A Posteriori Error‐Driven Space‐Time Adaptive FEM for Accurate Avascular Tumor and Drug Interaction Modeling

ABSTRACT This study presents a space‐time adaptive finite element method (FEM) integrated with a posteriori error analysis to simulate avascular tumor growth and drug interactions in prostate cancer. The model is based on a nonlinear, coupled system of reaction‐diffusion equations, incorporating discontinuities in time to represent drug interactions. We perform a residual‐based a posteriori error analysis, providing precise error estimates for both spatial and temporal components. The approach numerically ensures boundedness and positivity for the phase variable, nutrient, and prostate‐specific antigen (PSA) concentrations. Furthermore, we establish and prove the reliable error estimators for space and time discretization. Numerical tests, including cases with and without drug interactions, show that the estimators accurately capture tumor growth. A comparison of solution convergence using uniform grids and Adaptive Mesh Refinement (AMR) demonstrates the superior efficiency of AMR, significantly reducing computational effort in the absence of drug effects and achieving a substantial reduction in computational costs when drug effects are present. This highlights the effectiveness of AMR in optimizing both accuracy and computational efficiency, making it an essential tool for modeling such complex systems.