I would like to solve the biharmonic equation on with on the boundary. I applied the weak form and then used Legendre-Galerkin spectral method. The Dirichlet condition can easily be incorporated but what about the other essential Neumann boundary condition? Can I use the method of Lagrange-multipliers for Neumann-BCs, that is append the system of equations with
Since I use a nodal Galerkin method, I cannot choose the basis functions to fulfil the boundary conditions, opposed to the modal Galerkin method.