  The finite element method(FEM) is applied to bound leading eigenvalues of
Laplace operator over polygonal domain. Compared with classical numerical
methods, most of which can only give concrete eigenvalue bounds over special
domain of symmetry, our proposed algorithm can provide concrete eigenvalue
bounds for domain of arbitrary shape, even in the case that eigenfunction has
singularity.
  The problem of eigenvalue estimation is solved in two steps. First, we
construct computable a priori error estimation for FEM solution of Poisson's
problem even for non-convex domain with re-entrant corner. Second, a new
computable lower and upper bounds is developed for eigenvalues. As the interval
arithmetic is implemented in the FEM computation, the desired eigenvalue bounds
can be expected to be mathematically correct. The Lehmann's theorem is also
adopted to sharpen the eigenvalue bounds with high precision. At the end of
this paper, we illustrate several computation examples, such as the case of
L-shaped domain and crack domain, to demonstrate the efficiency and flexibility
of proposed method.
