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Laplace: 2x2 grid 10/15/20

Short answer

Using standard five-point Laplace formula and Gauss-Seidal iteration, converged interior values: u1=18.3333, u2=21.6667, u3=26.6667, u4=28.3333.

Solve uxx + uyy = 0 for the following square mesh with boundary conditions as shown in figure below. [Figure: interior points u1 (top-left), u2 (top-right), u3 (bottom-left), u4 (bottom-right). Top boundary left to right: 0 (corner), 10, 15, 20 (corner). Left boundary top to bottom: 0 (corner), 15, 25, 35 (corner). Right boundary top to bottom: 20 (corner), 25, 30, 35 (corner). Bottom boundary left to right: 35 (corner), 35, 35, 35 (corner).]

Interior points: u₁ (top-left), u₂ (top-right), u₃ (bottom-left), u₄ (bottom-right). 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 beside the points: u₁: top 10, left 15; u₂: top 15, right 25; u₃: left 25, bottom 35; u₄: right 30, bottom 35.

Equations

u₁ = ¼(25 + u₂ + u₃)
u₂ = ¼(40 + u₁ + u₄)
u₃ = ¼(60 + u₁ + u₄)
u₄ = ¼(65 + u₂ + u₃)

Iteration (Gauss–Seidel / Liebmann)

Initial value 24 for every unknown (close to the mean of the boundary values); each new value is used at once in the next equation.

iterationu₁u₂u₃u₄
024.000024.000024.000024.0000
118.250020.562525.562527.7812
217.781221.390626.390628.1953
318.195321.597726.597728.2988
418.298821.649426.649428.3247
518.324721.662426.662428.3312
618.331221.665626.665628.3328
718.332821.666426.666428.3332
818.333221.666626.666628.3333
918.333321.666626.666628.3333
1018.333321.666726.666728.3333

Converged to 4 decimals after 10 iterations.

Direct solution of the equations (check)

u₁ = 18.3333
u₂ = 21.6667
u₃ = 26.6667
u₄ = 28.3333

Answer: u₁ = 18.3333, u₂ = 21.6667, u₃ = 26.6667, u₄ = 28.3333

✓ Answer checked by Claude Opus 5.5 (high effort)

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