A Variant of the Nonlinear Multiscale Dynamic Diffusion Method

This paper presents a two-scale finite element formulation for a variant of the nonlinear Dynamic Diffusion (DD) method, applied to advection-diffusion-reaction problems. The approach, named here new-DD method, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization, and it is designed to be bounded. We use bubble functions to approximate the subgrid scale space, which are locally condensed on the resolved scales. The proposed methodology is solved by an iterative procedure that uses the bubble-enriched Galerkin solution as the correspondent initial approximation, which is automatically recovered wherever stabilization is not required. Since the artificial diffusion introduced by the new-DD method relies on a problem-depend parameter, we investigate alternative choices for this parameter to keep the accuracy of the method. We numerically evaluate stability and accuracy properties of the method for problems with regular solutions and with layers, ranging from advection-dominated to reaction-dominated transport problems.