An Adaptive in Space, Stabilized Finite Element Method via Residual Minimization for Linear and Nonlinear Unsteady Advection–Diffusion–Reaction Equations

Abstract

We construct a stabilized finite element method for linear and nonlinear unsteady advection–diffusion–reaction equations using the method of lines. We propose a residual minimization strategy that uses an ad-hoc modified discrete system that couples a time-marching schema and a semi-discrete discontinuous Galerkin formulation in space. This combination delivers a stable continuous solution and an on-the-fly error estimate that robustly guides adaptivity at every discrete time. We show the performance of advection-dominated problems to demonstrate stability in the solution and efficiency in the adaptivity strategy. We also present the method’s robustness in the nonlinear Bratu equation in two dimensions.

Keywords

• residual minimization
• unsteady advection–diffusion equations
• discontinuous Galerkin
• implicit time-marching schemes
• adaptive mesh refinement