--- title: "Laplace: 3x3 grid 400/800/600/1200" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:laplace-3x3-0-400-800" source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-3x3-0-400-800 site: BCT Engineering --- # Laplace: 3x3 grid 400/800/600/1200 Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2073 Magh, 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-3x3-0-400-800 - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question 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.] ## 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. ## 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 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 ```text 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) ```text 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** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. u1 = ? A. (400 + u2 + u4)/4 B. (600 + u2 + u4)/4 C. (1200 + u2 + u4)/4 D. (1000 + u2 + u4)/4 Answer: D. Top 400 + left 600 = 1000. (Key point: Equation u1) 2. u4 = ? A. (1200 + 2u1 + u5)/4 B. (600 + 2u1 + u5)/4 C. (800 + 2u1 + u5)/4 D. (1200 + u1 + u5)/4 Answer: A. Left boundary 1200; neighbours u1 and u7 (=u1) and u5. (Key point: Equation u4) 3. Why are only four unknowns needed? A. Corners are zero B. Figure symmetric about both centre lines C. Laplace's equation is linear D. u5 is on the boundary Answer: B. Symmetric boundary conditions give symmetric solutions. (Key point: Symmetry) 4. What is u5? A. 700 B. 800 C. 750 D. 625 Answer: C. Solution: u1=625, u2=700, u4=800, u5=750. (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 200/300/400/500](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-3x3-200-300-400-500) — Numerical Method, chapter 7, asked 5× - [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: 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: 0≤x,y≤3 boundary](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-uxx-0-3-boundary) — 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 - [Laplace: five-point formula, 2x2](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Alaplace-five-point-iterative) — Numerical Method, chapter 7