Ekström, Sven-Erik; Furci, Isabella; Garoni, Carlo; Manni, Carla; Serra-Capizzano, Stefano; Speleers, Hendrik Are the eigenvalues of the B-spline isogeometric analysis approximation of \(-\Delta u=\lambda u\) known in almost closed form? (English) Zbl 1513.65454 Numer. Linear Algebra Appl. 25, No. 5, e2198, 34 p. (2018). 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 \(\theta_{j,n}<\theta(p)\). Cited in 12 Documents 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 Keywords:eigenvalues and eigenvectors; isogeometric analysis and B-splines; Laplacian eigenvalue problem; mass and stiffness matrices; polynomial interpolation and extrapolation; spectral symbol and asymptotic eigenvalue expansion × Cite Format Result Cite Review PDF Full Text: DOI References: [1] CottrellJA, HughesTJR, BazilevsY. Isogeometric analysis: Toward integration of CAD and FEA. Chichester, UK: John Wiley & Sons; 2009. · Zbl 1378.65009 [2] CottrellJA, RealiA, BazilevsY, HughesTJR. Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng. 2006;195(41-43):5257-5296. · Zbl 1119.74024 [3] EkströmS‐E, GaroniC, HughesTJR, RealiA, Serra‐CapizzanoS, SpeleersH. Symbol‐based analysis of finite element and isogeometric B‐spline discretizations of eigenvalue problems: exposition and review. In preparation. [4] HughesTJR, EvansJA, RealiA. Finite element and NURBS approximations of eigenvalue, boundary‐value, and initial‐value problems. Comput Methods Appl Mech Eng. 2014;272:290-320. · Zbl 1296.65148 [5] HughesTJR, RealiA, SangalliG. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p‐method finite elements with k‐method NURBS. Comput Methods Appl Mech Eng. 2008;197(49-50):4104-4124. · Zbl 1194.74114 [6] RealiA. An isogeometric analysis approach for the study of structural vibrations. J Earthq Eng. 2006;10:1-30. [7] DonatelliM, GaroniC, ManniC, Serra‐CapizzanoS, SpeleersH. Spectral analysis and spectral symbol of matrices in isogeometric collocation methods. Math Comput. 2016;85(300):1639-1680. · Zbl 1335.65096 [8] GaroniC. Spectral distribution of PDE discretization matrices from isogeometric analysis: the case of L^1 coefficients and non‐regular geometry. J Spectr Theory. 2018;8(1):297-313. · Zbl 1390.35211 [9] GaroniC, ManniC, PelosiF, Serra‐CapizzanoS, SpeleersH. On the spectrum of stiffness matrices arising from isogeometric analysis. Numer Math. 2014;127(4):751-799. · Zbl 1298.65172 [10] GaroniC, ManniC, Serra‐CapizzanoS, SesanaD, SpeleersH. Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Math Comput. 2017;86(305):1343-1373. · Zbl 1359.65252 [11] GaroniC, ManniC, Serra‐CapizzanoS, SesanaD, SpeleersH. Lusin theorem, GLT Sequences and matrix computations: an application to the spectral analysis of PDE discretization matrices. J Math Anal Appl. 2017;446(1):365-382. · Zbl 1352.65501 [12] GaroniC, Serra‐CapizzanoS. Generalized locally Toeplitz sequences: theory and applications (Volume I). Cham, Switzerland: Springer; 2017. · Zbl 1376.15002 [13] GaroniC, Serra‐CapizzanoS. Generalized locally Toeplitz sequences: theory and applications (Volume II). Cham, Switzerland: Springer; To be published. · Zbl 1448.47004 [14] DonatelliM, GaroniC, ManniC, Serra‐CapizzanoS, SpeleersH. Robust and optimal multi‐iterative techniques for IgA Galerkin linear systems. Comput Methods Appl Mech Eng. 2015;284:230-264. · Zbl 1425.65136 [15] DonatelliM, GaroniC, ManniC, Serra‐CapizzanoS, SpeleersH. Robust and optimal multi‐iterative techniques for IgA collocation linear systems. Comput Methods Appl Mech Eng. 2015;284:1120-1146. · Zbl 1425.65196 [16] DonatelliM, GaroniC, ManniC, Serra‐CapizzanoS, SpeleersH. Symbol‐based multigrid methods for Galerkin B‐spline isogeometric analysis. SIAM J Numer Anal. 2017;55(1):31-62. · Zbl 1355.65153 [17] deBoorC. A practical guide to splines. New York, NY: Springer; 2001. · Zbl 0987.65015 [18] SchumakerLL. Spline functions: basic theory. 