--- title: "Laplace: 2x2 grid 100/200/50" subject: "Numerical Method" chapter: 7 key: "semester_4/numerical_method:laplace-100-200-50-grid" source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-100-200-50-grid site: BCT Engineering --- # Laplace: 2x2 grid 100/200/50 Numerical Method · Chapter 7: Numerical solution of Partial differential Equation - Asked in: 2074 Bhadra, 10 marks; 2070 Magh, 10 marks (asked 2 times) - 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-100-200-50-grid - Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7 ## Question Find the values of u(x, y) satisfying the Laplace equation ∇²u = 0, at the pivotal points of the square region with boundary conditions as shown below. [Figure: square region with interior points u1 (top-left), u2 (top-right), u3 (bottom-left), u4 (bottom-right). Top boundary: 100 (corner, left), 100, 100, 100 (corner, right). Left boundary from top to bottom: 100 (corner), 200, 200, 100 (corner). Right boundary from top to bottom: 100 (corner), 50, 0, 0 (corner). Bottom boundary left to right: 100 (corner), 50, 0, 0 (corner).] ## Short answer Apply the five-point Laplace formula u=(sum of four neighbours)/4 and iterate by Gauss-Seidel. The interior values converge to u1=120.8333, u2=79.1667, u3=104.1667, and u4=45.8333. ## Answer Label the 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)] Reading the figure, the boundary values touching each point are: u₁: top 100, left 200; u₂: top 100, right 50; u₃: left 200, bottom 50; u₄: right 0, bottom 0. Corner values are not used by the five-point formula. #### Equations ```text u₁ = ¼(300 + u₂ + u₃) u₂ = ¼(150 + u₁ + u₄) u₃ = ¼(250 + u₁ + u₄) u₄ = ¼(0 + u₂ + u₃) ``` #### Iteration (Gauss–Seidel / Liebmann) Initial value 88 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 | 88.0000 | 88.0000 | 88.0000 | 88.0000 | | 1 | 119.0000 | 89.2500 | 114.2500 | 50.8750 | | 2 | 125.8750 | 81.6875 | 106.6875 | 47.0938 | | 3 | 122.0938 | 79.7969 | 104.7969 | 46.1484 | | 4 | 121.1484 | 79.3242 | 104.3242 | 45.9121 | | 5 | 120.9121 | 79.2061 | 104.2061 | 45.8530 | | 6 | 120.8530 | 79.1765 | 104.1765 | 45.8383 | | 7 | 120.8383 | 79.1691 | 104.1691 | 45.8346 | | 8 | 120.8346 | 79.1673 | 104.1673 | 45.8336 | | 9 | 120.8336 | 79.1668 | 104.1668 | 45.8334 | | 10 | 120.8334 | 79.1667 | 104.1667 | 45.8334 | | 11 | 120.8334 | 79.1667 | 104.1667 | 45.8333 | | 12 | 120.8333 | 79.1667 | 104.1667 | 45.8333 | Converged to 4 decimals after 12 iterations. #### Direct solution of the equations (check) ```text u₁ = 120.8333 u₂ = 79.1667 u₃ = 104.1667 u₄ = 45.8333 ``` **Answer: u₁ = 120.8333, u₂ = 79.1667, u₃ = 104.1667, u₄ = 45.8333** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the standard five-point formula for Laplace's equation state? A. Each interior value is the sum of its neighbours B. Each interior value is the average of its four neighbours C. Each interior value equals the average of the corner values D. Each interior value is the average of all boundary values Answer: B. u(i,j) = 1/4[u(i-1,j)+u(i+1,j)+u(i,j-1)+u(i,j+1)]. (Key point: Five-point formula) 2. What is the equation for u1 (top 100, left 200)? A. u1 = (100 + u2 + u3)/4 B. u1 = (400 + u2 + u3)/4 C. u1 = (300 + u2 + u3)/4 D. u1 = (300 + u2 + u4)/4 Answer: C. Known neighbours 100 + 200 = 300; the other neighbours are u2 and u3. (Key point: Equation for u1) 3. Are the corner boundary values used in the five-point formula? A. Yes, in every equation B. Yes, only for u1 C. Only when the mesh is 2x2 with symmetry D. No, they are not neighbours of any interior point Answer: D. Corner points are diagonal to the interior points. (Key point: Corners) 4. How are the equations solved iteratively? A. Gauss-Seidel (Liebmann): each new value is used immediately B. Newton-Raphson on each point C. Using only the old iteration's values for all points D. Using RK-4 Answer: A. Gauss-Seidel uses updated values straight away until convergence. 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