Understanding avascular tumor growth and drug interactions through numerical analysis: A finite element method approach

This article establishes the existence of a fully discrete weak solution for the tumor growth model, which is described by a coupled non-linear reaction-diffusion system. This model incorporates crucial elements such as cellular proliferation, nutrient diffusion, prostate-specific antigen, and drug effects. We employ the finite element method for spatial discretization and the implicit Euler method for temporal discretization. Firstly, we analyzed the existence and uniqueness of the fully discretized tumor growth model. Additionally, stability bounds for the fully discrete coupled system are derived. Secondly, through multiple numerical simulations utilizing higher order finite element methods, we analyze tumor growth behavior both with and without drug interaction, yielding a more accurate numerical solution. Furthermore, we compare CPU time efficiency across different time marching methods and explore various preconditioners to optimize computational performance.