- Asked 6/13
- 2068 Bhadra · 10 marks
- 2068 Magh · 10 marks
Laplace: 3x3 grid 500/1000/500
Short answer
Use the five-point Laplace formula with Gauss-Seidel iteration; symmetry reduces the unknowns. The values are u1=u3=u7=u9=937.5, u2=u8=1000.0, u4=u6=1250.0, and u5=1125.0.
Solve the elliptic equation uxx + uyy = 0 for the square mesh with the boundary values as shown. [Figure: 3x3 interior points u1 u2 u3 / u4 u5 u6 / u7 u8 u9. Top boundary left to right: 0 (corner), 500, 1000, 500, 0 (corner). Bottom boundary: 0 (corner), 500, 1000, 500, 0 (corner). Left boundary top to bottom: 1000, 2000, 1000. Right boundary top to bottom: 1000, 2000, 1000.]
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 500, 1000, 500 (above u₁, u₂, u₃) and the same along the bottom (below u₇, u₈, u₉); left 1000, 2000, 1000 (beside u₁, u₄, u₇) and the same along the right (beside u₃, u₆, u₉). Corner values are not used.
Equations
u₁ = ¼(1500 + u₂ + u₄)
u₂ = ¼(1000 + 2u₁ + u₅)
u₄ = ¼(2000 + 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 1000 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 | 1000.0000 | 1000.0000 | 1000.0000 | 1000.0000 |
| 1 | 875.0000 | 937.5000 | 1187.5000 | 1062.5000 |
| 2 | 906.2500 | 968.7500 | 1218.7500 | 1093.7500 |
| 3 | 921.8750 | 984.3750 | 1234.3750 | 1109.3750 |
| 4 | 929.6875 | 992.1875 | 1242.1875 | 1117.1875 |
| 5 | 933.5938 | 996.0938 | 1246.0938 | 1121.0938 |
| 6 | 935.5469 | 998.0469 | 1248.0469 | 1123.0469 |
| 7 | 936.5234 | 999.0234 | 1249.0234 | 1124.0234 |
| 8 | 937.0117 | 999.5117 | 1249.5117 | 1124.5117 |
| 9 | 937.2559 | 999.7559 | 1249.7559 | 1124.7559 |
| 10 | 937.3779 | 999.8779 | 1249.8779 | 1124.8779 |
| … | … | … | … | … |
| 21 | 937.4999 | 999.9999 | 1249.9999 | 1124.9999 |
| 22 | 937.5000 | 1000.0000 | 1250.0000 | 1125.0000 |
Converged to 4 decimals after 22 iterations.
Direct solution of the equations (check)
u₁ = 937.5000
u₂ = 1000.0000
u₄ = 1250.0000
u₅ = 1125.0000
Answer: u₁ = 937.5000, u₂ = 1000.0000, u₃ = 937.5000, u₄ = 1250.0000, u₅ = 1125.0000, u₆ = 1250.0000, u₇ = 937.5000, u₈ = 1000.0000, u₉ = 937.5000
✓ Answer checked by Claude Opus 5.5 (high effort)
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