Two nodally bound-preserving and mass-conservative discontinuous Galerkin methods for reaction-diffusion problems

arXiv:2611.13588v1 Announce Type: new Abstract: We propose and analyze two high-order discontinuous Galerkin methods that preserve prescribed nodal bounds and total mass for reaction-diffusion problems with anisotropic and heterogeneous diffusion. Both methods combine a weighted interior-penalty prediction step with a convex correction in either the dG energy or the $L^2$ metric. We construct a pointwise bound-preserving and cellwise mass-preserving comparison function with optimal local approximation properties. This construction yields optimal dG-energy error estimates for the energy correction and, under a dual regularity assumption, optimal $L^2$ and dG-energy error estimates for the $L^2$ correction on quasi-uniform meshes. We further establish a general a posteriori estimate that combines a predictor estimator with the computable correction norm. The resulting estimators are globally reliable and efficient up to data oscillation, with parameter-dependent efficiency constants. Numerical experiments compare the two corrections and illustrate their use in adaptive computations.