--- title: "Elliptic: 8 cm plate, 50/30/10" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:poisson-8cm-plate" source: https://www.bctengineer.com/q/semester_4/numerical_method/poisson-8cm-plate site: BCT Engineering --- # Elliptic: 8 cm plate, 50/30/10 Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2075 Baisakh, 10 marks - Video: [Chapter 10.03: Lesson: Elliptic PDEs: Gauss-Seidel Method](https://www.youtube.com/watch?v=7AMJORc0bGI) — numericalmethodsguy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/numerical_method/poisson-8cm-plate - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Solve the elliptic equation ∇²u = 0 in the square plate of size 8cm × 8cm if the boundary values are given 50 on one side of the plate and 30 on its opposite side. On the other sides the values are given 10. Assume the square grids of size 2cm × 2cm. ## Short answer Using standard five-point finite difference approximation for Laplace's equation on a 2 cm mesh, solved by Gauss-Seidel iteration. Converged interior nodal values: u1=28.5714, u2=33.0357, u3=28.5714, u4=21.2500, u5=25.0000, u6=21.2500, u7=21.4286, u8=24.4643, u9=21.4286. ## Answer Assume 50 on the top side, 30 on the bottom side and 10 on the left and right sides. With 2 cm grids, the 8 cm plate has 3 × 3 = 9 interior points, numbered 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)] #### Equations ```text u₁ = ¼(60 + u₂ + u₄) u₂ = ¼(50 + 2u₁ + u₅) u₄ = ¼(10 + u₁ + u₅ + u₇) u₅ = ¼(0 + u₂ + 2u₄ + u₈) u₇ = ¼(40 + u₄ + u₈) u₈ = ¼(30 + u₅ + 2u₇) ``` The plate is symmetric from left to right, so u₁ = u₃, u₄ = u₆ and u₇ = u₉. The unknowns are u₁, u₂, u₄, u₅, u₇ and u₈. #### Iteration (Gauss–Seidel / Liebmann) Initial value 25 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 | u₇ | u₈ | |---|---|---|---|---|---|---| | 0 | 25.0000 | 25.0000 | 25.0000 | 25.0000 | 25.0000 | 25.0000 | | 1 | 27.5000 | 32.5000 | 21.8750 | 25.3125 | 21.7188 | 24.6875 | | 2 | 28.5938 | 33.1250 | 21.4062 | 25.1562 | 21.5234 | 24.5508 | | 3 | 28.6328 | 33.1055 | 21.3281 | 25.0781 | 21.4697 | 24.5044 | | 4 | 28.6084 | 33.0737 | 21.2891 | 25.0391 | 21.4484 | 24.4839 | | 5 | 28.5907 | 33.0551 | 21.2695 | 25.0195 | 21.4384 | 24.4741 | | 6 | 28.5812 | 33.0455 | 21.2598 | 25.0098 | 21.4335 | 24.4692 | | 7 | 28.5763 | 33.0406 | 21.2549 | 25.0049 | 21.4310 | 24.4667 | | 8 | 28.5739 | 33.0382 | 21.2524 | 25.0024 | 21.4298 | 24.4655 | | 9 | 28.5726 | 33.0369 | 21.2512 | 25.0012 | 21.4292 | 24.4649 | | 10 | 28.5720 | 33.0363 | 21.2506 | 25.0006 | 21.4289 | 24.4646 | | … | … | … | … | … | … | … | | 14 | 28.5715 | 33.0358 | 21.2500 | 25.0000 | 21.4286 | 24.4643 | | 15 | 28.5714 | 33.0357 | 21.2500 | 25.0000 | 21.4286 | 24.4643 | | 16 | 28.5714 | 33.0357 | 21.2500 | 25.0000 | 21.4286 | 24.4643 | Converged to 4 decimals after 15 iterations (iteration 16 gives the same values). #### Direct solution of the equations (check) ```text u₁ = 28.5714 u₂ = 33.0357 u₄ = 21.2500 u₅ = 25.0000 u₇ = 21.4286 u₈ = 24.4643 ``` **Answer: u₁ = 28.5714, u₂ = 33.0357, u₃ = 28.5714, u₄ = 21.2500, u₅ = 25.0000, u₆ = 21.2500, u₇ = 21.4286, u₈ = 24.4643, u₉ = 21.4286** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How many interior points does an 8 cm plate with 2 cm grids have? A. 16 B. 4 C. 25 D. 9 (3x3) Answer: D. 4 intervals per side gives 3 interior points per row. (Key point: Grid) 2. What is the equation for u1 (top 50, left 10)? A. u1 = (60 + u2 + u4)/4 B. u1 = (50 + u2 + u4)/4 C. u1 = (100 + u2 + u4)/4 D. u1 = (60 + u2 + u5)/4 Answer: A. 50 + 10 = 60. (Key point: Equation u1) 3. Which symmetry is used? A. Top-bottom: u1=u7 B. Left-right: u1=u3, u4=u6, u7=u9 C. All nine equal D. None; nine unknowns are needed Answer: B. Left and right sides both equal 10, but top and bottom differ. (Key point: Symmetry) 4. What is u5? A. 30.0000 B. 21.25 C. 25.0000 D. 28.5714 Answer: C. Direct solution of the equations gives u5 = 25. (Key point: Result) ## Related in Numerical Method - [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× - [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: ∇²u = -10(x²+y²+10)](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-10-x2-y2-10) — Numerical Method, chapter 7 - [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× ## Related in other subjects - [Solve one dimensional heat equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-one-dimensional-heat-equation) — Mathematics III, chapter 4 - [Solve one dimensional heat equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-one-dimensional-heat-equation-three) — Mathematics III, chapter 4 - [Footing size from plate load test curve](https://www.bctengineer.com/question.md?key=semester_5%2Ffoundation_engineering%3Aplate-load-footing-size-3200kn) — Foundation Engineering, chapter 6 - [Solve one dimensional heat equation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-one-dimensional-heat-equation-two) — Mathematics III, chapter 4