2D Poisson equation. Weighted Jacobi relaxation

n it ω

This GPU based script draws 10 cross-sections u_i,n/2 after every 2it weighted Jacobi iterations. f_i,j = sin(πi/n) sin(πj/n).

We consider 2D Poisson equation with fixed borders u_border = 0
Δu(x,y) = f(x,y).
On square n×n grid we get
u_i-1,j + u_i+1,j + u_i,j-1 + u_i,j+1 - 4u_i = h²f_i,j.
Weighted Jacobi iterations are
u_i,j^new = (1 - ω)u_i,j + ω(u_i-1,j + u_i+1,j + u_i,j-1 + u_i,j+1 - h²f_i,j )/4 (*)
Transformation (*) has eigenvectors (we put f_i,j = 0)
u_i,j^(m,k) = sin(πmi/n) sin(πkj/n)
with eigenvalues
λ_m,k = 1 - ω (sin²πm/2n + sin²πk/2n),
λ_n,n = 1 - 2ω, λ_n/2,n/2 = λ_0,n = 1 - ω, λ_0,n/2 = 1 - ω/2.
From |λ_n,n| = |λ_0,n/2| we get ω = 4/5 and λ_0,n/2 = 3/5.
At last for small m,k we get approximately
λ_m,k = 1 - ωπ²(m² + k²)/4n²,
so relaxation is very slow for long wave harmonics. As since convergence is faster for small n we need coarse grids to accelerate relaxation. We consider multigrid algorithm next.

Simulations on GPU updated 6 Aug 2012