Question 16.11: Solve the Dirichlet problem ∇²u = 0 in ℜ, where ℜ is the squ...
Solve the Dirichlet problem
∇²u = 0 in ℜ, where ℜ is the square 0 ≤ x, y ≤ 0.75.
Given that u = x + y on the boundary of the square ℜ. Take h = 0.25.
The node points are as follows
\begin{aligned} &x_0=0, x_1=0.25, x_2=0.5, x_3=0.75 \\ &y_0=0, y_1=0.25, y_2=0.5, y_3=0.75 \end{aligned}
Let u_{i j}=u\left(x_i, y_j\right)
The boundary condition u = x + y provides the following results
\begin{array}{ll} u_{10}=u_{01}=0.25 & u_{02}=u_{20}=0.5 \\ u_{13}=u_{31}=1.0 & u_{32}=u_{23}=1.25 \end{array}
The standard 5-points formula provides following simplified set of linear equations
\begin{array}{ll} \text { At }(1,1) & 4 u_{11}-u_{21}-u_{12}=0.5 \\ \text { At }(2,1) & 4 u_{21}-u_{11}-u_{22}=1.5 \\ \text { At }(1,2) & 4 u_{12}-u_{22}-u_{11}=1.5 \\ \text { At }(2,2) & 4 u_{22}-u_{12}-u_{21}=2.5 \end{array}
The following table contains 6 iterations \left[\begin{array}{llll} u_{11} & u_{12} & u_{21} & u_{22} \end{array}\right] of Gauss–Seidel method with initial approximation \left[\begin{array}{llll} 0 & 0 & 0 & 0 \end{array}\right]^{ T } for the above diagonally dominant system.
| Iteration 3 | Iteration 2 | Iteration 1 |
| 0.457031 0.728516 0.728516 0.989258 |
0.328125 0.664062 0.664062 0.957031 |
0.125000 0.406250 0.406250 0.828125 |
| Iteration 6 | Iteration 5 | Iteration 4 |
| 0.499329 0.749664 0.749664 0.999832 |
0.497314 0.748657 0.748657 0.999329 |
0.489258 0.744629 0.744629 0.997314 |
The exact answer is as follows
u_{11}=0.5, u_{21}=0.75, u_{12}=0.75, u_{22}=1.00
