What does the standard five-point formula for Laplace's equation state?

Each interior value is the average of its four neighbours

Each interior value is the sum of its neighbours

Each interior value equals the average of the corner values

Each interior value is the average of all boundary values

Are the corner boundary values used in the five-point formula?

No, they are not neighbours of any interior point

Only when the mesh is 2x2 with symmetry

How are the equations solved iteratively?

Gauss-Seidel (Liebmann): each new value is used immediately

Using only the old iteration's values for all points

Asked 2/13
2074 Bhadra · 10 marks
2070 Magh · 10 marks

Laplace: 2x2 grid 100/200/50

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.

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).]

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

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)

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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Laplace: 2x2 grid 100/200/50

Equations

Iteration (Gauss–Seidel / Liebmann)

Direct solution of the equations (check)

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