Generalized Spectrum of Second Order Differential Operators

0532921 - ÚI 2021 RIV US eng J - Journal Article
Gergelits, Tomáš - Nielsen, B. F. - Strakoš, Z.
Generalized Spectrum of Second Order Differential Operators.
SIAM Journal on Numerical Analysis. Roč. 58, č. 4 (2020), s. 2193-2211. ISSN 0036-1429. E-ISSN 1095-7170
Institutional support: RVO:67985807
Keywords : second order PDEs * generalized eigenvalues * spectrum * tensors * preconditioning
OECD category: Pure mathematics
Impact factor: 3.212, year: 2020
Method of publishing: Limited access
http://dx.doi.org/10.1137/20M1316159

We analyze the spectrum of the operator Delta(-1)[Delta . (K del u)], where Delta denotes the Laplacian and K = K(x, y) is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition K = Q Lambda Q(T), where Q = Q(x, y) is an orthogonal matrix and Lambda = Lambda(x, y) is a diagonal matrix. More precisely, provided that K is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of A. The involved domain is assumed to be bounded and Lipschitz, and both homogeneous Dirichlet and homogeneous Neumann boundary conditions are considered. We study operators defined on infinite dimensional Sobolev spaces. Our theoretical investigations are illuminated by numerical experiments, using discretized problems. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).
Permanent Link: http://hdl.handle.net/11104/0311292