How to numerically compute the eigenvalues of the Laplacian in a triangular domain with Dirichlet boundary condition?

Question

Consider an arbitrary triangle. Now impose the homogeneous Dirichlet boundary condition. How to compute the eigenvalues and eigenvectors of the Laplacian $-\nabla^2 = - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} $ numerically?

Answer 1

This is the setup for the standard finite element approach to the problem:

\begin{eqnarray*} & & -\nabla^2 u = \lambda u \\ & \Rightarrow & (\forall v \in V) \, -(\nabla^2 u) v = \lambda u v \\ & \Rightarrow & (\forall v \in V) \, \int_\Delta -(\nabla^2 u) v dx = \int_\Delta \lambda u v dx \\ & \Rightarrow & (\forall v \in V) \, \int_\Delta \nabla u \cdot \nabla v dx - \int_{\partial \Delta} v (\nabla u \cdot dn) = \int_\Delta \lambda u v dx \\ & \Rightarrow & (\forall v \in V) \, \int_\Delta \nabla u \cdot \nabla v dx = \int_\Delta \lambda u v dx \end{eqnarray*}

where $\Delta$ is the triangle. In the first transformation we multiply by test functions from a chosen set $V$ which satisfy a homogeneous Dirichlet condition. In the second transformation we integrate both sides. In the third transformation we integrate by parts. In the last transformation we use the boundary condition of the test functions.

If we now choose to have $u$ approximated by continuous piecewise linear functions on a finite element mesh of the triangle which satisfy the boundary condition, and use test functions $v$ of the same type, then the last equation is an ordinary matrix eigenvalue problem.