Three-dimensional extension of the SWIP-PolyDG analysis

Extend the stability and error analysis of the symmetric weighted interior penalty polytopal discontinuous Galerkin method for hierarchically coupled reaction–diffusion systems with cubic reaction terms to three spatial dimensions, where the available Sobolev embeddings do not generally provide the higher integrability required to control the cubic terms.

Background

The paper develops a semi-discrete SWIP-PolyDG method for hierarchically coupled multi-species reaction–diffusion systems with reaction terms up to cubic order. The analysis is established in two spatial dimensions because the relevant discrete Sobolev embeddings provide control of all finite Lebesgue norms in terms of the discrete H¹-type norm. This control is used to estimate products generated by the cubic interactions.

In three dimensions, the Sobolev embedding is restricted to exponents up to 6 and therefore does not generally supply the higher integrability needed by the proof. The authors consequently leave the extension of their analysis to three dimensions unresolved. A later conclusion reiterates this issue as an open challenge.