1. Elliptic Partial Differential Equations – Direct Method
Transforming Numerical Methods Education for STEM Undergraduates
11/26/2018 1

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5. Physical Example of an Elliptic PDE

6. Discretizing the Elliptic PDE

7. Example: Direct 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 by using the direct method.

8. Example: Direct Method
We can discretize the plate by taking,

9. Example: Direct Method
The nodal temperatures at the boundary nodes are given by:

10. Example: Direct Method
Here we develop the equation for the temperature at the node (2,3)
i=2 and j=3

11. Example: Direct Method
We can develop similar equations for every interior node leaving us with an equal number of equations and unknowns.
Question: How many equations would this generate?

12. Example: Direct Method
We can develop similar equations for every interior node leaving us with an equal number of equations and unknowns.
Question: How many equations would this generate? Answer: 12
Solving yields:

13. The End

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This material is based upon work supported by the National Science Foundation under Grant # Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

16. The End - Really