A numerical solution of parabolic quasi-variational inequality nonlinear using Newton-multigrid method

Iranian Journal of Numerical Analysis and Optimization

مقاله 1، دوره 14، Issue 4 - شماره پیاپی 31، اسفند 2024، صفحه 991-1015   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2024.86954.1395

نویسندگان

M. Bahi^{* 1}؛ M. Beggas¹؛ N. Nesba¹؛ A. Imtiaz²

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

²Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional, Kajang, Selangor, Malaysia.

چکیده

In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear 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 interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem. 

کلیدواژه‌ها

Newton’s method؛ Multigrid method؛ Parabolic variational inequality؛ Finite element method؛ Hamilton–Jacobi–Bellman equation

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