Optimal L^ -error analysis for parabolic systems of quasi-variational inequalities with nonlinear source terms

This paper is devoted to the numerical analysis of a system of parabolic quasi-variational inequalities with nonlinear source terms and solution-dependent obstacles. We consider a vector-valued problem defined on a bounded domain and governed by coercive elliptic operators and Lipschitz continuous nonlinearities. We establish existence, uniqueness, comparison, and monotonicity properties for the continuous system. A parabolic Bensoussan–Lions type iterative scheme is introduced and shown to converge geometrically in the strong norm L^ ( (0, T) ). We then construct a fully discrete scheme based on implicit Euler time discretization and finite element approximation in space, preserving the stability and monotonicity of the continuous problem. Under suitable assumptions, we derive an optimal L^-error estimate of order O (h^2| h|^2 + t). To the best of our knowledge, this is the first optimal maximum-norm error estimate for fully discrete parabolic systems of quasi-variational inequalities with solution-dependent obstacles. Numerical experiments are provided to validate the theoretical results and confirm the efficiency and accuracy of the proposed methods.