Elliptic: 8 cm plate, 50/30/10

Assume 50 on the top side, 30 on the bottom side and 10 on the left and right sides. With 2 cm grids, the 8 cm plate has 3 × 3 = 9 interior points, numbered 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)]

Equations

u₁ = ¼(60 + u₂ + u₄)
u₂ = ¼(50 + 2u₁ + u₅)
u₄ = ¼(10 + u₁ + u₅ + u₇)
u₅ = ¼(0 + u₂ + 2u₄ + u₈)
u₇ = ¼(40 + u₄ + u₈)
u₈ = ¼(30 + u₅ + 2u₇)

The plate is symmetric from left to right, so u₁ = u₃, u₄ = u₆ and u₇ = u₉. The unknowns are u₁, u₂, u₄, u₅, u₇ and u₈.

Iteration (Gauss–Seidel / Liebmann)

Initial value 25 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 15 iterations (iteration 16 gives the same values).

Direct solution of the equations (check)

u₁ = 28.5714
u₂ = 33.0357
u₄ = 21.2500
u₅ = 25.0000
u₇ = 21.4286
u₈ = 24.4643

Answer: u₁ = 28.5714, u₂ = 33.0357, u₃ = 28.5714, u₄ = 21.2500, u₅ = 25.0000, u₆ = 21.2500, u₇ = 21.4286, u₈ = 24.4643, u₉ = 21.4286