TY - JOUR

T1 - Non-Hermitian perturbations of Hermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

AU - Barbarino, Giovanni

AU - Serra-Capizzano, Stefano

N1 - Publisher Copyright:
© 2020 John Wiley & Sons, Ltd.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - This article concerns the spectral analysis of matrix-sequences which can be written as a non-Hermitian perturbation of a given Hermitian matrix-sequence. The main result reads as follows. Suppose that for every n there is a Hermitian matrix Xn of size n and that {Xn}n∼λf, that is, the matrix-sequence {Xn}n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f; if (Formula presented.) with ‖·‖2 being the Schatten 2 norm, then {Xn+Yn}n∼λf. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix-sequences {Xn}n and {Yn}n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix-sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix-sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article.

AB - This article concerns the spectral analysis of matrix-sequences which can be written as a non-Hermitian perturbation of a given Hermitian matrix-sequence. The main result reads as follows. Suppose that for every n there is a Hermitian matrix Xn of size n and that {Xn}n∼λf, that is, the matrix-sequence {Xn}n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f; if (Formula presented.) with ‖·‖2 being the Schatten 2 norm, then {Xn+Yn}n∼λf. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix-sequences {Xn}n and {Yn}n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix-sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix-sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article.

KW - approximation of PDEs

KW - perturbation results

KW - preconditioning

KW - spectral distribution in the Weyl sense

UR - http://www.scopus.com/inward/record.url?scp=85080978672&partnerID=8YFLogxK

U2 - 10.1002/nla.2286

DO - 10.1002/nla.2286

M3 - Article

AN - SCOPUS:85080978672

VL - 27

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 3

M1 - e2286

ER -