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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?

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  • $\begingroup$ You should probably state what is intended by Dirichlet boundary conditions for the reader's sake. $\endgroup$ Commented Jul 23, 2014 at 21:28

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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.

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