Coupled Model and Grid Adaptivity in Hierarchical Reduction of Elliptic Problems

In this paper we propose a surrogate model for advection-diffusionreaction problems characterized by a dominant direction in their dynamics. We resort to a hierarchical-model reduction where we couple a modal representation of the transverse dynamics with a finite element approximation along the mainstream. This different treatment of the dynamics entails a surrogate model enhancing a purely 1D description related to the leading direction. The coefficients of the finite element expansion along this direction introduce a generally non-constant description of the transversal dynamics. Aim of this paper is to provide an automatic adaptive approach to locally select the dimension of the modal expansion as well as the finite element step in order to satisfy a prescribed tolerance on a goal functional of interest.