  Certain applications that analyze damping effects require the solution of
quadratic eigenvalue problems (QEPs). We use refined isogeometric analysis
(rIGA) to solve quadratic eigenproblems. rIGA discretization, while conserving
desirable properties of maximum-continuity isogeometric analysis (IGA), reduces
the interconnection between degrees of freedom by adding low-continuity basis
functions. This connectivity reduction in rIGA's algebraic system results in
faster matrix LU factorizations when using multifrontal direct solvers. We
compare computational costs of rIGA versus those of IGA when employing Krylov
eigensolvers to solve quadratic eigenproblems arising in 2D vector-valued
multifield problems. For large problem sizes, the eigencomputation cost is
governed by the cost of LU factorization, followed by costs of several
matrix-vector and vector-vector multiplications, which correspond to Krylov
projections. We minimize the computational cost by introducing C^0 and C^1
separators at specific element interfaces for our rIGA generalizations of the
curl-conforming Nedelec and divergence-conforming Raviart-Thomas finite
elements. Let p be the polynomial degree of basis functions; the LU
factorization is up to O((p-1)^2) times faster when using rIGA compared to IGA
in the asymptotic regime. Thus, rIGA theoretically improves the total
eigencomputation cost by O((p-1)^2) for sufficiently large problem sizes. Yet,
in practical cases of moderate-size eigenproblems, the improvement rate
deteriorates as the number of computed eigenvalues increases because of
multiple matrix-vector and vector-vector operations. Our numerical tests show
that rIGA accelerates the solution of quadratic eigensystems by O(p-1) for
moderately sized problems when we seek to compute a reasonable number of
eigenvalues.
