1. Elliptic Partial Differential Equations – Gauss-Seidel Method
Transforming Numerical Methods Education for STEM Undergraduates
1/18/2019 1

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5. Physical Example of an Elliptic PDE

6. Discretizing the Elliptic PDE

7. The Gauss-Seidel Method
Recall the discretized equation
This can be rewritten as
For the Gauss-Seidel Method, this equation is solved iteratively for all interior nodes until a pre-specified tolerance is met.

8. Example 2: Gauss-Seidel Method
Consider a plate that is subjected to the boundary conditions shown below. Find the temperature at the interior nodes using a square grid with a length of using the Gauss-Siedel method. Assume the initial temperature guess at all interior nodes to be

9. Example: Gauss-Seidel Method
We can discretize the plate by taking

10. Example: Gauss-Seidel Method
The nodal temperatures at the boundary nodes are given by:

11. Example: Gauss-Seidel Method
Now we can begin to solve for the temperature at each interior node using
Assume all internal nodes temperature guess of zero.
Iteration #1
i=1 and j=1
i=1 and j=2

12. Example: Gauss-Seidel Method
After the first iteration, the temperatures are as follows. These will now be used as the nodal temperatures for the second iteration.

13. Example: Gauss-Seidel Method
Iteration #2
i=1 and j=1

14. Example: Gauss-Seidel Method
The figures below show the temperature distribution and absolute relative error distribution in the plate after two iterations:
Temperature Distribution
Absolute Relative Approximate
Error Distribution

15. Example: Gauss-Seidel Method
Node
Temperature Distribution in the Plate (°C)
Number of Iterations 1 2 10
9.2773

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19. The End - Really