A gradient-enhanced approach for stable finite element approximations of reaction-convection-diffusion problems

Contributions Author #1 (Soheil Firooz): Conceptualization, methodology, implementation, simulations, data curation, writing—original draft. Author #2 (B. Daya Reddy): Supervision, conceptual discussions, review & editing. Author #3 (Paul Steinmann): Supervision, conceptual discussions, review & editing. Data collection: period and details From 10.01.2025 to 30.04.2025 using a custom finite element code developed by the first author. Data corresponds to the numerical experiments and figures reported in the article. Funding sources Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 460333672 – CRC 1540 Exploring Brain Mechanics (sub-project C01). Data structure and information Put a description of the data structure Hint: JTCAM curation policy First things first: JTCAM data editors are here to help! Second: the JTCAM data curation policy can be found here: https://zenodo.org/communities/jtcam/curation-policy JTCAM editors wish to follow the FAIR principles regarding the open data and algorithms: https://en.wikipedia.org/wiki/FAIR_data The intention is to publish data relevant to the JTCAM paper. This includes: Raw data used to produce Figures Experimental data Simulation Input and Output Algorithms (software, scripts) This dataset contains the numerical results underlying the figures in the publication. The data are organized into subdirectories for one-dimensional and two-dimensional benchmark problems, and into a dedicated folder for the figure datasets. In essence, this Open Data approach will allow reproducibility and an easy analysis by another research group. Data structure and information Authors are required to provide one README.md file per directory to help navigation through the data Here the authors are expected to provide a full description of the directories hierarchy with a consise description. For instance it could be: Results/files structure: drawing1.svg Schematic illustration of the examples and boundary conditions. example1/ Contains the individual plots and data files that together form the corresponding figure in the manuscript. example2/ Contains the individual plots (PNG images) that together form the corresponding figure in the manuscript. example3/ Contains the individual plots (PNG images) that together form the corresponding figure in the manuscript. example4/ Contains the individual plots (PNG images) that together form the corresponding figure in the manuscript. example5/ Contains the individual plots (PNG images) that together form the corresponding figure in the manuscript. example6/ Contains the individual plots (PNG images) that together form the corresponding figure in the manuscript. computational_time/ Contains the plots for the computational time comparison presented in the manuscript. Paper Description We develop a micromorphic-based approach for finite element stabilization of reaction-convection-diffusion equations, by gradient enhancement of the field of interest via introducing an auxiliary variable. The well-posedness of the coupled-field approach is established, together with an error estimate. Through a set of 1D and 2D numerical examples the high accuracy and enhanced stability of the approach in approximating solutions associated with complex problems is demonstrated, for situations of varying reactivity and convection.