Error Estimates of Generic Discretisation of Reaction-Diffusion System with Constraints
Yahya Alnashri
TL;DR
This work tackles a constrained parabolic reaction–diffusion system arising in biofilm growth by embedding the model in a gradient discretisation framework that unifies conforming and non-conforming discretisations. It provides the first general error estimates for approximating the constrained semilinear PDE, proving existence and uniqueness of the discrete solution under mild time-step restrictions and deriving convergence rates that depend on interpolation/consistency quantities $S_{\mathcal{D}}$, flux-compatibility $W_{\mathcal{D}}$, and the nonlinearity term ${\mathbb M}_{\mathcal{D}}$. With additional regularity, the authors show first-order convergence in space and time for common low-order gradient discretisations, and they discuss optimal orders for $S$, $W_{\mathcal{D}}$, and ${\mathbb M}_{\mathcal{D}}$ (e.g., $O(h)$ and $O(h^{2})$ respectively). Numerical experiments using a mixed finite-volume scheme, including a manufactured-solution test, validate the theoretical rates and demonstrate the framework’s applicability to non-conforming methods. Overall, the paper delivers rigorous error control for a broad class of discretisations of constrained reaction–diffusion systems and guides practical scheme selection for biofilm-type models.
Abstract
In this paper, we study a parabolic reaction diffusion system with constraints that model biofilm growth. Within a unified framework encompassing multiple numerical schemes, we derive the first general convergence rates for approximating this model using both conforming and non conforming discretisation methods. Under standard assumptions on the time discretisation, we establish the existence and uniqueness of the discrete solution. Numerical experiments are conducted using a mixed finite volume scheme that fits within the proposed unified framework. A test case with an analytical solution is designed to confirm our theoretical convergence rates.