Normal.Derivs.tex

During the discussion above we have been concerned with transforming a basis on
a local triangle to that of a basis on a reference triangle. However, we have
not considered the implications that results from the assumption of always
rotating the normal derivative $\dfrac{\pi}{2}$ counterclockwise.  If we always
rotate $\dfrac{\pi}{2}$ counterclockwise there will be a jump discontinuity in
the basis function along that edge.  This results from assuming
$\dfrac{\partial\varphi_i}{\partial\mathbf{n}_i} = 1$ on triangle $I$ while at
the same time assuming $\dfrac{\partial\varphi_i}{\partial\mathbf{n}_i} = -1$ on
the adjacent triangle $J$. The potentional issue can be seen graphically in
\autoref{fig:Normals}. This is what causes the discontinuity along shared edges.
To address this one has two options: one can multiply the value of $\varphi_i$ by
negative one on the triangle $J$, without changing $\varphi_i$ on triangle
$I$, or one can avoid the issue all together by numbering the nodes in such a way
that this mismatching of normal derivatives doesn't occur.

\input{Normal.Cont.tex}