TITLE: What is the purpose of computing the eigenvalue of a PDE problem?
QUESTION [22 upvotes]: I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one).  But when I solve partial differential equations using a finite difference scheme, I'm generally more interested in the solution, its stability, and its convergence.  What insight can I gain from solving an eigenvalue problem from the same PDE?  
Say, for example, the eigenvalue problem  
$\nabla^2u=\lambda u$  in $\Omega$
$u=0$ on $\partial \Omega$  
Without a context to a physical problem, does solving this eigenvalue problem provide any insight into other related problems such as  
$\nabla^2u=0$  in $\Omega$
$u=0$ on $\partial \Omega$
or 
$\nabla^2u=f$  in $\Omega$
$u=0$ on $\partial \Omega$ ?

REPLY [5 votes]: Solving a time dependent equation like the heat equation
$$
u_t-\nabla^2u=0,\quad x\in\Omega,\quad t>0
$$
with boundary condition $u(x,t)=0$ if $x\in\partial\Omega$, $t>0$, and initial condition $u(x,0)=u_0(x)$, $x\in\Omega$, by separation of variables (that is, looking for solutions of the form $u(x,t)=T(t)X(x)$) leads in a natural way to the eigenvalue problem. The solution can be expanded (under some conditions on $\Omega$ and $U_0$) as
$$
u(x,t)=\sum_{n}e^{-\lambda_nt}X_n(t),
$$
where $\lambda_n$ are the eigenvalues and $X_n$ the associated eigenfunctions.