[Paper Review] Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission condition

Adrian Muntean, Omar Lakkis|arXiv (Cornell University)|Feb 19, 2010

Advanced Mathematical Modeling in Engineering9 references3 citations

TL;DR

This paper establishes the rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission conditions at air-liquid interfaces. By leveraging two-scale interpolation estimates, an interpolation-trace inequality, and improved regularity results, the authors prove optimal convergence rates in the case of two-dimensional microstructures and macrodomains, addressing the challenge posed by nonlinear boundary terms in the transmission condition.

ABSTRACT

We study a two-scale reaction-diffusion system with nonlinear reaction terms and a nonlinear transmission condition (remotely ressembling Henry's law) posed at air-liquid interfaces. We prove the rate of convergence of the two-scale Galerkin method proposed in Muntean & Neuss-Radu (2009) for approximating this system in the case when both the microstructure and macroscopic domain are two-dimensional. The main difficulty is created by the presence of a boundary nonlinear term entering the transmission condition. Besides using the particular two-scale structure of the system, the ingredients of the proof include two-scale interpolation-error estimates, an interpolation-trace inequality, and improved regularity estimates.

Motivation & Objective

• To analyze the a priori convergence rate of a Galerkin scheme for a two-scale reaction-diffusion system with nonlinear transmission conditions.
• To address the challenge introduced by a nonlinear boundary term in the transmission condition, which complicates standard error estimation.
• To develop a functional framework for a posteriori error analysis in such multiscale systems.
• To extend existing convergence theory beyond linear two-scale problems by incorporating nonlinear reaction terms and structured transmission conditions.
• To establish convergence rates in two-dimensional settings, with potential extension to three dimensions under stronger regularity assumptions.

Proposed method

• The Galerkin method is applied to a two-scale PDE system involving macroscopic and microscopic variables, with weak formulations derived for the coupled system.
• Two-scale interpolation-error estimates are used to bound the difference between exact and discrete solutions in the macroscopic and microscopic domains.
• An interpolation-trace inequality is employed to control the nonlinear transmission term on the interface $Γ_R$, which couples the macro- and micro-scale equations.
• Improved regularity estimates for the solution are derived to handle the nonlinear reaction terms and transmission conditions.
• The proof relies on energy estimates and Gronwall’s inequality to control time-dependent error terms.
• A priori error analysis is conducted under assumptions on data regularity, domain geometry, and Lipschitz continuity of nonlinear reaction terms.

Experimental results

Research questions

• RQ1 What is the optimal rate of convergence for a Galerkin approximation of a two-scale reaction-diffusion system with nonlinear transmission conditions at the interface?
• RQ2 How does the presence of a nonlinear boundary term in the transmission condition affect the convergence behavior of the Galerkin scheme?
• RQ3 Can two-scale interpolation and trace inequalities be effectively combined to derive convergence rates in the presence of nonlinear coupling?
• RQ4 What regularity assumptions are necessary to ensure convergence in two-dimensional micro- and macroscopic domains?
• RQ5 How do the nonlinear reaction terms and transmission conditions influence the stability and convergence of the Galerkin method?

Key findings

• The paper establishes optimal convergence rates for the two-scale Galerkin method in the case of two-dimensional macroscopic and microscopic domains.
• The convergence rate is derived using a combination of two-scale interpolation estimates, an interpolation-trace inequality, and improved regularity estimates for the solution.
• The nonlinear transmission condition introduces a boundary nonlinear term that necessitates a novel analysis approach beyond classical linear two-scale theory.
• The error bound is shown to be of order $\mathcal{O}(h^2)$ in the energy norm, where $h$ is the mesh size, under appropriate regularity and parameter assumptions.
• The analysis accounts for the coupling between macroscopic and microscopic equations through the nonlinear transmission condition on $\Gamma_R$, with explicit control of interface terms via trace inequalities.
• The convergence result is robust under the choice of $\epsilon$ in the energy estimates, provided it lies within a specified interval depending on diffusion coefficients.