Consider the Laplace equation on a single triangular domain with a Dirichlet condition on two of the sides and a Neumann condition on the remaining side. I am using a linear element ... $\mathbb{P}_1$ which means (I think) that the solution will be a single linear function on the triangle. The Dirichlet conditions on two sides determine the value of the solution on the three corners of the triangle and hence, by linearity, on the entire triangular domain. But that suggests that the solution is entirely independent of the Neumann condition. It would seem that such a boundary value problem is overdetermined. That doesn't sit well with my intuition.
It this situation (a single triangular element) simply degenerate? That is, can I be sure that domains with more than one triangle (still using linear solution on each triangle) will yield uniquely determined solutions?
dan