  Anisotropic mesh adaptation is studied for the linear finite element solution
of eigenvalue problems with anisotropic diffusion operators. The M-uniform mesh
approach is employed with which any nonuniform mesh is characterized
mathematically as a uniform one in the metric specified by a metric tensor.
Bounds on the error in the computed eigenvalues are established for
quasi-M-uniform meshes. Numerical examples arising from the Laplace-Beltrami
operator on parameterized surfaces, nonlinear diffusion, thermal diffusion in a
magnetic field in plasma physics, and the Laplacian operator on an L-shape
domain are presented. Numerical results show that anisotropic adaptive meshes
can lead to more accurate computed eigenvalues than uniform or isotropic
adaptive meshes. They also confirm the second order convergence of the error
that is predicted by the theoretical analysis.
  The effects of approximation of curved boundaries on the computation of
eigenvalue problems is also studied in two dimensions. It is shown that the
initial mesh used to define the geometry of the physical domain should contain
at least \sqrt{N} boundary points to keep the effects of boundary approximation
at the level of the error of the finite element approximation, where N is the
number of the elements in the final adaptive mesh. Only about N^{1/3} boundary
points in the initial mesh are needed for boundary value problems. This implies
that the computation of eigenvalue problems is more sensitive to the boundary
approximation than that of boundary value problems.
