Compute Eigenfunctions in an L-shaped Region

Specify an L-shaped region.

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L = Polygon[{{1, 0}, {2, 0}, {2, 2}, {0, 2}, {0, 1}, {1, 1}}];

Specify a Laplacian operator.

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\[ScriptCapitalL] = Laplacian[u[x, y], {x, y}];

Specify a Dirichlet boundary condition.

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\[ScriptCapitalB] = DirichletCondition[u[x, y] == 0., True];

Compute the eigenfunctions in the L-shaped region.

In[4]:=

{vals, funs} = NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x, y], {x, y} \[Element] L, 6];

Inspect the eigenvalues.

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vals

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Visualize the eigenfunctions.

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Plot3D[#, {x, y} \[Element] L, PlotPoints -> 75, Mesh -> None, PlotStyle -> Directive[Orange, Specularity[White, 30]], BoxRatios -> {1, 1, 0.8}] & /@ funs

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