PDE Finite Element Method

This is the partial differential equation finite element method by using the Ritz-Galerkin methods.

We use the finite element method of triangular dissection to solve practical problems, and the related computational algorithms all use the Gaussian numerical integration method in numerical analysis.

Question:

Here are the simulation problems solved by the algorithm.

$$ \left{\begin{array}{c} -\nabla \cdot(\nabla u)+u=-e^{x+y},(x, y) \in[-1,1] \times[-1,1] \\ \mathrm{u}=\mathrm{e}^{-1+y}, \mathrm{x}=-1 \\ \mathrm{u}=\mathrm{e}^{1+y}, \mathrm{x}=1 \\ \nabla \mathrm{u} \cdot \overrightarrow{\mathrm{n}}=-\mathrm{e}^{\mathrm{x}-1}, \mathrm{y}=-1 \\ \mathrm{u}=\mathrm{e}^{\mathrm{x}+1}, \mathrm{y}=1 \end{array}\right. $$

In addition, we compare the errors between the finite element solution and the exact solution at different parametrizations for different mesh dissections.

Code

If you want to use this code in your machine, you should fork my work and download the code in your computer. Then, you just need to run "PDE_main.m". Absolulately, you can change the parameters to get the different results.

Result

This are some figures of the results.

• The solutions:

• The Grid sectioning_diagram:

• The FEM result:

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Thank you!