  In this paper new innovative fourth order compact schemes for Robin and
Neumann boundary conditions have been developed for boundary value problems of
elliptic PDEs in two and three dimensions.
  Different from traditional finite difference operator approach, which may not
work for flux type of boundary conditions, carefully designed undetermined
coefficient methods are utilized in developing high order compact (HOC)
schemes. The new methods not only can be utilized to design HOC schemes for
flux type of boundary conditions but can also be applied to general elliptic
PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic
equations with linear boundary conditions.
  In the new developed HOC methods, the coefficient matrices are generally
M-matrices, which guarantee the discrete maximum principle for well-posed
problems, so the convergence of the HOC methods. The developed HOC methods are
versatile and can cover most of high order compact schemes in the literature.
The HOC methods for Robin boundary conditions and for anisotropic diffusion and
advection equations with Robin or even Dirichlet boundary conditions are likely
the first ones that have ever been developed. With the help of pseudo-inverse,
or SVD solutions, we have also observed that the developed HOC methods usually
have smaller error constants compared with traditional HOC methods when
applicable. Non-trivial examples with large wave numbers and oscillatory
solutions are presented to confirm the performance of the new HOC methods.
