An effective finite element Newton method for 2D p-Laplace equation with particular initial iterative function

Zhendong Luo,Fei Teng

doi:10.1186/s13660-016-1223-9

Zhendong Luo, Fei Teng

Open Access

https://doi.org/10.1186/s13660-016-1223-9

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Abstract

In this article, a functional minimum problem equivalent to the p-Laplace equation is introduced, a finite element-Newton iteration formula is established, and a well-posed condition of iterative functions satisfied is provided. According to the well-posed condition, an effective initial iterative function is presented. Using the effective particular initial function and Newton iterations with the iterative step length equal to 1, an effective particular sequence of iterative functions is obtained. With the decreasing properties of gradient modulus of subdivision finite element, it has been proved that the function sequence converges to the solution of finite element formulation of p-Laplace equation. Moreover, a discussion on local convergence rate of iterative functions is provided. In summary, the iterative method based on the effective particular initial function not only makes up the shortage of the Newton algorithm, which requires an exploratory reduction in the iterative step length, but also retains the benefit of fast convergence rate, which is verified with theoretical analysis and numerical experiments.

Highlights

Let ⊂ R be a bounded and connected domain
The Newton iteration of the p-Laplace equation is not discussed in detail in their study, which is very dependent on selection of initial iteration function and requires an exploratory reduction in the iterative step length
6 Some numerical experiments we present some experiments of the Newton method to solve the p-Laplace equation based on two different initial iterative functions so as to validate the conclusion of theoretical analysis in the preceding section

Summary

Introduction

Let ⊂ R be a bounded and connected domain. Consider the following p-Laplace equation with Dirichlet boundary.Problem I Find u such that div(|∇u|p– ∇u) = f , (x, y) ∈ , ( . )u = , (x, y) ∈ ∂ , where p > , the source term f is smooth enough to ensure validity of the following analysis and does not vanish on any nonzero measure set K (K ⊂ ).The p-Laplace equation is a tool for researching the special theory of Sobolev spaces [ ], but is an important mathematical model of many physical processes andLuo and Teng Journal of Inequalities and Applications (2016) 2016:281 other applied sciences; for example, it can be used to describe a variety of nonlinear media such as phase transitions in water and ice at transition temperature [ ], elasticity [ ], population models [ ], the non-Newtonian fluid movement in the boundary layer [ ], and digital image processing [ ]. The initial iterative function need to be selected to satisfy the well-posed condition, which will be discussed in detail in Section .

Results

Conclusion