Question 16.13: The Laplace equation ∇²u = uxx + uyy = 0  is defined over th...

Since we have to compute the values of u(x, y) at the nodal points of the triangular region {x = 0; y = 0; x + y = 5} with mess length 1. Therefore, mesh points are given by

\begin{aligned} &x_0=0, x_1=1, x_2=2, x_3=3, x_4=4, x_5=5 \\ &y_0=0, y_1=1, y_2=2, y_3=3, y_4=4, y_5=5 \end{aligned}

Let u_{i j}=u\left(x_i, y_j\right) . We have to compute the values of u_{11}, u_{21}, u_{31}, u_{12}, u_{22}, u_{13} .

The following boundary conditions are defined

i) u = 0 over the edges x = 0, y = 0
ii) u = 25 − x² − y² at the edge x + y = 5

So, we have

\begin{aligned} &u_{00}=u_{10}=u_{20}=u_{30}=u_{40}=u_{50}=0 \\ &u_{01}=u_{02}=u_{03}=u_{04}=u_{05}=0 \\ &u_{14}=u_{41}=8 \\ &u_{23}=u_{32}=12 \end{aligned}

It is easy to apply standard 5-points formula for calculations at each nodal point to get following equations

\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}-4 u_{31}=-20 \\ \text { At }(1,2) & u_{22}+u_{11}+u_{13}-4 u_{12}=0 \\ \text { At }(2,2) & u_{12}+u_{21}-4 u_{22}=-24 \\ \text { At }(1,3) & u_{12}-4 u_{13}=-20 \end{array}

We get following iterations of Gauss–Seidel method for u_{11}, u_{21}, u_{31}, u_{12}, u_{22}, u_{13}

The final solution is given by

u_{11}=2, u_{21}=4, u_{31}=6, u_{12}=4, u_{22}=8, u_{13}=6

Note: The Laplace equation and boundary conditions are symmetrical about the line y = x. So, the problem can also be discussed with following symmetry consideration

u_{21}=u_{12} \text { and } u_{31}=u_{13}