Compute Eigenfunctions in an L-shaped Region
Specify an L-shaped region.
In[1]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_38.png)
L = Polygon[{{1, 0}, {2, 0}, {2, 2}, {0, 2}, {0, 1}, {1, 1}}];
Specify a Laplacian operator.
In[2]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_39.png)
\[ScriptCapitalL] = Laplacian[u[x, y], {x, y}];
Specify a Dirichlet boundary condition.
In[3]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_40.png)
\[ScriptCapitalB] = DirichletCondition[u[x, y] == 0., True];
Compute the eigenfunctions in the L-shaped region.
In[4]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_41.png)
{vals, funs} =
NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]},
u[x, y], {x, y} \[Element] L, 6];
Inspect the eigenvalues.
In[5]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_42.png)
vals
Out[5]=
![](assets.en/compute-eigenfunctions-in-an-l-shaped-region/O_18.png)
Visualize the eigenfunctions.
In[6]:=
![Click for copyable input](assets.en/compute-eigenfunctions-in-an-l-shaped-region/In_43.png)
Plot3D[#, {x, y} \[Element] L, PlotPoints -> 75, Mesh -> None,
PlotStyle -> Directive[Orange, Specularity[White, 30]],
BoxRatios -> {1, 1, 0.8}] & /@ funs
Out[6]=
![](assets.en/compute-eigenfunctions-in-an-l-shaped-region/O_19.png)