[Paper Review] Convergence Rates in Parabolic Homogenization with Time-Dependent Periodic Coefficients

Jun Geng, Zhongwei Shen|arXiv (Cornell University)|Apr 22, 2016

Advanced Mathematical Modeling in Engineering17 references3 citations

TL;DR

This paper establishes sharp O(ε) convergence rates in L² for weak solutions of second-order parabolic systems with rapidly oscillating, time-dependent periodic coefficients. Using energy estimates and corrector-based duality arguments, it proves that the difference between the heterogeneous solution and its homogenized limit decays at rate ε in L²(Ω×(0,T)), under minimal regularity assumptions on the coefficient matrix and domain.

ABSTRACT

For a family of second-order parabolic systems with bounded measurable, rapidly oscillating and time-dependent periodic coefficients, we investigate the sharp convergence rates of weak solutions in $L^2$. Both initial-Dirichlet and initial-Neumann problems are studied.

Motivation & Objective

• To establish sharp convergence rates in L² for parabolic systems with bounded measurable, rapidly oscillating, and time-dependent periodic coefficients.
• To analyze both initial-Dirichlet and initial-Neumann boundary value problems under minimal regularity assumptions.
• To derive quantitative L² convergence estimates that are optimal (i.e., O(ε)) without requiring smoothness of the coefficient matrix.
• To extend homogenization theory to time-dependent periodic structures by controlling the influence of time-scale separation via energy and duality techniques.

Proposed method

• Formulate the parabolic system ∂ₜ + ℒₑuₑ = F with coefficients A(x/ε, t/ε²), periodic in both space and time.
• Use duality pairing between L²(0,T;H¹₀(Ω)) and its dual to estimate the L² norm of the difference uₑ − u₀.
• Introduce corrector functions zₑ and auxiliary solutions vₑ, v₀ to decompose the error and control boundary and initial layer effects.
• Apply energy estimates and weighted L² bounds on ∇u₀ in thin time layers (t−ε²,t) to control transient oscillations.
• Leverage Lemmas 3.5, 4.1, and 4.2 to bound error terms involving gradients and time derivatives via dual solutions.
• Use duality arguments with adjoint problems to reduce the L² error estimate to bounds on dual solutions and their gradients.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate in L² for weak solutions of parabolic systems with time-dependent periodic coefficients?
• RQ2 How does the presence of time-scale oscillations (t/ε²) affect the convergence rate compared to purely spatial homogenization?
• RQ3 Can sharp O(ε) convergence rates be established without assuming smoothness of the coefficient matrix?
• RQ4 How do initial and boundary conditions influence the convergence rate, especially in the Neumann and Dirichlet cases?
• RQ5 What role do time-localized L² norms of ∇u₀ play in the convergence estimate, and how can they be controlled?

Key findings

• The paper establishes sharp O(ε) convergence rates in L²(Ω×(0,T)) for both initial-Dirichlet and initial-Neumann problems.
• The convergence estimate is of the form ‖uₑ − u₀‖_{L²(Ω×(0,T))} ≤ Cε{‖u₀‖_{L²(0,T;H²(Ω))} + ‖F‖_{L²(Ω×(0,T))} + sup_{ε²<t<T}(ε⁻¹∫_{t−ε²}^{t}∫_Ω|∇u₀|²)^{1/2}}.
• In the case g=0 and h=0, the estimate simplifies to ‖uₑ − u₀‖_{L²(Ω×(0,T))} ≤ Cε‖F‖_{L²(Ω×(0,T))}, showing optimal dependence on the forcing term.
• When g=0 and h∈H¹(Ω), the bound becomes ‖uₑ − u₀‖_{L²(Ω×(0,T))} ≤ Cε{‖F‖_{L²(Ω×(0,T))} + ‖h‖_{H¹(Ω)}}.
• The dependence on the time-localized gradient of u₀ is essential and cannot be removed, reflecting the influence of fast time oscillations.
• The proof technique extends to the Neumann case via duality and adjoint problems, confirming the same rate holds under identical assumptions.