Poisson: uxx + uyy = -81xy

For Poisson's equation ∇²u = f, the five-point formula is

u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1) − 4u(i,j) = h²f(x,y)

Here f = −81xy and h = 1/3, so h²f = −9xy and

u(i,j) = ¼[ sum of four neighbours + 9xy ]

Set up

Interior points (h = 1/3, y increases upward): u₁ = u(1/3, 2/3), u₂ = u(2/3, 2/3), u₃ = u(1/3, 1/3), u₄ = u(2/3, 1/3). Boundary: u = 0 on x = 0 and y = 0; u = 100 on x = 1 and y = 1.

9xy at u₁ = 9(1/3)(2/3) = 2,  u₂ = 9(2/3)(2/3) = 4,  u₃ = 9(1/3)(1/3) = 1,  u₄ = 9(2/3)(1/3) = 2

u₁ = ¼(100 + 0 + u₂ + u₃ + 2)   = ¼(102 + u₂ + u₃)
u₂ = ¼(100 + 100 + u₁ + u₄ + 4) = ¼(204 + u₁ + u₄)
u₃ = ¼(0 + 0 + u₁ + u₄ + 1)     = ¼(1 + u₁ + u₄)
u₄ = ¼(100 + 0 + u₂ + u₃ + 2)   = ¼(102 + u₂ + u₃)

(u₁ and u₄ have the same equation, so u₁ = u₄.)

Iteration (Gauss–Seidel)

Start with u = 51 for all four points.

Converged to 4 decimals after 8 iterations.

Direct solution (check)

With u₁ = u₄ = a, u₂ = b, u₃ = c: a = ¼(102 + b + c), b = ¼(204 + 2a), c = ¼(1 + 2a). Substituting gives a = 51.0833, b = 76.5417, c = 25.7917.

Answer: u₁ = 51.0833, u₂ = 76.5417, u₃ = 25.7917, u₄ = 51.0833