On the boundedness of discrete solutions to a haptotaxis model

The haptotaxis model provides a fundamental framework for describing the directed migration of cells in response to gradients of extracellular matrix components, and it plays a central role in modelling tumour invasion and tissue remodelling. In this work, we propose and analyse a nonlinear control volume finite element (CVFE) scheme for the haptotaxis system, incorporating anisotropic diffusion to represent directionally dependent transport, and coupling it with tissue degradation and production mechanisms. The proposed scheme satisfies a discrete maximum principle without requiring any restriction on the transmissibility coefficients, thereby guaranteeing the boundedness and stability of the numerical solution. By employing compactness arguments, we prove the convergence of the discrete solutions towards a weak solution of the continuous problem. Finally, two- and three-dimensional numerical experiments illustrate the accuracy and robustness of the proposed method, successfully capturing complex cell invasion patterns driven by haptotactic and diffusive interactions.