Convergence analysis of a new dynamic diffusion method

Abstract 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 O ( h 1 / 2 ) in the energy norm, and numerical experiments have led to optimal convergence rates in the L 2 ( Ω ) , H 1 ( Ω ) , and energy norms.