Spectral analysis of matrices from isogeometric methods

Abstract: When discretizing a linear PDE by a linear numerical method, the computation of the numerical solution reduces to solving a linear system. The size of this system grows when we refine the discretization mesh. We are then in the presence of a sequence of linear systems with increasing size. It is usually observed in practice that the corresponding sequence of discretization matrices enjoys an asymptotic spectral distribution. Roughly speaking this means that there exists a function, say f, such that the eigenvalues of the considered sequence of matrices behave like a sampling of f over an equispaced grid on the domain of f, up to some outliers.

Isogeometric analysis is a well-established paradigm for the analysis of problems governed by PDEs. It provides a design-through-analysis connection by exploiting a common representation model. This connection is achieved by using the functions adopted in CAD systems not only to describe the domain geometry, but also to represent the numerical solution of the differential problem. In its original formulation IgA is based on (tensor-product) B-splines and their rational extension, the so-called NURBS [2].

In this talk we review the main spectral properties of discretization matrices arising from isogeometric methods, based on d-variate NURBS of given degrees and applied to general second-order elliptic differential problems defined on a d-dimensional domain [4,5], discussing the differences and the similarities with the FEM case [6]. We also discuss the relation between outliers and convergence to eigenfunctions of classical differential operators under k-refinement.

The provided spectral information can be exploited for designing iterative solvers [3] with convergence speed independent of the fineness parameters and also substantially independent of the degrees of the used NURBS, [1].

The talk is based on joint works with C. Garoni, F. Pelosi, E. Sande, H. Speleers, S. Serra-Capizzano.

References

[1] N. Collier, L. Dalcin, D. Pardo, V.M. Calo The cost of continuity: Performance of iterative solvers on isogeometric finite elements, SIAM Journal on Scientific Computing, 35 A767-A784, 2013 [2] J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009.

[3] M. Donatelli, C. Garoni, C. Manni, S. Serra-Capizzano, H. Speleers, Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, SIAM Journal on Numerical Analysis, 55, 31-62, 2017.

[4] C. Garoni, C. Manni, F. Pelosi, S. Serra-Capizzano, H. Speleers, On the spectrum of stiffness matrices arising from isogeometric analysis, Numerische Mathematik, 127, 751-799, 2014.

[5] C. Garoni, C. Manni, S. Serra-Capizzano, H. Speleers, NURBS in isogeometric discretization methods: A spectral analysis, Numerical Linear Algebra with Application}, 2020;27:e2318.

[6] C. Garoni, H. Speleers, S-E. Ekstrom, A. Reali, S. Serra-Capizzano, T.J.R. Hughes, Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: Exposition and review, Archives of Computational Methods in Engineering, 26, 1639-1690, 2019.

[7] E. Sande, C. Manni, H. Speleers: Sharp error estimates for spline approximation: explicit constants, n-widths, and eigenfunction convergence, Mathematical Models and Methods in Applied Sciences}, 29, 1175--1205, 2019

computational engineering, finance, and sciencemathematical softwarenumerical analysiscomputational physics

Audience: researchers in the topic

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