3rd ed.Cambridge, UK: Cambridge University Press; 2007. · Zbl 1123.41008 [19] BozzoE, Di FioreC. On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decomposition. SIAM J Matrix Anal Appl. 1995;16(1):312-326. · Zbl 0819.65017 [20] AhmadF, Al‐AidarousES, AlrehailiDA, EkströmS‐E, FurciI, Serra‐CapizzanoS. Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?Numer Algorithm. 2018;78(3):867-893. · Zbl 1398.65055 [21] EkströmS‐E, GaroniC, Serra‐CapizzanoS. Are the eigenvalues of banded symmetric Toeplitz matrices known in almost closed form?Exper Math. 2017. https://doi.org/10.1080/10586458.2017.1320241 · Zbl 1405.15037 · doi:10.1080/10586458.2017.1320241 [22] BogoyaJM, BöttcherA, GrudskySM, MaximenkoEA. Eigenvalues of Hermitian Toeplitz matrices with smooth simple‐loop symbols. J Math Anal Appl. 2015;422(2):1308-1334. · Zbl 1302.65086 [23] BogoyaJM, GrudskySM, MaximenkoEA. Eigenvalues of Hermitian Toeplitz matrices generated by simple‐loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: Advances and applications. Cham, Switzerland: Birkhäuser, 2017; p. 179-212. · Zbl 1468.15007 [24] BöttcherA, GrudskySM, MaximenkoEA. Inside the eigenvalues of certain Hermitian Toeplitz band matrices. J Comput Appl Math. 2010;233(9):2245-2264. · Zbl 1195.15009 [25] ChenH, JiaS, XieH. Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem. Appl Numer Math. 2011;61(4):615-629. · Zbl 1209.65126 [26] YinX, XieH, JiaS, GaoS. Asymptotic expansions and extrapolations of eigenvalues for the Stokes problem by mixed finite element methods. J Comput Appl Math. 2008;215(1):127-141. · Zbl 1149.65090 [27] EkströmS‐E, GaroniC. A matrix‐less and parallel interpolation-extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices. Numer Algorithm. 2018. https://doi.org/10.1007/s11075-018-0508-0 · Zbl 1455.65051 · doi:10.1007/s11075-018-0508-0 [28] GaroniC, Serra‐CapizzanoS. Generalized locally Toeplitz sequences: a spectral analysis tool for discretized differential equations. To appear. · Zbl 1433.65132 [29] BiniD, CapovaniM. Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra Appl. 1983;52-53:99-126. · Zbl 0549.15005 [30] SangalliG, TaniM. Isogeometric preconditioners based on fast solvers for the Sylvester equation. SIAM J Sci Comput. 2016;38(6):A3644-A3671. · Zbl 1353.65035 [31] StoerJ, BulirschR. Introduction to numerical analysis. 3rd ed. New York, NY: Springer; 2002. · Zbl 1004.65001 [32] BrezinskiC, Redivo ZagliaM. Extrapolation methods: Theory and practice. North Holland, The Netherlands: Elsevier; 1991. · Zbl 0744.65004 [33] Serra‐CapizzanoS. Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 2003;366:371-402. · Zbl 1028.65109 [34] Serra‐CapizzanoS. The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 2006;419(1):180-233. · Zbl 1109.65032 [35] ChuiCK. An introduction to wavelets. London, UK: Academic Press; 1992. · Zbl 0925.42016 [36] Serra‐CapizzanoS. On the extreme spectral properties of Toeplitz matrices generated by L^1 functions with several minima/maxima. BIT Numer Math. 1996;36(1):135-142. · Zbl 0851.15008 [37] BhatiaR. Matrix analysis. New York, NY: Springer; 1997. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.