A numerical method for solving two-dimensional nonlinear parabolic problems based on a preconditioning operator

• Published in: Mathematical modelling and analysis Vol. 25; no. 4; pp. 531 - 545
• Main Authors: Salehi Shayegan, Amir Hossein, Zakeri, Ali, Hosseini, Seyed Mohammad
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 13.10.2020
• Subjects: Approximation; Boundary value problems; Error analysis; Finite element analysis; Finite element method; Ill-conditioned problems (mathematics); Iterative methods; Mathematical analysis; Mathematical research; Meshless methods; Methods; nonlinear parabolic problems; Nonlinear systems; Nonlinear theories; Numerical analysis; Numerical methods; Preconditioning; preconditioning operator; Sobolev space; Sobolev space gradient method; Sobolev spaces; WEB-spline finite element method; Iran; WEB-spline finite element method; nonlinear parabolic problems; Sobolev space gradient method; preconditioning operator
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2020.4310

‎This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods.