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

Abstract

‎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.

Keywords

• ‎Sobolev space gradient method
• WEB-spline finite element method
• preconditioning operator
• nonlinear parabolic problems