Convergence analysis of a new dynamic diffusion method

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 priorierror estimates. We theoretically show that the new DD method has convergence rate of O(h(1/2)) in the energy norm, and numerical experiments have led to optimal convergence rates in the L-2(Omega), Pi(1)(Omega), and energy norms.

Automated summary

This study introduces a new variant of the nonlinear multiscale Dynamic Diffusion (DD) method, which provides additional stability through a dynamic nonlinear operator acting in all scales. The paper proves the existence of discrete solutions, stability, and a priori error estimates, and theoretically shows that the new DD method has a convergence rate of O(h(1/2)) in the energy norm. Numerical experiments also confirm optimal convergence rates in various norms.