  We introduce a boundary penalization technique to improve the spectral
approximation of isogeometric analysis (IGA). The technique removes the
outliers appearing in the high-frequency region of the approximate spectrum
when using the $C^{p-1}, p$-th ($p\ge3$) order isogeometric elements. We focus
on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate
the idea and then use the tensor-product structure to generate the stiffness
and mass matrices for multiple dimensional problems. To remove the outliers, we
penalize the product of the higher-order derivatives from both the solution and
test spaces at the domain boundary. Intuitively, we construct a better
approximation by weakly imposing features of the exact solution. Effectively,
we add terms to the variational formulation at the boundaries with minimal
extra computational cost. We then generalize the idea to remove the outliers
for the isogeometric analysis to the Neumann eigenvalue problem (for $p\ge2$).
The boundary penalization does not change the test and solution spaces. In the
limiting case when the penalty goes to infinity, we perform the dispersion
analysis of $C^2$ cubic elements for Dirichlet eigenvalue problem and $C^1$
quadratic elements for Neumann eigenvalue problem. We obtain the analytical
eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments
show optimal convergence rates for the eigenvalues and eigenfunctions of the
discrete operator.
