Question 16.12: The Laplace equation ∇²u = uxx + uyy = 0  is defined over th...

The nodal points of the square region {0 ≤ x ≤ 0.6; 0 ≤ y ≤ 0.6} with mess length 0.2 are given by

\begin{aligned} &x_0=0, x_1=0.2, x_2=0.4, x_3=0.6 \\ &y_0=0, y_1=0.2, y_2=0.4, y_3=0.6 \end{aligned}

The Dirichlet condition (u = 0 ) is given at x = 0, y = 0, y = 0.6, and Neumann condition \left\lgroup\frac{\partial u}{\partial x}=1 \right\rgroup is given at the edge, x = 0.6. The values of u(x, y) have to be determined at the mess points and boundary (x = 0.6).

Let u_{i j}=u\left(x_i, y_j\right) . Now, we have to compute the values u_{11}, u_{12}, u_{21}, u_{22}, u_{31}, u_{32} .

To replace the derivative boundary condition at x = 0.6 with central difference, we have to extend the boundary at x = 0.6.

On replacing the boundary condition \frac{\partial u}{\partial x}=1 \text { at any point }\left(x_i, y_j\right) with central difference, we get

\left.\frac{\partial u}{\partial x}\right|_{\left(x_i, y_j\right)}=\frac{u_{i+1, j}-u_{i-1, j}}{2 h}=1

At the edge x = 0.6(i = 3) and for j = 1, 2, we have

\frac{u_{4,1}-u_{2,1}}{2(0.2)}=1 \quad \Rightarrow u_{4,1}=0.4+u_{2,1}           (16.76)

\frac{u_{4,2}-u_{2,2}}{2(0.2)}=1 \quad \Rightarrow u_{4,2}=0.4+u_{2,2}           (16.77)

On applying standard 5-points formula for calculations at each nodal point, we have

\begin{array}{ll} \text { At }(1,1) & u_{21}+u_{12}-4 u_{11}=0 \\ \text { At }(2,1) & u_{31}+u_{11}+u_{22}-4 u_{21}=0 \\ \text { At }(3,1) & u_{21}+u_{32}+u_{41}-4 u_{31}=0 \end{array}

u_{21}+u_{32}+\left(u_{21}+0.4\right)-4 u_{31}=0          From Eq. (16.76)

\begin{array}{ll} \text { At }(1,2) & u_{22}+u_{11}-4 u_{12}=0 \\ \text { At }(2,2) & u_{12}+u_{21}+u_{32}-4 u_{22}=0 \\ \text { At }(3,2) & u_{31}+u_{22}+u_{42}-4 u_{32}=0 \end{array}

u_{31}+u_{22}+\left(0.4+u_{22}\right)-4 u_{32}=0          From Eq. (16.77)

Equations (16.76) and (16.77) are used to replace values of u_{41} \text { and } u_{42} from equations at nodes (3, 1) and (3, 2). Now, we are left with following six equations

\begin{aligned} &u_{11}=\frac{1}{4}\left(u_{21}+u_{12}\right) \\ &u_{21}=\frac{1}{4}\left(u_{31}+u_{11}+u_{22}\right) \\ &u_{31}=\frac{1}{4}\left(u_{21}+u_{32}+u_{21}+0.4\right) \\ &u_{12}=\frac{1}{4}\left(u_{22}+u_{11}\right) \\ &u_{22}=\frac{1}{4}\left(u_{12}+u_{21}+u_{32}\right) \\ &u_{32}=\frac{1}{4}\left(u_{31}+u_{22}+0.4+u_{22}\right) \end{aligned}

These equations are diagonally dominant, so we can apply Gauss–Seidel method. On solving this system by Gauss–Seidel with initial approximation [0,0,0,0,0,0]^{ T } , we get following iterations

The final solution of the Laplace equation is given by