Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation

2023/12/26 by Yu. M. Meshkova, Meshkova, Yulia
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods

paper · pdf · doi:10.48550/arxiv.2312.15887

Abstract

Let O⊂ ℝ^d be a bounded domain of class C^1,1. In L₂(O;ℂⁿ), we consider a matrix elliptic second order differential operator A_D,ε with the Dirichlet boundary condition. Here ε >0 is a small parameter. The coefficients of the operator A_D,ε are periodic and depend on x/ε. The principal terms of approximations for the operator cosine and sine functions are given in the (H²→ L₂)- and (H¹→ L₂)-operator norms, respectively. The error estimates are of the precise order O(ε) for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation (∂ _t²+A_D,ε)u_ε =F.

Related

• Homogenization of initial boundary value problems for parabolic systems with periodic coefficients
• On operator error estimates for homogenization of hyperbolic systems with periodic coefficients
• Homogenization of hyperbolic equations with periodic coefficients
• Homogenization of the Neumann problem for elliptic systems with periodic coefficients
• Operator error estimates for homogenization of the elliptic dirichlet problem in a bounded domain