Rate of convergence for a Galerkin scheme approximating a two-scale reaction-diffusion system with nonlinear transmission condition
This provides theoretical convergence guarantees for a numerical method used to model air-liquid interface phenomena, but the result is incremental as it extends prior work to a specific nonlinear case.
The authors prove the rate of convergence for a two-scale Galerkin method applied to a reaction-diffusion system with nonlinear transmission conditions, achieving an error estimate in the two-dimensional case.
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.