Stabilized weak-gradient discontinuous finite elements with optimal error estimates for second-order elliptic PDEs

Authors

Aymen Laadhari, Department of Mathematics, College of Computing and Mathematical Sciences, Khalifa University of Science and Technology, P.O. Box: 127788, Abu Dhabi, United Arab EmiratesFollow

Abstract

This work introduces an accurate finite element approach employing a new stabilized discrete weak gradient, designed for second-order elliptic problems on arbitrary conforming meshes. We formulate the approach within a discontinuous Galerkin framework and derive a consistent and coercive bilinear form. Appropriate error analysis on a model problem confirms optimal convergence. Building on the core analysis, we extend the method to more challenging settings, including time-dependent heterogeneous scenarios and a biophysically realistic optimal-control model of photobleaching in the budding yeast cell. We further illustrate the versatility of the weak-gradient construction by applying it to an unsteady level-set equation relevant to multiphase flow modeling. The implementation supports high-order polynomial spaces, varied boundary conditions, and parallel execution with both direct and iterative linear solvers.

Recommended Citation

Laadhari, Aymen (2025) "Stabilized weak-gradient discontinuous finite elements with optimal error estimates for second-order elliptic PDEs," Mathematical Modelling and Numerical Simulation with Applications: Vol. 5: Iss. 3, Article 8.
DOI: https://doi.org/10.53391/2791-8564.1007
Available at: https://mmnsa.researchcommons.org/journal/vol5/iss3/8