An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems

TL;DR

This paper introduces a residual minimization-based stabilized finite element method for complex advection-diffusion-reaction problems, enabling robust adaptive strategies and stable solutions even in challenging heterogeneous and anisotropic scenarios.

Contribution

It presents a novel residual minimization framework that ensures stability and adaptivity for complex advection-diffusion-reaction problems with heterogeneous and anisotropic properties.

Findings

Method delivers stable solutions in extreme scenarios.

Comparable performance to classical DG methods on each mesh.

Enables solving on coarse meshes with adaptive refinement.

Abstract

In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a saddle-point problem that delivers a stable discrete solution and a robust error estimate that can drive mesh adaptivity. In this work, we demonstrate the efficiency of the method in extreme scenarios, delivering stable solutions. The quality and performance of the solutions are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on each mesh. Meanwhile, this technique allows us to solve on coarse meshes and adapt the solution to achieve a user-specified solution quality.