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August 2017
Abstract:In this paper we consider the B-spline IgA approximation of the second-order eigenvalue problem -Delta u = λ u on Omega =(0,1)d, with zero Dirichlet boundary conditions and with Delta = sumj=1dpartial2/partial xj2, dge 1. We use B-splines of degree p=(p1,...,pd) and 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 of d=1, pge 1, regularity Cp-1, with coefficient matrix Ln[p] having size N(n,p)=n+p-2. In previous works, it has been established that the normalized sequence {n-2Ln[p]}n has a canonical distribution in the eigenvalue sense and the so-called spectral symbol ep(theta) has been identified.
In this paper we provide numerical evidence of a precise asymptotic expansion for the eigenvalues, which obviously begins with the function ep, up to the largest nprm out=p+rm mod(p,2)-2 eigenvalues which behave as outliers. More precisely, for every integer alphage 0, every n, every pge 3 and every j=1,...,hat N=N(n,p)-nprm out=n-rm mod(p,2), the following asymptotic expansion holds: begin{align*} λj(n-2 Ln[p])=ep(thetaj,n,p)+sumk=1alphack(p)(thetaj,n,p)hk+Ej,n,alpha(p), end{align*} where: • the eigenvalues of n-2Ln[p] are arranged in nondecreasing order and ep is increasing; • {ck(p)}k=1,2,... is a sequence of functions from [0,pi] to mathbb R which depends only on ep; • for any pge 3 and k, there exists bartheta (p,k)>0 such that ck(p) vanishes (at least numerically) on the whole nontrivial interval [0,bartheta (p,k)], so that the formula is exact, up to machine precision, for a large portion of the small eigenvalues; • h=1/n and thetaj,n,p=jpi/n=jpi h, j=1,..., n-rm mod(p,2); • Ej,n,alpha(p)=O(halpha +1) is the remainder (the error), which satisfies the inequality |Ej,n,alpha(p)|le Calpha halpha +1 for some constant Calpha depending only on alpha and ep. For the case p=1,2 the complete structure of the eigenvalues and eigenvectors is identified exactly. Indeed, for such values of p, the matrices Ln[p] belong to Toeplitz -minus- Hankel algebras and this is also the reason why there are no outliers, that is nprm out=0.
Moreover, for pge 3 and based on the eigenvalue asymptotics for n-2Ln[p], 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 pge 3 and thetaj,n,p belonging to the interval [0,bartheta (p)], bartheta (p)=infkbartheta (p,k), the value ep(thetaj,n,p) coincides with λj(n-2 Ln[p]), up to machine precision.
Such expansions are of the same type studied in the literature for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)} and of a sequence of preconditioned Toeplitz matrices {Tn-1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, 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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