Question 16.15: Solve the Poisson equation uxx+ uyy  = x² + y² for a thin re...

The edges of the square region are x = 0, x = 2, y = 0, y = 2 with the mess length 0.5. So, the values of x and y are as follows

\begin{array}{ll} x & y \\ x_0=0 & y_0=0 \\ x_1=0.5 & y_1=0.5 \\ x_2=1 & y_2=1 \\ x_3=1.5 & y_3=1.5 \end{array}

x_4=2 \quad y_4=2             (16.81)

Let the value of u\left(x_i, y_j\right)=u_{i j} , we have

\begin{aligned} &u_{01}=0, \quad u_{02}=0, u_{03}=0 \quad(x=0) \\ &u_{41}=0, \quad u_{42}=0, u_{43}=0 \quad(x=2) \\ &u_{10}=100, \quad u_{20}=100, u_{30}=100 \quad(y=0) \\ & \end{aligned}

u_{14}=100, \quad u_{24}=100, u_{34}=100 \quad(y=2)               (16.82)

Let us solve the Poisson equation u_{x x}+u_{y y}=x^2+y^2 with symmetry consideration. Since the boundary conditions and the Poisson equation are symmetrical around the lines x =1 and y =1; so we can assume the values about these lines are equal; i.e.

u_{11}=u_{31}, u_{12}=u_{32}, u_{13}=u_{33}       (Symmetry about x = 1)

u_{11}=u_{13}, u_{21}=u_{23}, u_{31}=u_{33}           (Symmetry about y = 1)            (16.83)

So, we need to find values of u_{11}, u_{21}, u_{12}, u_{22} only.

The Poisson equation \nabla^2 u=u_{x x}+u_{y y}=x^2+y^2 \text { at point }\left(x_i, y_j\right) is given by

\frac{u_{i+1, j}-2 u_{i, j}+u_{i-1, j}}{h^2}+\frac{u_{i, j+1}-2 u_{i, j}+u_{i, j-1}}{h^2}=x_i^2+y_j^2

(or)      u_{i+1, j}+u_{i-1, j}+u_{i, j+1}+u_{i, j-1}-4 u_{i, j}=h^2\left(x_i^2+y_j^2\right)

We need to find values of u_{11}, u_{21}, u_{12}, u_{22}, so

Using symmetries from Eq. (16.81), the values of x_i, y_j, u_{i j} from Eqs. (16.82) and (16.83), the set of Eqs. (16.84) becomes

On solving this system of linear equations with the help of Gauss–Seidel method for u_{11}, u_{21}, u_{12}, u_{22} , we have

The Final solution is given by

\begin{aligned} &u_{11}=u_{13}=u_{31}=u_{33}=49.842991 \\ &u_{21}=u_{23}=62.249241 \\ &u_{12}=u_{32}=37.249241 \\ &u_{22}=49.624241 \end{aligned}