--- title: "Laplace: 2x2 grid 10/15/20" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:laplace-2x2-10-15-20-25" source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-2x2-10-15-20-25 site: BCT Engineering --- # Laplace: 2x2 grid 10/15/20 Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2070 Bhadra, 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-2x2-10-15-20-25 - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Solve 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 left to right: 0 (corner), 10, 15, 20 (corner). Left boundary top to bottom: 0 (corner), 15, 25, 35 (corner). Right boundary top to bottom: 20 (corner), 25, 30, 35 (corner). Bottom boundary left to right: 35 (corner), 35, 35, 35 (corner).] ## Short answer Using standard five-point Laplace formula and Gauss-Seidal iteration, converged interior values: u1=18.3333, u2=21.6667, u3=26.6667, u4=28.3333. ## Answer 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 15; u₂: top 15, right 25; u₃: left 25, bottom 35; u₄: right 30, bottom 35. #### Equations ```text u₁ = ¼(25 + u₂ + u₃) u₂ = ¼(40 + u₁ + u₄) u₃ = ¼(60 + u₁ + u₄) u₄ = ¼(65 + u₂ + u₃) ``` #### Iteration (Gauss–Seidel / Liebmann) Initial value 24 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 | 24.0000 | 24.0000 | 24.0000 | 24.0000 | | 1 | 18.2500 | 20.5625 | 25.5625 | 27.7812 | | 2 | 17.7812 | 21.3906 | 26.3906 | 28.1953 | | 3 | 18.1953 | 21.5977 | 26.5977 | 28.2988 | | 4 | 18.2988 | 21.6494 | 26.6494 | 28.3247 | | 5 | 18.3247 | 21.6624 | 26.6624 | 28.3312 | | 6 | 18.3312 | 21.6656 | 26.6656 | 28.3328 | | 7 | 18.3328 | 21.6664 | 26.6664 | 28.3332 | | 8 | 18.3332 | 21.6666 | 26.6666 | 28.3333 | | 9 | 18.3333 | 21.6666 | 26.6666 | 28.3333 | | 10 | 18.3333 | 21.6667 | 26.6667 | 28.3333 | Converged to 4 decimals after 10 iterations. #### Direct solution of the equations (check) ```text u₁ = 18.3333 u₂ = 21.6667 u₃ = 26.6667 u₄ = 28.3333 ``` **Answer: u₁ = 18.3333, u₂ = 21.6667, u₃ = 26.6667, u₄ = 28.3333** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. u2 = ? A. (25 + u1 + u4)/4 B. (40 + u1 + u4)/4 C. (35 + u1 + u4)/4 D. (45 + u1 + u4)/4 Answer: B. Top 15 + right 25 = 40. (Key point: Equation u2) 2. u3 = ? A. (50 + u1 + u4)/4 B. (70 + u1 + u4)/4 C. (60 + u1 + u4)/4 D. (35 + u1 + u4)/4 Answer: C. Left 25 + bottom 35 = 60. (Key point: Equation u3) 3. Why start with 24? A. It is required for the formula to work B. It is the exact answer C. Initial guess must be zero D. It is close to the mean of boundary values, which speeds convergence Answer: D. The initial guess affects only speed, not the limit. (Key point: Initial guess) 4. What is u4? A. 28.3333 B. 18.3333 C. 21.6667 D. 26.6667 Answer: A. Solution: 18.3333, 21.6667, 26.6667, 28.3333. (Key point: Result) ## Close variants set in other exams - [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: 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: 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× ## 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 - [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 ## Related in other subjects - [Laplace: i1, i2 in unenergised two-mesh network](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Alaplace-two-mesh-unenergised-100v-10ohm) — Electrical Circuit Theory, chapter 4, asked 3× - [Mesh analysis, power in 15Ω](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Amesh-power-dissipation-15-ohm-400v-200v) — Basic Electrical Engineering, chapter 2