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

Abstract

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 streamlineupwind/Petrov-Galerkin (SUPG) method, further enhanced with a residual-based discontinuity-capturing operator (YZ beta 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 FE[iCS 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-YZ beta 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.