Question 16.16: Solve the following Poisson problem ∇²u = e^x + y in ℜ, wher...

Numerical Methods: Fundamentals and Applications

Solve the following Poisson problem

$\nabla^2 u=e^{x+y}$ in ℜ, where ℜ is the square 0 ≤ x, y ≤ 0.75.

Given that u = x² + y² on the boundary of the square ℜ. Take h = 0.25.

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The node points are as follows

$\begin{aligned} &x_0=0, x_1=0.25, x_2=0.5, x_3=0.75 \\ &y_0=0, y_1=0.25, y_2=0.5, y_3=0.75 \end{aligned}$

The boundary condition u = x² + y² provides the following values

$\begin{array}{ll} u_{10}=u_{01}=0.0625 & u_{02}=u_{20}=0.25 \\ u_{13}=u_{31}=0.625 & u_{32}=u_{23}=0.8125 \end{array}$

The system of linear equations is obtained by applying standard 5-points formula. Then, Gauss–Seidel method provides following 6 iterations

Iteration 1
0.457299	0.187044	0.187044	0.005489
Iteration 2
0.526151	0.324749	0.324749	0.099011
Iteration 3
0.543365	0.359176	0.359176	0.167863
Iteration 4
0.547668	0.367782	0.367782	0.185077
Iteration 5
0.548744	0.369934	0.369934	0.189380
Iteration 6
0.549013	0.370472	0.370472	0.190456

After 6 iterations, the solution is as follows

u_{11}=0.190456 \quad u_{12}=0.370472 \quad u_{21}=0.370472 \quad u_{22}=0.549013

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