A numerical solution of parabolic quasi variational inquality non-linear using Newton-Multigrid method

Iranian Journal of Numerical Analysis and Optimization

مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 24 اردیبهشت 1403

نوع مقاله: Review Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.86954.1395

نویسندگان

Mostafa Bahi^{* 1}؛ Mohammed Beggas¹؛ Nourelhouda Nesba¹؛ Imtiaz Ahmad²

¹Department of Mathematics, Faculty of exact Sciences, University of EL-OUED, Algeria.

²Department of Mathematics, University of Swabi, Khyber Pakhtunkhwa, Pakistan

چکیده

In this article, we apply three numerical methods to study the uniform convergence of the Newton-Multigrid method for parabolic quasi-variational inequalities with a non-linear right-hand side. To discretize the problem, we utilize a finite element method for the operator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton-Jacobi-Bellman equation. For linearizing the problem on the coarse grid, we employ Newton's method as an external iteration to obtain the Jacobian system. On the smooth grid, we apply the multi-grid method as an interior iteration of the Jacobian system. Finally, we provide proof of the uniform convergence of the Newton-Multigrid method for parabolic quasi-variational inequalities with a nonlinear right hand, by giving a numerical example of this problem.

کلیدواژه‌ها

Newton' s method؛ Multigrid method؛ Parabolic Variational In- equality؛ Finite element method؛ Hamilton-Jacobi-Bellman equation