# Multigrid for Robin boundary conditions

I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary, $$- \Delta u = f \text{ in } \Omega$$ and $$\frac{\partial u}{\partial n} - au= g \text{ on } \partial \Omega.$$ I use a finite difference discretization with central differences (a standard five point stencil) for the interior points, and eliminate the boundary conditions so that, for example, at a point $x_C$ on the left boundary with east, south and north neighbors $x_E, x_S, x_N$ the resulting equation is

$$\frac{1}{h^2}\left( 4u(x_C)-u(x_S)-u(x_N)-2u(x_E)\right) - 2\frac{a}{h}u(x_C) = \frac{2}{h}g(x_C)+f(x_C).$$

In chapter 5 of the book 'Multigrid' of Trottenberg et. al., it is explained that if this approach is followed for a Neumann problem ($a=0$) and the full-weight restriction operator $R$ is modified on the boundary points, e.g., on a vertical boundary the modified stencil is $$\frac{1}{16}\left[\begin{array}{cc} 2 &2 \\ 4&4 \\2&2 \end{array}\right]_{h}^{2h}$$ then a relaxation method can be applied directly on the equations for the boundary points with the eliminated boundary condition (no need of transferring the equation and boundary condition separately). I am using the interpolation operator $I=4R^T$ and Galerkin coarse grid systems.

My question is: Is this multigrid approach correct (modified FW operator on the boundaries + relaxation on the equation with eliminated boundary conditions and Galerkin coarse systems) also for more general Robin boundary conditions, provided a good relaxation method is used?