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.

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.

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.

This are some figures of the results.

• The solutions:

• The Grid sectioning_diagram:

• The FEM result:

If you think this work is uesful for yourself, please give me a star.

Thank you!