Multigrid method for noncoercive parabolic variational inequality

In this article, our work is focused on the proof of the uniform convergence of the multigrid method for parabolic variational inequality with a noncoercive operator and its numerical solution. To discretize the problem, we utilize a finite element scheme for the noncoercive operator and Euler scheme for the time. To obtain the system discretization of our problem, we reformulate the parabolic variational inequality as a Hamilton–Jacobi–Bellman equation. On the smooth grid, we apply the multigrid method as an interior iteration on the linear system. Finally, we provide a proof of the uniform convergence of the multigrid method for parabolic variational inequality with a noncoercive operator, providing a numerical example of this problem.