Are the eigenvalues of the B‐spline isogeometric analysis approximation of − Δ u = λ u known in almost closed form?

Summary We consider the B‐spline isogeometric analysis approximation of the Laplacian eigenvalue problem −Δ u  =  λ u 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 matrixL n [ p ]arising from the isogeometric analysis approximation based on B‐splines of degree p of the unidimensional problem − u ′ ′ = λ u . Here, n is the mesh fineness parameter, and the size ofL n [ p ]is N ( n , p ) =  n  +  p  − 2. In previous works, it was established that the normalized sequence{ n − 2L n [ p ] } nenjoys an asymptotic spectral distribution described by a function e p ( 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 e p ( θ ). In particular, we show that e p ( θ ) is monotone increasing on [0, π ] for all p  ≥ 1 and that e p ( θ )→ θ 2 uniformly on [0, π ] as p → ∞ . For p  = 1 and p  = 2, we show thatL n [ p ]belongs to a matrix algebra associated with a fast unitary sine transform, and we compute eigenvalues and eigenvectors ofL n [ p ] . In both cases, the eigenvalues are given by e p ( θ j , n ), j  = 1,…, n  +  p  − 2, where θ j , n  =  j π / n . For p  ≥ 3, we provide numerical evidence of a precise asymptotic expansion for the eigenvalues of n − 2L n [ p ] , excluding the largestn p out = 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 ) − n p out ,λ jn − 2L n [ p ]= e p ( θ j , n ) + ∑ k = 1 αc k [ p ] ( θ j , n ) h k + E j , n , α[ p ] , where the eigenvalues of n − 2L n [ p ]are arranged in ascending order, λ 1 ( n − 2L n [ p ] ) ≤ … ≤ λ n + p − 2 ( n − 2L n [ p ] ) ;{ c k [ p ] } k = 1 , 2 , …is a sequence of functions from [0, π ] to R , which depends only on p ; h  = 1/ n and θ j , n  =  j π h for j  = 1,…, n ; andE j , n , α [ p ] = O ( h α + 1 ) is the remainder, which satisfies | E j , n , α [ p ] | ≤ C α[ p ]h α + 1for some constantC α [ p ]depending only on α and p . We also provide a proof of this expansion for α  = 0 and j  = 1,…, N ( n , p ) −(4 p  − 2), where 4 p  − 2 represents a theoretical estimate of the number of outliersn p out .We show through numerical experiments that, for p  ≥ 3 and k  ≥ 1, there exists a point θ (  p , k ) ∈ (0, π ) such thatc k [ p ] ( 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 y p , is strictly positive, and the equation λ j ( n − 2L n [ p ] ) = e p ( θ j , n ) holds numerically whenever θ j , n  <  θ (  p ), where θ (  p ) is a point in (0, y p ] 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 ofL n [ p ] , excluding then p outoutliers. 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 θ j , n  <  θ (  p ).