- 2071 Magh · 10 marks
Laplace: 2x2 grid 3000/2000/500
Short answer
Using standard five-point Laplace formula and Gauss-Seidal iteration, converged interior values are: u1=1958.3333, u2=1541.6667, u3=1291.6667, u4=708.3333.
Calculate the value of u(x, y) satisfying the Laplace equation ∇²u = 0 at the interior points of the square region with boundary conditions shown in figure below. [Figure: square region with interior points u1 (top-left), u2 (top-right), u3 (bottom-left), u4 (bottom-right). Top boundary: 3000, 3000, 3000 (left to right). Left boundary top to bottom: 3000, 2000, 2000, 1000. Right boundary: 500 (next to u2 row), 0 (next to u4 row), 0 (bottom). Bottom boundary: 500, 0.]
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)]
Reading the figure: top 3000, 3000 (above u₁, u₂); left 2000, 2000 (beside u₁, u₃); right 500, 0 (beside u₂, u₄); bottom 500, 0 (below u₃, u₄). Corner values are not used.
Equations
u₁ = ¼(5000 + u₂ + u₃)
u₂ = ¼(3500 + u₁ + u₄)
u₃ = ¼(2500 + u₁ + u₄)
u₄ = ¼(0 + u₂ + u₃)
Iteration (Gauss–Seidel / Liebmann)
Initial value 1375 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 | u₃ | u₄ |
|---|---|---|---|---|
| 0 | 1375.0000 | 1375.0000 | 1375.0000 | 1375.0000 |
| 1 | 1937.5000 | 1703.1250 | 1453.1250 | 789.0625 |
| 2 | 2039.0625 | 1582.0312 | 1332.0312 | 728.5156 |
| 3 | 1978.5156 | 1551.7578 | 1301.7578 | 713.3789 |
| 4 | 1963.3789 | 1544.1895 | 1294.1895 | 709.5947 |
| 5 | 1959.5947 | 1542.2974 | 1292.2974 | 708.6487 |
| 6 | 1958.6487 | 1541.8243 | 1291.8243 | 708.4122 |
| 7 | 1958.4122 | 1541.7061 | 1291.7061 | 708.3530 |
| 8 | 1958.3530 | 1541.6765 | 1291.6765 | 708.3383 |
| 9 | 1958.3383 | 1541.6691 | 1291.6691 | 708.3346 |
| 10 | 1958.3346 | 1541.6673 | 1291.6673 | 708.3336 |
| … | … | … | … | … |
| 13 | 1958.3334 | 1541.6667 | 1291.6667 | 708.3333 |
| 14 | 1958.3333 | 1541.6667 | 1291.6667 | 708.3333 |
Converged to 4 decimals after 14 iterations.
Direct solution of the equations (check)
u₁ = 1958.3333
u₂ = 1541.6667
u₃ = 1291.6667
u₄ = 708.3333
Answer: u₁ = 1958.3333, u₂ = 1541.6667, u₃ = 1291.6667, u₄ = 708.3333
✓ Answer checked by Claude Opus 5.5 (high effort)
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