A discontinuity-capturing SUPG finite element framework for simulating haptotaxis-driven cancer invasion

This study presents a computational framework for simulating haptotaxis-driven cancer invasion dynamics, governed by time-dependent, nonlinear, and coupled partial differential equations (PDEs) incorporating cross-diffusion terms. In convection-dominated regimes, conventional Galerkin finite element methods (GFEM) typically suffer from numerical instabilities, such as spurious oscillations and nonphysical (negative) densities. To overcome such numerical challenges, we propose a stabilized finite element formulation based on the streamline-upwind/Petrov–Galerkin (SUPG) method, further enhanced with a residual-based discontinuity-capturing operator (YZβ technique) to ensure numerical robustness near sharp gradients. Time discretization is performed using the Crank–Nicolson scheme, and the implementation is carried out within the open-source FEniCS computing platform. The performance of the proposed formulation is assessed across four established haptotaxis models. Numerical results demonstrate that, unlike standard GFEM and classical SUPG formulations, the combined SUPG-YZβ strategy effectively eliminates nonphysical oscillations while preserving solution accuracy. The proposed method offers a reliable and computationally efficient tool for simulating tumor progression in two-dimensional settings and contributes to the broader field of mathematical biology and oncology by enabling stable simulations of invasion dynamics and treatment responses.