Existence and uniqueness results of solutions for a new class of parabolic varitional inequalities related with impulse control problem

In this work, we establish novel existence and uniqueness results for parabolic quasi-variational inequalities (PQVIs) by developing a structured four-phase numerical framework . The proposed methodology combines spatial discretization via finite element methods (FEM) with a semi-implicit time-stepping scheme to enhance stability and convergence. We construct a discrete iterative algorithm by linking the underlying variational system to a fixed-point formulation and further enrich it using a monotone iterative scheme inspired by Bensoussan’s approach. To ensure mathematical rigor, we first present precise definitions and assumptions on the operators, coefficients, and source terms. Subsequently, the algorithm is fully developed and its consistency and stability are rigorously justified. A comprehensive convergence analysis is then provided, extending classical results to the FEM setting, and highlighting the impact of variational inconsistencies inherent to the discretization. Under strengthened regularity assumptions on the source term f , we derive refined error estimates in appropriate Sobolev norms, e.g., <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mo stretchy="false">‖</m:mo> <m:mi>u</m:mi> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>h</m:mi> </m:mrow> </m:msub> <m:msub> <m:mrow> <m:mo stretchy="false">‖</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>C</m:mi> <m:msup> <m:mrow> <m:mi>h</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> u-uₕ_H¹(Ω)≤ Chα , where u and u h denote the continuous and discrete solutions, respectively. Our findings provide both theoretical validation and practical guidance for implementing FEM in challenging fractional, nonlocal, and impulse-control-related PQVIs.