Reyna: A minimal overhead polytopal discontinuous Galerkin finite element library

Partial differential equations (PDEs) underpin much of engineering, physics and applied science, providing the basis for simulations across a wide range of physical and biological systems.Many high-performance numerical schemes are implemented in compiled languages, which deliver speed but at the cost of accessibility.These implementations are often difficult to adapt or extend, limiting rapid experimentation, the development of new methods and their use in training.Reyna is a Python package for solving second-order PDEs with non-negative characteristic form using the polytopal discontinuous Galerkin finite element method (DGFEM).It offers a flexible and approachable workflow while avoiding the complexity and steep learning curve of HPC-focused codebases.Reyna is designed with a modular structure and an optimised implementation aiming at balancing clarity with performance.It provides a practical platform for developing and testing finite element methods and for producing concise instructional examples, lowering the barrier to experimentation whilst retaining state-of-the-art numerical capability. Statement of NeedDGFEMs offer flexibility and high-order accuracy, and polytopal elements extend this adaptability further through general meshing and refined discretisations.Yet most available implementations are embedded in complex C++ or Fortran frameworks, see MFEM by Anderson et al. (2021), Feel++ by Prud'Homme et al. (2012), Dune-FEM-DG by Dedner et al. (2017) (now supporting polygonal meshes), deal.II by Bangerth et al. (2007) and FEMPAR by Badia et al. (2018), for example, which require substantial technical overhead to adapt or extend.Packages such as FEniCS by Alnæs et al. (2015) and Firedrake by Rathgeber et al. (2016) exist with Python interfaces, but extensive knowledge is required, limiting their use for prototyping and teaching.Quail by Ching et al. (2022) provides a Python DGFEM package aimed at this purpose, but it is restricted to triangular and quadrilateral meshes.Reyna provides a lightweight, vectorised Python framework that makes polytopal DGFEMs accessible without sacrificing efficiency.It enables rapid experimentation, straightforward extension and practical use in both research and education by leveraging established scientific Python libraries and its modular architecture.