---
title: "Laplace: 2x2 grid 3000/2000/500"
subject: "Numerical Method"
chapter: 7
key: "semester_4/numerical_method:laplace-2x2-3000-grid"
source: https://www.bctengineer.com/q/semester_4/numerical_method/laplace-2x2-3000-grid
site: BCT Engineering
---

# Laplace: 2x2 grid 3000/2000/500

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

- Asked in: 2071 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-2x2-3000-grid
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-7

## Question

Calculate the value of u(x, y) satisfying the Laplace equation ∇²u = 0 at the interior points of the square region with boundary conditions shown in figure below.
[Figure: square region with interior points u1 (top-left), u2 (top-right), u3 (bottom-left), u4 (bottom-right). Top boundary: 3000, 3000, 3000 (left to right). Left boundary top to bottom: 3000, 2000, 2000, 1000. Right boundary: 500 (next to u2 row), 0 (next to u4 row), 0 (bottom). Bottom boundary: 500, 0.]

## Short answer

Using standard five-point Laplace formula and Gauss-Seidal iteration, converged interior values are: u1=1958.3333, u2=1541.6667, u3=1291.6667, u4=708.3333.

## 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)]

Reading the figure: top 3000, 3000 (above u₁, u₂); left 2000, 2000 (beside u₁, u₃); right 500, 0 (beside u₂, u₄); bottom 500, 0 (below u₃, u₄). Corner values are not used.

#### Equations
```text
u₁ = ¼(5000 + u₂ + u₃)
u₂ = ¼(3500 + u₁ + u₄)
u₃ = ¼(2500 + u₁ + u₄)
u₄ = ¼(0 + u₂ + u₃)
```

#### Iteration (Gauss–Seidel / Liebmann)
Initial value 1375 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 | 1375.0000 | 1375.0000 | 1375.0000 | 1375.0000 |
| 1 | 1937.5000 | 1703.1250 | 1453.1250 | 789.0625 |
| 2 | 2039.0625 | 1582.0312 | 1332.0312 | 728.5156 |
| 3 | 1978.5156 | 1551.7578 | 1301.7578 | 713.3789 |
| 4 | 1963.3789 | 1544.1895 | 1294.1895 | 709.5947 |
| 5 | 1959.5947 | 1542.2974 | 1292.2974 | 708.6487 |
| 6 | 1958.6487 | 1541.8243 | 1291.8243 | 708.4122 |
| 7 | 1958.4122 | 1541.7061 | 1291.7061 | 708.3530 |
| 8 | 1958.3530 | 1541.6765 | 1291.6765 | 708.3383 |
| 9 | 1958.3383 | 1541.6691 | 1291.6691 | 708.3346 |
| 10 | 1958.3346 | 1541.6673 | 1291.6673 | 708.3336 |
| … | … | … | … | … |
| 13 | 1958.3334 | 1541.6667 | 1291.6667 | 708.3333 |
| 14 | 1958.3333 | 1541.6667 | 1291.6667 | 708.3333 |

Converged to 4 decimals after 14 iterations.

#### Direct solution of the equations (check)
```text
u₁ = 1958.3333
u₂ = 1541.6667
u₃ = 1291.6667
u₄ = 708.3333
```

**Answer: u₁ = 1958.3333, u₂ = 1541.6667, u₃ = 1291.6667, u₄ = 708.3333**

## Check yourself

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

1. u1 = ?
   A. (6000 + u2 + u3)/4
   B. (3000 + u2 + u3)/4
   C. (2000 + u2 + u3)/4
   D. (5000 + u2 + u3)/4
   Answer: D. Top 3000 + left 2000. (Key point: Equation u1)

2. u4 = ?
   A. (u2 + u3)/4
   B. (500 + u2 + u3)/4
   C. (u1 + u2)/4
   D. (1000 + u2 + u3)/4
   Answer: A. Right and bottom are both 0. (Key point: Equation u4)

3. How are equations solved?
   A. One RK-4 step
   B. Gauss-Seidel iteration until values stop changing, or directly
   C. Using Euler's method
   D. Binary search
   Answer: B. Iteration converges in about 14 steps. (Key point: Method)

4. What is u1?
   A. 1541.6667
   B. 1291.6667
   C. 1958.3333
   D. 708.3333
   Answer: C. The solution is 1958.3333, 1541.6667, 1291.6667, 708.3333. (Key point: Result)

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