An Improved Finite Element Method with Unconditional Positivity for a Reaction-diffusion HIV Infection Model with Spatial Heterogeneity

This paper develops an improved finite element method for the reaction-diffusion HIV infection model with spatial heterogeneity. By introducing the h-inner product, the positivity of numerical solutions is unconditionally guaranteed without mesh constraints, overcoming a key limitation of classical methods. The boundedness, convergence, and global stability threshold of the disease-free equilibrium are established. The linearly implicit Euler method is adopted to avoid step-size restrictions of explicit schemes while reducing computational cost of implicit ones. The full-discrete scheme achieves first-order convergence in space and time, preserves positivity and boundedness, and shares the same stability threshold as the semi-discrete system. This work provides reliable theoretical foundations for the efficient and structure-preserving simulation of spatially heterogeneous epidemic models.