Abstract
This work aims to provide a comprehensive and unified numerical analysis for a nonlinear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish the existence of an exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and nonconforming numerical schemes.
Highlights
1 Introduction Nonlinear parabolic variational inequalities and PDEs are useful tools to model the coupled biochemical interactions of microbial cells, which are crucial to numerous applications, especially in the medical field and food production [16, 19, 21]
We consider here a nonlinear parabolic system consisting of PDEs and variational inequalities:
An L2-error estimate is provided in different studies, such as [5], by using a finite difference in time
Summary
Nonlinear parabolic variational inequalities and PDEs are useful tools to model the coupled biochemical interactions of microbial cells, which are crucial to numerous applications, especially in the medical field and food production [16, 19, 21]. Definition 3.4 (Consistency) If D is a gradient discretization, let SD : K → [0, ∞) and SD : H01( ) → [0, ∞) be defined by Definition 3.5 (Limit-conformity) If D is a gradient discretization, let WD : Hdiv( ) := {ψ ∈ L2( )d : divψ ∈ L2( )} → [0, +∞) be defined by
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