--- title: "Laplace: 3x3 grid 200/300/400/500" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:laplace-3x3-200-300-400-500" source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-3x3-200-300-400-500 site: BCT Engineering --- # Laplace: 3x3 grid 200/300/400/500 Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2072 Asoj, 10 marks - Video: [Numerical Solution of Laplace Equation Using Finite Difference MethodFDM](https://www.youtube.com/watch?v=Z5Mx_Wh9E40) — Keshav Jadhav - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-3x3-200-300-400-500 - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Solve the Laplace equation uxx + uyy = 0 for the square mesh with boundary values as shown in the figure. [Figure: 3x3 interior points u1 u2 u3 / u4 u5 u6 / u7 u8 u9. Top boundary left to right: 0 (corner), 200, 300, 200, 0 (corner). Bottom boundary: 0 (corner), 200, 300, 200, 0 (corner). Left boundary top to bottom: 400, 500, 400. Right boundary top to bottom: 400, 500, 400.] ## Short answer Using standard five-point Laplace formula and symmetry, Gauss-Seidal iteration gives converged values: u1=325.0000, u2=325.0000, u3=325.0000, u4=375.0000, u5=350.0000, u6=375.0000, u7=325.0000, u8=325.0000, u9=325.0000. ## Answer 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 200, 300, 200 (above u₁, u₂, u₃) and the same along the bottom (below u₇, u₈, u₉); left 400, 500, 400 (beside u₁, u₄, u₇) and the same along the right (beside u₃, u₆, u₉). Corner values are not used. #### Equations ```text u₁ = ¼(600 + u₂ + u₄) u₂ = ¼(300 + 2u₁ + u₅) u₄ = ¼(500 + 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 333 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 | 333.0000 | 333.0000 | 333.0000 | 333.0000 | | 1 | 316.5000 | 316.5000 | 366.5000 | 341.5000 | | 2 | 320.7500 | 320.7500 | 370.7500 | 345.7500 | | 3 | 322.8750 | 322.8750 | 372.8750 | 347.8750 | | 4 | 323.9375 | 323.9375 | 373.9375 | 348.9375 | | 5 | 324.4688 | 324.4688 | 374.4688 | 349.4688 | | 6 | 324.7344 | 324.7344 | 374.7344 | 349.7344 | | 7 | 324.8672 | 324.8672 | 374.8672 | 349.8672 | | 8 | 324.9336 | 324.9336 | 374.9336 | 349.9336 | | 9 | 324.9668 | 324.9668 | 374.9668 | 349.9668 | | 10 | 324.9834 | 324.9834 | 374.9834 | 349.9834 | | … | … | … | … | … | | 18 | 324.9999 | 324.9999 | 374.9999 | 349.9999 | | 19 | 325.0000 | 325.0000 | 375.0000 | 350.0000 | Converged to 4 decimals after 19 iterations. #### Direct solution of the equations (check) ```text u₁ = 325.0000 u₂ = 325.0000 u₄ = 375.0000 u₅ = 350.0000 ``` **Answer: u₁ = 325.0000, u₂ = 325.0000, u₃ = 325.0000, u₄ = 375.0000, u₅ = 350.0000, u₆ = 375.0000, u₇ = 325.0000, u₈ = 325.0000, u₉ = 325.0000** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. u2 = ? A. (200 + 2u1 + u5)/4 B. (300 + 2u1 + u5)/4 C. (500 + 2u1 + u5)/4 D. (300 + u1 + u5)/4 Answer: B. Top boundary above u2 is 300. (Key point: Equation u2) 2. u4 = ? A. (400 + 2u1 + u5)/4 B. (300 + 2u1 + u5)/4 C. (500 + 2u1 + u5)/4 D. (900 + 2u1 + u5)/4 Answer: C. Left boundary beside u4 is 500. (Key point: Equation u4) 3. u1 = ? A. (700 + u2 + u4)/4 B. (500 + u2 + u4)/4 C. (600 + u2 + u5)/4 D. (600 + u2 + u4)/4 Answer: D. 200 + 400 = 600. (Key point: Equation u1) 4. What are u1 and u5? A. 325 and 350 B. 375 and 350 C. 350 and 325 D. 325 and 375 Answer: A. Solution: u1=u2=325, u4=375, u5=350. (Key point: Result) ## Close variants set in other exams - [Laplace: 3x3 grid 500/1000/500](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-3x3-0-500-1000) — Numerical Method, chapter 7, asked 6× - [Laplace: 3x3 grid 400/800/600/1200](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-3x3-0-400-800) — Numerical Method, chapter 7, asked 6× - [Laplace: 2x2 grid 10/15/20](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-2x2-10-15-20-25) — Numerical Method, chapter 7, asked 6× ## Related in Numerical Method - [Laplace: 2x2 grid 100/200/50](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-100-200-50-grid) — Numerical Method, chapter 7, asked 2× - [Laplace: 2x2 grid 3000/2000/500](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-2x2-3000-grid) — Numerical Method, chapter 7 - [Elliptic: 2x2 grid 10/20/40/50](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Aelliptic-2x2-10-20-40-50) — Numerical Method, chapter 7, asked 5× - [Laplace: five-point formula, 2x2](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-five-point-iterative) — Numerical Method, chapter 7 - [Poisson: uxx + uyy = -81xy](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-81xy) — Numerical Method, chapter 7