Asymptotic behavior of a system of parabolic quasivariational inequalities

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In this work, we investigated non-coercive parabolic quasi-variational inequality (qvi) systems associated with Hamilton–Jacobi–Bellman (HJB) equations. Our methodology consists of reformulating QVI systems directly into HJB equations and solving them, which represents a reversed perspective compared to most previous studies. We focused on the analysis of the asymptotic behavior of these systems in both coercive and non-coercive cases, by considering continuous and discrete models. The discrete approximation was developed using finite element and finite difference methods. Our approach relies on overlapping domain decomposition techniques of the alternating Schwarz type, first applied to two subdomains and then generalized to the case of q subdomains. Within this framework, we constructed two monotone sequences of lower and upper solutions and proved that they converge monotonically and geometrically to the unique discrete solution, while providing a sharp error estimate in the L∞ norm. Finally, we conducted several numerical experiments to study the asymptotic behavior of the computed solutions. The results demonstrated the stability and accuracy of the method, confirmed its effciency in approximating the dynamics of parabolic Hamilton–Jacobi–Bellman equations, and underscored the crucial role of verlapping in accelerating convergence.

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