I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using Geometric Multigrid. An illustration is shown in 2-D (apologies for the hand drawn figure). Crosses indicate unknowns, red squares indicate ghost points for Neumann boundary, D indicates Dirichlet, 9-pt stencil is shown in green lines. It is given that $\frac{\partial u}{\partial n}=0$ at Neumann boundaries.
- If I use a 9-pt stencil in 2-D to approximate variable $u$ at various mesh points, what will be the value of the point at the
Green questions mark
? - Should I be using a 5-pt stencil at the Neumann-Neumann boundary intersection?
- Will using a 5-pt stencil at the corner point not deteriorate the
convergence rate
in Multigrid? - If I use the 9-pt stencil at the top-left boundary, are we not taking a Dirichlet point which is outside the physical domain ?