Automatically adaptive, stabilized finite element method via residual minimization for heterogeneous, anisotropic advection–diffusion–reaction problems

Abstract

We describe a stable finite element formulation for advection–diffusion–reaction problems that allows for the easy implementation of robust automatic adaptivity. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. We apply a general stabilized finite element framework (Calo et al., 2020) that seeks a discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a sequence of saddle-point problems that are discretely stable and deliver a robust error estimate that drives mesh adaptivity. In this work, we demonstrate the method’s efficiency in extreme scenarios, where the solutions’ quality and performance are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on a particular mesh. We focus on the practical implementation of the adaptive technology to a broad range of engineering applications, from singularly perturbed linear advection-dominated diffusion to highly non-linear problems. This technique starts from coarse meshes and adapts itself to achieve a user-specified solution quality.