Numerical approximation of the spectrum of self-adjoint continuously\n invertible operators

2021/03/01 by Tomáš Gergelits, Gergelits, Tomáš, Bjørn Fredrik Nielsen +3
Computer Science · Mathematics · #35J99 #65F08 #65F15 #65N12 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2103.00849

openalex publication_date 2021/03/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

This paper deals with the generalized spectrum of continuously invertible\nlinear operators defined on infinite dimensional Hilbert spaces. More\nprecisely, we consider two bounded, coercive, and self-adjoint operators\n bcA, B: V\↦ V ^#, where V ^# denotes the dual of V, and\ninvestigate the conditions under which the whole spectrum of\n bcB^-1 bcA:V\↦ V can be approximated to an arbitrary accuracy by\nthe eigenvalues of the finite dimensional discretization\n bcB_n^-1 bcA_n. Since bcB^-1 bcA is continuously invertible,\nsuch an investigation cannot use the concept of uniform (normwise) convergence,\nand it relies instead on the pointwise (strong) convergence of\n bcB_n^-1 bcA_n to bcB^-1 bcA.\n The paper is motivated by operator preconditioning which is employed in the\nnumerical solution of boundary value problems. In this context, bcA,\n bcB: H₀¹(\Ω) \↦ H^-1(\Ω) are the standard\nintegral/functional representations of the differential operators -\∇\n\⋅ (k(x)\∇ u) and -\∇ \⋅ (g(x)\∇ u), respectively, and\nk(x) and g(x) are scalar coefficient functions. The investigated question\ndiffers from the eigenvalue problem studied in the numerical PDE literature\nwhich is based on the approximation of the eigenvalues within the framework of\ncompact operators.\n This work follows the path started by the two recent papers published in\n[SIAM J. Numer. Anal., 57 (2019), pp.~1369-1394 and 58 (2020), pp.~2193-2211]\nand addresses one of the open questions formulated at the end of the second\npaper.\n

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