Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems

• Published in: Applied mathematical modelling Vol. 38; no. 2; pp. 775 - 783
• Main Authors: Zakeri, Ali, Shayegan, Amir Hossein Salehi
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2014
• Subjects: Approximation; Convergence; Finite element method; Mathematical analysis; Mathematical models; Preconditioning operator; Quasilinear elliptic boundary value problems; Sobolev space; Sobolev space gradient method; Two dimensional; WEB-spline finite element method; WEB-spline finite element method; Preconditioning operator; Quasilinear elliptic boundary value problems; Sobolev space gradient method
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2013.06.018

In this paper, a two-dimensional quasilinear elliptic problem of the form -divF(x,▽u)=g(x) is considered. This problem is ill-conditioned and we therefore propose a modified iterative algorithm based on coupling of the Sobolev space gradient method and WEB-spline finite element method. Applying the preconditioned iterative method, which has been already provided by Farago and Karatson (2001) [1] reduces the our considered problem to a sequence of linear Poisson’s problems. Then the WEB-spline finite element method is applied to the approximate solution of these Poisson’s problems. In this sense, a convergence theorem is proved and the advantages of this technique than the gradient finite element method (GFEM) is also described. Finally, the presented method is tested on some examples and compared with GFEM. It is shown that the gradient WEB-spline finite element method gives better test results.