaluate the integral. I understand that quadratic test functions have nodes at the vertices and at the mid-point of each side of each triangle . My question relates to how to deal with this at assembly. If the coordinates of the vertices of each triangle are in p, and the connectivity matrix is t, then what is the procedure to extend these for the additional nodes located on the sides of each triangle?
This wonderful reference http://www.colorado.edu/engineering/CAS/courses.d/AFEM.d/AFEM.Ch23.d/AFEM.Ch23.pdf '23.2.7. Six Node Quadratic Interpolation' describes the quadratic elements perfectly, but I can't figure out how to integrate the midway points into my node list?