Are the eigenvalues of the B-spline isogeometric analysis approximation of −Δu = λu known in almost closed form?

We consider the B-spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δu = λu over the d-dimensional hypercube (0,1)d. By using tensor-product arguments, we show that the eigenvalue–eigenvector structure of the resulting discretization matrix is completely determined by the eigenvalue–eigenvector structure of the matrix L[p]n arising from the isogeometric analysis approximation based on B-splines of degree p of the unidimensional problem −u′′ = u. Here, n is the mesh fineness parameter, and the size of L[p]n is N(n,p) = n + p − 2. In previous works, it was established that the normalized sequence {n−2L[p]n}n enjoys an asymptotic spectral distribution described by a function ep(θ), the so-called spectral symbol. The contributions of this paper can be summarized as follows: We prove several important analytic properties of the spectral symbol ep(θ). In particular, we show that ep(θ) is monotone increasing on [0,π] for all p ≥ 1 and that ep(θ)→θ2 uniformly on [0,π] as p→∞. For p = 1 and p = 2, we show that L[p]n belongs to a matrix algebra associated with a fast unitary sine transform, and we compute eigenvalues and eigenvectors of L[p]n. In both cases, the eigenvalues are given by ep(θj,n), j = 1,…,n + p − 2, where θj,n = jπ/n. For p ≥ 3, we provide numerical evidence of a precise asymptotic expansion for the eigenvalues of n−2L[p]n, excluding the largest noutp = p − 2 +mod( p, 2) eigenvalues (the so-called outliers). More precisely, we numerically show that for every p ≥ 3, every integer α ≥ 0, every n, and every j= 1, … ,N(n, p) − noutp, (Formula presented.) where the eigenvalues of n−2L[p]n are arranged in ascending order, λ1(n−2L[p]n)≤ … ≤ n+p−2(n−2L[p]n); {c[p]k } k=1,2,… is a sequence of functions from [0,π] to R, which depends only on p; h = 1/n and θj,n = jπh for j = 1,…,n; and E[p],j, n, α = O(h+1) is the remainder, which satisfies |E[p],j, n, α|≤ C[p]α hα+1 for some constant C[p]α depending only on α and p. We also provide a proof of this expansion for α = 0 and j = 1,…,N(n,p) −(4p − 2), where 4p − 2 represents a theoretical estimate of the number of outliers noutp. We show through numerical experiments that, for p ≥ 3 and k ≥ 1, there exists a point θ(p,k) ∈ (0,π) such that c[p]k(θ) vanishes on [0,θ(p,k)]. Moreover, as it is suggested by the numerics of this paper, the infimum of θ(p,k) over all k ≥ 1, say yp, is strictly positive, and the equation λ1(n−2L[p]n) holds numerically whenever θj,n < θ(p), where θ(p) is a point in (0,yp] which grows with p. For p ≥ 3, based on the asymptotic expansion in the above item 3, we propose a parallel interpolation–extrapolation algorithm for computing the eigenvalues of L[p]n, excluding the noutp outliers. The performance of the algorithm is illustrated through numerical experiments. Note that, by the previous item 4, the algorithm is actually not necessary for computing the eigenvalues corresponding to points θj,n < θ(p).