- Asked 5/13
- 2069 Bhadra · 12 marks
Elliptic: 2x2 grid 10/20/40/50
Short answer
Using the standard five-point finite difference approximation for Laplace's equation and Gauss-Seidel iteration, the interior nodal values converge to u1 = 20.0000, u2 = 30.0000, u3 = 30.0000, u4 = 40.0000.
Solve the elliptic equation 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: 10, 20 (left to right). Left boundary: 10, 20 (top to bottom). Right boundary: 40, 50 (top to bottom). Bottom boundary: 40, 50 (left to right).]
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 10; u₂: top 20, right 40; u₃: left 20, bottom 40; u₄: right 50, bottom 50.
Equations
u₁ = ¼(20 + u₂ + u₃)
u₂ = ¼(60 + u₁ + u₄)
u₃ = ¼(60 + u₁ + u₄)
u₄ = ¼(100 + u₂ + u₃)
Iteration (Gauss–Seidel / Liebmann)
Initial value 30 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 | 30.0000 | 30.0000 | 30.0000 | 30.0000 |
| 1 | 20.0000 | 27.5000 | 27.5000 | 38.7500 |
| 2 | 18.7500 | 29.3750 | 29.3750 | 39.6875 |
| 3 | 19.6875 | 29.8438 | 29.8438 | 39.9219 |
| 4 | 19.9219 | 29.9609 | 29.9609 | 39.9805 |
| 5 | 19.9805 | 29.9902 | 29.9902 | 39.9951 |
| 6 | 19.9951 | 29.9976 | 29.9976 | 39.9988 |
| 7 | 19.9988 | 29.9994 | 29.9994 | 39.9997 |
| 8 | 19.9997 | 29.9998 | 29.9998 | 39.9999 |
| 9 | 19.9999 | 30.0000 | 30.0000 | 40.0000 |
| 10 | 20.0000 | 30.0000 | 30.0000 | 40.0000 |
| 11 | 20.0000 | 30.0000 | 30.0000 | 40.0000 |
Converged to 4 decimals after 11 iterations.
Direct solution of the equations (check)
u₁ = 20.0000
u₂ = 30.0000
u₃ = 30.0000
u₄ = 40.0000
Answer: u₁ = 20.0000, u₂ = 30.0000, u₃ = 30.0000, u₄ = 40.0000
✓ Answer checked by Claude Opus 5.5 (high effort)
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