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Anisotropic mesh adaptation is studied for the linear finite element solutionof eigenvalue problems with anisotropic diffusion operators. The M-uniform meshapproach is employed with which any nonuniform mesh is characterizedmathematically as a uniform one in the metric specified by a metric tensor.Bounds on the error in the computed eigenvalues are established forquasi-M-uniform meshes. Numerical examples arising from the Laplace-Beltramioperator on parameterized surfaces, nonlinear diffusion, thermal diffusion in amagnetic field in plasma physics, and the Laplacian operator on an L-shapedomain are presented. Numerical results show that anisotropic adaptive meshescan lead to more accurate computed eigenvalues than uniform or isotropicadaptive meshes. They also confirm the second order convergence of the errorthat is predicted by the theoretical analysis.  The effects of approximation of curved boundaries on the computation ofeigenvalue problems is also studied in two dimensions. It is shown that theinitial mesh used to define the geometry of the physical domain should containat least \sqrt{N} boundary points to keep the effects of boundary approximationat the level of the error of the finite element approximation, where N is thenumber of the elements in the final adaptive mesh. Only about N^{1/3} boundarypoints in the initial mesh are needed for boundary value problems. This impliesthat the computation of eigenvalue problems is more sensitive to the boundaryapproximation than that of boundary value problems.

A study on anisotropic mesh adaptation for finite element approximation of eigenvalue problems with anisotropic diffusion operators