Are the eigenvalues of the B-spline IgA approximation of −∆u = λu known in almost closed form?

In this paper we consider the B-spline IgA approximation of the second-order eigenvalue problem −∆u = λu on Ω = (0, 1), with zero Dirichlet boundary conditions and with ∆ = ∑d j=1 ∂ ∂xj , d ≥ 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, p ≥ 1, regularity Cp−1, with coefficient matrix L n having size N(n, p) = n+ p− 2. In previous works, it has been established that the normalized sequence {n−2L n }n has a canonical distribution in the eigenvalue sense and the so-called spectral symbol ep(θ) 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 n out p = p + mod(p, 2) − 2 eigenvalues which behave as outliers. More precisely, for every integer α ≥ 0, every n, every p ≥ 3 and every j = 1, . . . , N̂ = N(n, p)−n p = n−mod(p, 2), the following asymptotic expansion holds: λj(n L n ) = ep(θj,n,p) + α ∑ k=1 c (p) k (θj,n,p)h k + E (p) j,n,α, where: • the eigenvalues of n−2L n are arranged in nondecreasing order and ep is increasing; • {c k }k=1,2,... is a sequence of functions from [0, π] to R which depends only on ep; • for any p ≥ 3 and k, there exists θ̄(p, k) > 0 such that c k vanishes (at least numerically) on the whole nontrivial interval [0, θ̄(p, k)], so that the formula is exact, up to machine precision, for a large portion of the small eigenvalues; • h = 1 n and θj,n,p = jπ n = jπh, j = 1, . . . , n−mod(p, 2); • E j,n,α = O(h ) is the remainder (the error), which satisfies the inequality |E j,n,α| ≤ Cαh α+1 for some constant Cα depending only on α 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 L [p] n belong to Toeplitz -minusHankel algebras and this is also the reason why there are no outliers, that is n p = 0. Moreover, for p ≥ 3 and based on the eigenvalue asymptotics for n−2L n , we devise an extrapolation algorithm for computing the eigenvalues of the discrete problem with a high level of accuracy and with a relatively negligible