---
title: "Elliptic: 2x2 grid 10/20/40/50"
subject: "Numerical Method"
chapter: 7
key: "semester_4/numerical_method:elliptic-2x2-10-20-40-50"
source: https://www.bctengineer.com/q/semester_4/numerical_method/elliptic-2x2-10-20-40-50
site: BCT Engineering
---

# Elliptic: 2x2 grid 10/20/40/50

Numerical Method · Chapter 7: Numerical solution of Partial differential Equation

- Asked in: 2069 Bhadra, 12 marks
- Video: [Lecture 3 Finite Difference Method to Solve Elliptic Equation](https://www.youtube.com/watch?v=aQx-URrIn3w) — Vipul Patel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/elliptic-2x2-10-20-40-50
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7

## Question

Solve the elliptic equation 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: 10, 20 (left to right). Left boundary: 10, 20 (top to bottom). Right boundary: 40, 50 (top to bottom). Bottom boundary: 40, 50 (left to right).]

## Short answer

Using the standard five-point finite difference approximation for Laplace's equation and Gauss-Seidel iteration, the interior nodal values converge to u1 = 20.0000, u2 = 30.0000, u3 = 30.0000, u4 = 40.0000.

## 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 10; u₂: top 20, right 40; u₃: left 20, bottom 40; u₄: right 50, bottom 50.

#### Equations
```text
u₁ = ¼(20 + u₂ + u₃)
u₂ = ¼(60 + u₁ + u₄)
u₃ = ¼(60 + u₁ + u₄)
u₄ = ¼(100 + u₂ + u₃)
```

#### Iteration (Gauss–Seidel / Liebmann)
Initial value 30 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 | 30.0000 | 30.0000 | 30.0000 | 30.0000 |
| 1 | 20.0000 | 27.5000 | 27.5000 | 38.7500 |
| 2 | 18.7500 | 29.3750 | 29.3750 | 39.6875 |
| 3 | 19.6875 | 29.8438 | 29.8438 | 39.9219 |
| 4 | 19.9219 | 29.9609 | 29.9609 | 39.9805 |
| 5 | 19.9805 | 29.9902 | 29.9902 | 39.9951 |
| 6 | 19.9951 | 29.9976 | 29.9976 | 39.9988 |
| 7 | 19.9988 | 29.9994 | 29.9994 | 39.9997 |
| 8 | 19.9997 | 29.9998 | 29.9998 | 39.9999 |
| 9 | 19.9999 | 30.0000 | 30.0000 | 40.0000 |
| 10 | 20.0000 | 30.0000 | 30.0000 | 40.0000 |
| 11 | 20.0000 | 30.0000 | 30.0000 | 40.0000 |

Converged to 4 decimals after 11 iterations.

#### Direct solution of the equations (check)
```text
u₁ = 20.0000
u₂ = 30.0000
u₃ = 30.0000
u₄ = 40.0000
```

**Answer: u₁ = 20.0000, u₂ = 30.0000, u₃ = 30.0000, u₄ = 40.0000**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. u1 = ?
   A. (10 + u2 + u3)/4
   B. (30 + u2 + u3)/4
   C. (20 + u2 + u3)/4
   D. (60 + u2 + u3)/4
   Answer: C. Top 10 + left 10 = 20. (Key point: Equation u1)

2. u4 = ?
   A. (90 + u2 + u3)/4
   B. (50 + u2 + u3)/4
   C. (80 + u2 + u3)/4
   D. (100 + u2 + u3)/4
   Answer: D. Right 50 + bottom 50. (Key point: Equation u4)

3. Why do u2 and u3 have the same equation?
   A. Both have boundary sum 60 and neighbours u1 and u4
   B. They are boundary points
   C. Because Laplace's equation forces all equal
   D. They are corners
   Answer: A. Hence u2 = u3 = 30. (Key point: Equal values)

4. What is u4?
   A. 30
   B. 40
   C. 20
   D. 50
   Answer: B. Solution: 20, 30, 30, 40. (Key point: Result)

## Close variants set in other exams

- [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×
- [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×

## 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: 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: 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
- [Elliptic: 8 cm plate, 50/30/10](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Apoisson-8cm-plate) — Numerical Method, chapter 7

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