I am simulating $\nabla^{2}u=0$ with mixed Dirichlet-Neumann boundary conditions on 2-D
using 2-Grid
method. Dirichlet conditions are applied to left and bottom boundary with $u=1+x+y$ and Neuman conditions are applied to top and right boundary with $\frac{\partial u}{\partial n}=1$. The solution converges for 4x4
grid without using 2-grid
. With 2-grid, please let me know if the following are correct (as I am unable to converge with 2-grid).
I do
2
iterations on the fine grid (using Pure Jacobi), calculate residual, use injection to restrict to Coarse grid (2x2
). Is it okay to use Pure Jacobi or do we always use Damped (weighted) Jacobi ?With error matrix $e=0$ as initial guess, I solve the residual equation $Ae =r$ till full convergence, then interpolate/prolongate the error back to fine grid. Should the residual equation be solved till convergence or would 2-3 iterations be sufficient ? ( The error equation does converge fully.)
Where exactly do I have to use the norm of an error/residual/whatever ? Since I know the true solution on the Fine Grid is $u = 1 + i*h + j*h$ where
h=1/4
fori,j
vary from0
to4
, I am using it as convergence criterion on the fine grid i.e. till $error = u_{new} - (1 + i*h + j*h)$ is greater than the tolerance of $10^{-5}$ keep iterating.