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arXiv · 2609.34625

A weighted polygonal discontinuous Galerkin method for hierarchically coupled reaction-diffusion systems with cubic interactions

Mattia Corti
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Abstract

We develop and analyse a symmetric weighted interior penalty polytopal discontinuous Galerkin (SWIP-PolyDG) method for multi-species reaction--diffusion systems with nonlinear reaction terms of up to cubic order. Such terms arise naturally when higher-order interactions are incorporated into population models, allowing non-additive effects among multiple species to influence local growth and conversion mechanisms. The proposed framework accommodates heterogeneous and possibly anisotropic diffusion tensors on general polygonal meshes. To control the nonlinear coupling, we consider a hierarchical block structure in the reaction operator, whereby each population block depends only on its own variables and on those associated with preceding blocks. In addition, the cubic self-interactions within each block are assumed to have a dissipative diagonal structure. Under these hypotheses and proceeding recursively over the hierarchy of population blocks, we derive local-in-time stability estimates in two spatial dimensions for the semi-discrete formulation in both the $L^2(Ω)$-and dG-norms. We further establish an a priori error estimate in a combined $L^2(Ω)$-dG energy norm for sufficiently regular solutions for the semi-discrete formulation. Finally, numerical experiments confirm the predicted convergence behaviour and illustrate the robustness of the method under heterogeneous diffusion.

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Mattia Corti. 2026-09-28. A weighted polygonal discontinuous Galerkin method for hierarchically coupled reaction-diffusion systems with cubic interactions. https://arxiv.org/abs/2609.34625

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