Analytical and numerical studies of a cancer invasion model with nonlocal diffusion

Abstract

This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H + ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient.