  The Laplace eigenvalue problem on circular sectors has eigenfunctions with
corner singularities. Standard methods may produce suboptimal approximation
results. To address this issue, a novel numerical algorithm that enhances
standard isogeometric analysis is proposed in this paper by using a
single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal
convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the
results show that smooth splines possess a superior approximation constant
compared to their $C^0$-continuous counterparts for the lower part of the
Laplace spectrum. This is an extension of previous findings about excellent
spectral approximation properties of smooth splines on rectangular domains to
circular sectors. In addition, graded meshes prove to be particularly
advantageous for an accurate approximation of a limited number of eigenvalues.
The novel algorithm applied here has a drawback in the singularity of the
isogeometric parameterization. It results in some basis functions not belonging
to the solution space of the corresponding weak problem, which is considered a
variational crime. However, the approach proves to be robust. Finally, a
hierarchical mesh structure is presented to avoid anisotropic elements, omit
redundant degrees of freedom and keep the number of basis functions
contributing to the variational crime constant, independent of the mesh size.
Numerical results validate the effectiveness of hierarchical mesh grading for
the simulation of eigenfunctions with and without corner singularities.
