Homogenization of elliptic problems: error estimates in dependence of the spectral parameter

2014/06/29 by T. A. Suslina, Suslina, Tatiana
Computer Science · Engineering · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1406.7530

openalex publication_date 2014/06/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider a strongly elliptic differential expression of the form b(D)^* g(x/ε) b(D), ε >0, where g(x) is a matrix-valued function in \mathbb R^d assumed to be bounded, positive definite and periodic with respect to some lattice; b(D)=∑_l=1^d b_l D_l is the first order differential operator with constant coefficients. The symbol b(ξ) is subject to some condition ensuring strong ellipticity. The operator given by b(D)^* g(x/ε) b(D) in L₂(\mathbb R^d;\mathbb Cⁿ) is denoted by A_ε. Let \mathcal O ⊂ \mathbb R^d be a bounded domain of class C^1,1. In L₂(\mathcal O;\mathbb Cⁿ), we consider the operators A_D,ε and A_N,ε given by b(D)^* g(x/ε) b(D) with the Dirichlet or Neumann boundary conditions, respectively. For the resolvents of the operators A_ε, A_D,ε, and A_N,ε in a regular point ζ we find approximations in different operator norms with error estimates depending on ε and the spectral parameter ζ.

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