August 2017

In this paper we consider the B-spline IgA approximation of the second-order eigenvalue problem

-Delta u = lambda uonOmega =(0,1), with zero Dirichlet boundary conditions and with^{d}Delta = sum,_{j=1}^{d}partial^{2}/partial x_{j}^{2}dge 1. We use B-splines of degreeand maximal smoothness and we consider the natural Galerkin approach. By using elementary tensor arguments, we show that the eigenvalue-eigenvector structure of the discrete problem can be reduced to the case ofp=(p_{1},...,p_{d})d=1, pge 1, regularityC, with coefficient matrix^{p-1}Lhaving size_{n}^{[p]}N(n,p)=n+p-2. In previous works, it has been established that the normalized sequence{nhas a canonical distribution in the eigenvalue sense and the so-called spectral symbol^{-2}L_{n}^{[p]}}_{n}ehas been identified._{p}(theta)In this paper we provide numerical evidence of a precise asymptotic expansion for the eigenvalues, which obviously begins with the function

e, up to the largest_{p}neigenvalues which behave as outliers. More precisely, for every integer_{p}^{rm out}=p+rm mod(p,2)-2alphage 0, everyn, everypge 3and everyj=1,...,hat N=N(n,p)-n, the following asymptotic expansion holds: begin{align*} lambda_{p}^{rm out}=n-rm mod(p,2)_{j}(n^{-2}L_{n}^{[p]})=e_{p}(theta_{j,n,p})+sum_{k=1}^{alpha}c_{k}^{(p)}(theta_{j,n,p})h^{k}+E_{j,n,alpha}^{(p)}, end{align*} where: • the eigenvalues ofnare arranged in nondecreasing order and^{-2}L_{n}^{[p]}eis increasing; •_{p}{cis a sequence of functions from_{k}^{(p)}}_{k=1,2,...}[0,pi]tomathbb Rwhich depends only one; • for any_{p}pge 3andk, there existsbartheta (p,k)>0such thatcvanishes (at least numerically) on the whole nontrivial interval_{k}^{(p)}[0,bartheta (p,k)], so that the formula is exact, up to machine precision, for a large portion of the small eigenvalues; •h=1/nandtheta,_{j,n,p}=jpi/n=jpi hj=1,..., n-rm mod(p,2); •Eis the remainder (the error), which satisfies the inequality_{j,n,alpha}^{(p)}=O(h^{alpha +1})|Efor some constant_{j,n,alpha}^{(p)}|le C_{\}alpha h^{alpha +1}Cdepending only on_{\}alphaalphaande. For the case_{p}p=1,2the complete structure of the eigenvalues and eigenvectors is identified exactly. Indeed, for such values ofp, the matricesLbelong to Toeplitz -minus- Hankel algebras and this is also the reason why there are no outliers, that is_{n}^{[p]}n._{p}^{rm out}=0Moreover, for

pge 3and based on the eigenvalue asymptotics forn, we devise an extrapolation algorithm for computing the eigenvalues of the discrete problem with a high level of accuracy and with a relatively negligible computational cost. However, the algorithm is not necessary for all the spectrum and indeed, for^{-2}L_{n}^{[p]}pge 3andthetabelonging to the interval_{j,n,p}[0,bartheta (p)],bartheta (p)=inf, the value_{k}bartheta (p,k)ecoincides with_{p}(theta_{j,n,p})lambda, up to machine precision._{j}(n^{-2}L_{n}^{[p]})Such expansions are of the same type studied in the literature for the eigenvalues of a sequence of Toeplitz matrices

{Tand of a sequence of preconditioned Toeplitz matrices_{n}(f)}{T, for_{n}^{-1}(g)T_{n}(f)}ftrigonometric polynomial,gnonnegative, not identically zero trigonometric polynomial.Extensive numerical experiments are discussed and further future research steps are illustrated at the end of the paper.

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