Given a triangulation in 2D, a point within each triangle $\vec{p}_k$ with known value at it: $u(\vec{p}_k) = u_k \in [0,1]$, and prescribed continuity at each edge of the triangulation (e.g. $C_{-1}$ at edge $e_{i}$ and $C_{2}$ at edge $e_{j}$), I want to construct a smooth function $u$, that has discontinuities only at edges with $C_{-1}$. I am aware that with only these conditions the solution is not unique, so I am really looking for research papers/any suggestions regarding this problem.
I have been looking into finite element methods, but I am unsure whether I can enforce such edge continuity conditions there.