- Asked 6/13
- 2073 Magh · 10 marks
Laplace: 3x3 grid 400/800/600/1200
Short answer
Using standard five-point Laplace formula and symmetry, Gauss-Seidal iteration yields converged values: u1=625.0000, u2=700.0000, u3=625.0000, u4=800.0000, u5=750.0000, u6=800.0000, u7=625.0000, u8=700.0000, u9=625.0000.
Solve the elliptic equation uxx + uyy = 0 for the square mesh with boundary values as shown in the figure below. [Figure: 3x3 interior points u1 u2 u3 / u4 u5 u6 / u7 u8 u9. Top boundary left to right: 0 (corner), 400, 800, 400, 0 (corner). Bottom boundary: 0 (corner), 400, 800, 400, 0 (corner). Left boundary top to bottom: 600, 1200, 600. Right boundary top to bottom: 600, 1200, 600.]
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 400, 800, 400 (above u₁, u₂, u₃) and the same along the bottom (below u₇, u₈, u₉); left 600, 1200, 600 (beside u₁, u₄, u₇) and the same along the right (beside u₃, u₆, u₉). Corner values are not used.
Equations
u₁ = ¼(1000 + u₂ + u₄)
u₂ = ¼(800 + 2u₁ + u₅)
u₄ = ¼(1200 + 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 667 for every unknown (close to the mean of the boundary values); each new value is used at once in the next equation.
| iteration | u₁ | u₂ | u₄ | u₅ |
|---|---|---|---|---|
| 0 | 667.0000 | 667.0000 | 667.0000 | 667.0000 |
| 1 | 583.5000 | 658.5000 | 758.5000 | 708.5000 |
| 2 | 604.2500 | 679.2500 | 779.2500 | 729.2500 |
| 3 | 614.6250 | 689.6250 | 789.6250 | 739.6250 |
| 4 | 619.8125 | 694.8125 | 794.8125 | 744.8125 |
| 5 | 622.4062 | 697.4062 | 797.4062 | 747.4062 |
| 6 | 623.7031 | 698.7031 | 798.7031 | 748.7031 |
| 7 | 624.3516 | 699.3516 | 799.3516 | 749.3516 |
| 8 | 624.6758 | 699.6758 | 799.6758 | 749.6758 |
| 9 | 624.8379 | 699.8379 | 799.8379 | 749.8379 |
| 10 | 624.9189 | 699.9189 | 799.9189 | 749.9189 |
| … | … | … | … | … |
| 20 | 624.9999 | 699.9999 | 799.9999 | 749.9999 |
| 21 | 625.0000 | 700.0000 | 800.0000 | 750.0000 |
Converged to 4 decimals after 21 iterations.
Direct solution of the equations (check)
u₁ = 625.0000
u₂ = 700.0000
u₄ = 800.0000
u₅ = 750.0000
Answer: u₁ = 625.0000, u₂ = 700.0000, u₃ = 625.0000, u₄ = 800.0000, u₅ = 750.0000, u₆ = 800.0000, u₇ = 625.0000, u₈ = 700.0000, u₉ = 625.0000
✓ Answer checked by Claude Opus 5.5 (high effort)
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