Numerical Study of Approximation Techniques for the Temporal Weights to the DWR Method

TL;DR

This paper numerically investigates various approximation techniques for temporal weights in the DWR method applied to convection-dominated, time-dependent convection-diffusion equations, comparing accuracy, efficiency, and stability.

Contribution

It introduces a comparison between higher-order finite elements and a cost-efficient reconstruction approach for temporal weights in the DWR method.

Findings

Higher-order finite elements improve accuracy but at increased computational cost.

Reconstruction approach offers a more efficient alternative with acceptable accuracy.

Numerical results highlight trade-offs between stability and efficiency.

Abstract

This work presents a numerical investigation of different approximation techniques for the temporal weights used in the Dual Weighted Residual (DWR) method applied to a time-dependent convection-diffusion equation which is assumed to be convection-dominated. It is a continuation of a previous work by the authors where spatial weights were compared for a steady-state case. A higher-order finite elements approach is compared to a more cost-efficient higher-order reconstruction approach. Numerical examples point out the results regarding accuracy, efficiency and stability reasons.