This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization. We prove existence of discrete solutions, stability, and a priori error estimates. We theoretically show that the new DD method has convergence rate of $\mathcal{O}(h^{1/2})$ in the energy norm, and numerical experiments have led to optimal convergence rates in the $L^2(\mathrm{\Omega })$, $H^1(\mathrm{\Omega })$, and energy norms.

本文介绍了 Arruda 等人最初提出的对流-扩散-反应方程的非线性多尺度动态扩散 (DD) 方法的一种变体的数值分析。[1] 最近由 Valli 等人研究。[2]。新的 DD 方法基于双尺度方法，通过在离散化的所有尺度上起作用的非线性算子，在局部和动态地引入了额外的稳定性。我们证明了离散解的存在、稳定性和先验误差估计。我们理论上表明，新的 DD 方法的收敛速度为$\mathcal{哦}(H^{1/2})$ 在能量范数中，数值实验导致了最佳收敛速度 ${升}^2(\mathrm{\Omega })$, $H^1(\mathrm{\Omega })$, 和能量规范。