Neumann boundary conditions, where the value of the normal derivative is known
For most applications in quantitative finance the h can be chosen equal to zero.
The inclusion of boundary condition of the Neumann type is accomplished by using the surface integral (see the beginning of the FEM blog post series).
It can be evaluated once the element shape functions are known. Here, we demonstrate the calculation for the linear triangular. Depending on the side of the triangle where the Neumann boundary condition is specified we obtain the results
for sides ij, jk, ik respectively.
The post excerpts a chapter of the book A Workout in Computational Finance.