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enter image description here 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.

  1. 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?
  2. Should I be using a 5-pt stencil at the Neumann-Neumann boundary intersection?
  3. Will using a 5-pt stencil at the corner point not deteriorate the convergence rate in Multigrid?
  4. 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 ?
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  • $\begingroup$ I dont have an answer but curious to know the answer. How will 5-pt stencil help, since the neumann condition does not make sense at a corner. If (nx,ny) is the top-right corner point, how do you get the ghost value at (nx+1,ny) and (nx,ny+1) ? $\endgroup$ – cpraveen Nov 1 '17 at 13:42
  • $\begingroup$ @PraveenChandrashekar: Its given that $\frac{\partial u}{\partial n}=0$, so we can use it to find u(nx+1,ny) and u(nx,ny+1). For example, using central difference : $\frac{u(nx,ny+1) - u(nx,ny-1)}{2h}=0$, hence $u(nx,ny+1) = u(nx,ny-1)$. Simiarly we can find u(nx+1,ny). Thus, 5-pt stencil can be applied. $\endgroup$ – Gaurav Saxena Nov 1 '17 at 14:09
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    $\begingroup$ ok, so you take $u_x=0$ and $u_y=0$ at the top-right corner. Then you could do $u(nx+1,ny+1)=u(nx-1,ny-1)$ to get the corner ghost value ??? $\endgroup$ – cpraveen Nov 1 '17 at 14:19
  • $\begingroup$ @PraveenChandrashekar: How exactly do you derive this ? $\endgroup$ – Gaurav Saxena Nov 1 '17 at 14:48
  • $\begingroup$ If you take $u_x=u_y=0$ then every directional derivative is zero. Use this along the line joining $(nx-1,ny-1)$ to $(nx+1,ny+1)$. $\endgroup$ – cpraveen Nov 2 '17 at 3:10

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