Multi-goal-oriented anisotropic error control and mesh adaptivity for time-dependent convection-dominated problems

TL;DR

This paper introduces an anisotropic multi-goal error control method using the Dual Weighted Residual approach for time-dependent convection-diffusion-reaction equations, enabling efficient adaptive mesh refinement for multiple quantities of interest.

Contribution

It develops a novel anisotropic, multi-goal error estimation technique with directional indicators for adaptive mesh refinement in convection-dominated problems.

Findings

Efficiently captures layers with anisotropic meshes.

Robust error control for multiple goal quantities.

Validated on benchmark convection-dominated problems.

Abstract

In this work, we present an anisotropic multi-goal error control based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. This multi-goal oriented approach allows for an accurate and efficient error control with regard to several quantities of interest simultaneously. Using anisotropic interpolation and restriction operators, we obtain elementwise error indicators in space and time, where the spatial indicators are additionally separated with respect to the single directions. The directional error indicators quantify anisotropy of the solution with respect to the goals, and produce adaptive, anisotropic meshes that efficiently capture layers. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high P\'{e}clet numbers. Numerical examples show efficiency and robustness of the proposed approach for several goal quantities using established benchmarks for convection-dominated transport.