Laplace: 3x3 grid 200/300/400/500

Number the interior points row-wise: u₁ u₂ u₃ / u₄ u₅ u₆ / u₇ u₈ u₉. For Laplace's equation ∇²u = 0 on a square mesh, each interior value equals the mean of its four neighbours (standard five-point formula):

u(i,j) = ¼[u(i−1,j) + u(i+1,j) + u(i,j−1) + u(i,j+1)]

Boundary values next to the interior points: top 200, 300, 200 (above u₁, u₂, u₃) and the same along the bottom (below u₇, u₈, u₉); left 400, 500, 400 (beside u₁, u₄, u₇) and the same along the right (beside u₃, u₆, u₉). Corner values are not used.

Equations

u₁ = ¼(600 + u₂ + u₄)
u₂ = ¼(300 + 2u₁ + u₅)
u₄ = ¼(500 + 2u₁ + u₅)
u₅ = ¼(0 + 2u₂ + 2u₄)

The figure is symmetric about the vertical and horizontal centre lines, so u₁ = u₃ = u₇ = u₉, u₂ = u₈ and u₄ = u₆. Only u₁, u₂, u₄ and u₅ are unknown.

Iteration (Gauss–Seidel / Liebmann)

Initial value 333 for every unknown (close to the mean of the boundary values); each new value is used at once in the next equation.

Converged to 4 decimals after 19 iterations.

Direct solution of the equations (check)

u₁ = 325.0000
u₂ = 325.0000
u₄ = 375.0000
u₅ = 350.0000

Answer: u₁ = 325.0000, u₂ = 325.0000, u₃ = 325.0000, u₄ = 375.0000, u₅ = 350.0000, u₆ = 375.0000, u₇ = 325.0000, u₈ = 325.0000, u₉ = 325.0000