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Are the eigenvalues of the B-spline isogeometric analysis approximation of \(-\Delta u=\lambda u\) known in almost closed form? (English) Zbl 1513.65454

Summary: We consider the B-spline isogeometric analysis approximation of the Laplacian eigenvalue problem \(-\Delta u=\lambda u\) 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 matrix \(L_n^{[p]}\) arising from the isogeometric analysis approximation based on B-splines of degree \(p\) of the unidimensional problem \(-u''=\lambda u\). Here, \(n\) is the mesh fineness parameter, and the size of \(L_n^{[p]}\) is \(N(n,p)=n+p-2\). In previous works, it was established that the normalized sequence \(\{ n^{-2}L_n^{[p]}\}_n\) enjoys an asymptotic spectral distribution described by a function \(e_p(\theta )\), the so-called spectral symbol. The contributions of this paper can be summarized as follows:
1.
We prove several important analytic properties of the spectral symbol \(e_p(\theta)\). In particular, we show that \(e_p(\theta)\) is monotone increasing on \([0,\pi]\) for all \(p\geq 1\) and that \(e_p(\theta)\to\theta^2\) uniformly on \([0,\pi]\) as \(p\to\infty\).
2.
For \(p=1\) and \(p=2\), we show that \(L_n^{[p]}\) belongs to a matrix algebra associated with a fast unitary sine transform, and we compute eigenvalues and eigenvectors of \(L_n^{[p]}\). In both cases, the eigenvalues are given by \(e_p(\theta_{j,n})\), \(j=1,\dots,n+p-2\), where \(\theta_{j,n}=j\pi /n\).
3.
For \(p\geq 3\), we provide numerical evidence of a precise asymptotic expansion for the eigenvalues of \(n^{-2}L_n^{[p]}\), excluding the largest \(n_p^{\text{out}}=p-2+\bmod{(p,2)}\) eigenvalues (the so-called outliers). More precisely, we numerically show that for every \(p\geq 3\), every integer \(\alpha\geq 0\), every \(n\), and every \(j=1,\dots,N(n,p)-n_p^{\text{out}}\), \[\lambda_j\left(n^{-2}L_n^{[p]}\right)=e_p(\theta_{j,n})+\sum_{k=1}^{\alpha}c_k^{[p]}(\theta_{j,n})h^k+E_{j,n,\alpha}^{[p]},\] where
the eigenvalues of \(n^{-2}L_n^{[p]}\) are arranged in ascending order, \(\lambda_1(n^{-2}L_n^{[p]})\leq\dots\leq\lambda_{n+p-2}(n^{-2}L_n^{[p]})\);
\(\{ c_k^{[p]}\}_{k=1,2,\dots}\) is a sequence of functions from \([0,\pi]\) to \(\mathbb{R}\), which depends only on \(p\);
\(h=1/n\) and \(\theta_{j,n}=j\pi h\) for \(j=1,\dots,n\); and
\(E_{j,n,\alpha}^{[p]}=O(h^{\alpha+1})\) is the remainder, which satisfies \(|E_{j,n,\alpha}^{[p]}|\leq C_{\alpha}^{[p]}h^{\alpha+1}\) for some constant \(C_{\alpha}^{[p]}\) depending only on \(\alpha\) and \(p\). We also provide a proof of this expansion for \(\alpha=0\) and \(j=1,\dots,N(n,p)-(4p-2)\), where \(4p-2\) represents a theoretical estimate of the number of outliers \(n_p^{\text{out}}\).
4.
We show through numerical experiments that, for \(p\geq 3\) and \(k\geq 1\), there exists a point \(\theta(p,k)\in(0,\pi)\) such that \(c_k^{[p]}(\theta)\) vanishes on \([0,\theta(p,k)]\). Moreover, as it is suggested by the numerics of this paper, the infimum of \(\theta(p,k)\) over all \(k\geq 1\), say \(y_p\), is strictly positive, and the equation \(\lambda_j(n^{-2}L_n^{[p]})=e_p(\theta_{j,n})\) holds numerically whenever \(\theta_{j,n}<\theta(p)\), where \(\theta(p)\) is a point in \((0,y_p]\) which grows with \(p\).
5.
For \(p\geq 3\), based on the asymptotic expansion in the above item 3, we propose a parallel interpolation-extrapolation algorithm for computing the eigenvalues of \(L_n^{[p]}\), excluding the \(n_p^{\text{out}}\) 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 \(\theta_{j,n}<\theta(p)\).

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65D05 Numerical interpolation
65B05 Extrapolation to the limit, deferred corrections
15B05 Toeplitz, Cauchy, and related matrices
35C20 Asymptotic expansions of solutions to PDEs
65Y05 Parallel numerical computation
Full Text: DOI

References:

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