Question -error estimate finite elements


This is not an explicit FreeFEM question, but I hope you can indicate some useful reference, since it deals with the precision of the finite element method when the boundary condition changes.

Suppose we have a configuration where three types of boundary conditions are present for the Laplace problem: Neumann=0, Neumann=g and Dirichlet. It is known that depending on g it might happen that the solution is not in H^2, especially at the junction points between the different boundary conditions. Therefore, classical error estimates do not immediately apply.

FreeFEM can easily provide an approximation using P1 finite elements, for example. Can you indicate some reference in which the convergence rate of the error is studied in this case where multiple types of boundary conditions appear simultaneously in the PDE?

Thank you,