Eigenvalues and Eigenmodes of L-Shaped Membrane

This example shows how to calculate eigenvalues and eigenvectors. The eigenvalue problem is $-\Delta u=\lambda u$. This example computes all eigenmodes with eigenvalues smaller than 100.

Create a model and include this geometry. The geometry of the L-shaped membrane is described in the file `lshapeg`.

```model = createpde; geometryFromEdges(model,@lshapeg);```

Set zero Dirichlet boundary conditions on all edges.

```applyBoundaryCondition(model,"dirichlet", ... Edge=1:model.Geometry.NumEdges, ... u=0);```

Specify the coefficients for the problem: `d` = 1 and `c` = 1. All other coefficients are equal to zero.

`specifyCoefficients(model,m=0,d=1,c=1,a=0,f=0);`

Set the interval `[0 100]` as the region for the eigenvalues in the solution.

`r = [0 100];`

Create a mesh and solve the problem.

```generateMesh(model,Hmax=0.05); results = solvepdeeig(model,r);```

There are 19 eigenvalues smaller than 100.

`length(results.Eigenvalues)`
```ans = 19 ```

Plot the first eigenmode and compare it to the MATLAB's `membrane` function.

```u = results.Eigenvectors; pdeplot(model,XYData=u(:,1),ZData=u(:,1));```

```figure membrane(1,20,9,9)```

Eigenvectors can be multiplied by any scalar and remain eigenvectors. This explains the difference in scale that you see.

`membrane` can produce the first 12 eigenfunctions for the L-shaped membrane. Compare the 12th eigenmodes.

```figure pdeplot(model,XYData=u(:,12),ZData=u(:,12));```

```figure membrane(12,20,9,9)```