Approximation of PDE eigenvalue problems involving parameter dependent matrices
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Approximation of PDE eigenvalue problems involving parameter dependent matrices Daniele Boffi1,2,3 · Francesca Gardini2 · Lucia Gastaldi4 Received: 2 October 2020 / Revised: 2 October 2020 / Accepted: 22 October 2020 © The Author(s) 2020
Abstract We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form 𝖠x = 𝜆𝖡x , where the matrices 𝖠 and/or 𝖡 may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples we show that the presence of one (or both) parameters can produce unexpected results. Keywords Partial differential equations · Eigenvalue problem · Parameter dependent matrices · Virtual element method · Polygonal meshes Mathematics Subject Classification 65N30 · 65N25
1 Introduction Several schemes for the approximation of eigenvalue problems arising from partial differential equations lead to the algebraic form: find 𝜆 ∈ ℝ and x ∈ ℝn with x ≠ 0 such that * Francesca Gardini [email protected] http://www-dimat.unipv.it/gardini/ Daniele Boffi [email protected] http://cemse.kaust.edu.sa/people/person/daniele-boffi Lucia Gastaldi [email protected] http://lucia-gastaldi.unibs.it 1
King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia
2
Dipartimento di Matematica F. Casorati, Università di Pavia, Pavia, Italy
3
Department of Mathematics and System Analysis, Aalto University, Helsinki, Finland
4
DICATAM, Università di Brescia, Brescia, Italy
13
Vol.:(0123456789)
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𝖠x = 𝜆𝖡x,
(1)
where 𝖠 and 𝖡 are matrices in ℝn×n. We consider the case when the matrices 𝖠 and 𝖡 are symmetric and positive semidefinite and may depend on a parameter. This is a typical situation found in applications where elliptic partial differential equations are approximated by schemes that require suitable parameters to be tuned (for consistency and/or stability reasons). In this paper we discuss in particular applications arising from the use of the Virtual Element Method (VEM), see [5, 13, 15, 16, 20–22], where suitable parameters have to be chosen for the correct approximation. Similar situations are present, for instance, when a parameter-dependent stabilization is used for the approximation of discontinuous Galerkin formulations and when a penalty term is added to the discretization of the eigenvalue problem associated with Maxwell’s equations [2, 6–8, 10–12, 23, 26] In general, it may be not immediate to describe how the matrices 𝖠 and 𝖡 depend on the given parameters. For simplicity, we consider the case when the dependence is linear: under suitable assumptions it is easy to discuss how the computed spectrum varies with respect to the parameters. The description of the spectrum in the linear case is not surprising and is well known to a broad scientific community [14, 18, 19, 24, 25]. Nevertheless
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