The
p-Laplace equation is not only a tool for researching the special theory of Sobolev spaces [
1], but is also an important mathematical model of many physical processes and 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 [
2], elasticity [
3], population models [
4], the non-Newtonian fluid movement in the boundary layer [
5], and digital image processing [
6]. However, since the equation includes a very strong nonlinear factor, it is an important approach to solve the equation by numerical methods. A finite element method, combined with Newton iteration scheme, is one of most efficient numerical methods. Some posteriori error estimates for the finite element approximation of the
p-Laplace equation are developed by Carstensen
et al. [
7,
8]. Carstensen [
9,
10] applied these posteriori error estimates to a control method of solving the equation. The control method is based on the Newton iteration. However, 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 also requires an exploratory reduction in the iterative step length (the default step length is 1; see [
11]). Therefore, it is necessary to study how to select a suitable initial function. On the other hand, though the Newton algorithm has the advantage of very fast convergence rate near the solution (see [
11]), there are many factors (for instance, the ill-posed factor) to affect the convergence of Newton iterations for the
p-Laplace equation. Bermejo and Infante [
12] applied the Polak-Ribiere iterations to the multigrid algorithm for the
p-Laplace equation instead of Newton iterations due to the difficulties in computation relating to the ill-posed coefficient matrix. In order to overcome the ill-posed problem, it is necessary to develop a well-posed condition of the iteration functions. To the best of our knowledge, a well-posed condition of the iteration functions of finite element-Newton iterations for the
p-Laplace equation has not been provided so far. Therefore, in this paper, we aim mainly to establish a well-posed condition of the iterative functions of finite element-Newton iterations for the
p-Laplace equation and to provide theory analysis. To this end, we intend to transform the p-Laplace equation into a functional minimum problem (see Section 2 in [
12]) solved by Newton iterations. According to the well-posed condition, an effective particular initial function is selected, and an effective iterative function sequence is constructed. Besides, utilizing the gradient modulus and gradient direction of an element, we will discuss the factors affecting the convergence of Newton iterations.
To this end, we first introduce some special Sobolev spaces and two preparative definitions as follows. Let
$$\begin{aligned} W^{1,p}(\Omega)= \biggl\{ v; \int_{\Omega} \vert \nabla v \vert ^{p}< \infty \biggr\} , \qquad W_{0}^{1,p}(\Omega)= \bigl\{ v\in W^{1,p}( \Omega); v|_{\partial\Omega}=0 \bigr\} \end{aligned}$$
with inner product and norm
$$\begin{aligned} (u,v)= \int_{\Omega} u\cdot v{\,\mathrm{d}}x{\,\mathrm{d}}y\quad\mbox{and}\quad \Vert \nabla u \Vert _{p}= \biggl( \int_{\Omega} \vert \nabla u \vert ^{p}{\,\mathrm{d}}x{\,\mathrm{d}}y\biggr) ^{1/p} . \end{aligned}$$
In particular, we set
\(H^{1}_{0}(\Omega)=W_{0}^{1,2}(\Omega)\) when
\(p=2\). Since
\(p>2\) and Ω is bounded, the imbedding of
\(W_{0}^{1,p}(\Omega)\) into
\(H^{1}_{0}(\Omega)\) is continuous (see [
1]).
The paper is organized as follows. In Section
2, a functional minimum problem equivalent to the
p-Laplace equation is introduced, a finite element-Newton iteration formula is established, and the classical Newton algorithm is presented. In Section
3, we discuss the well-posed condition of iterative functions. Even though the initial function fails to satisfy the well-posed condition, after a sufficient number of Newton iterations with default step length are implemented, well-posed iterative functions can be always obtained, as will be seen in the well-posed theorem of Section
3. However, the iteration step number is often large. So an effective particular initial iterative function that satisfies the well-posed condition should be selected in order to make the iteration step number reduced greatly. This is related to the content of Section
5. In Section
4, an effective particular iterative function sequence is provided. These functions possess some properties that the gradient moduli on each subdivision finite element decrease monotonically and have a certain lower boundary. By using these properties we prove that this sequence converges to the solution of finite element formulation of the
p-Laplace equation and present some results about its convergence speed. In Section
5, considering the well-posed condition and properties mentioned, we select an effective particular initial iterative function, which results in the special iterative functions involved in Section
4 by finite element Newton iterations with default step length. In Section
6, some numerical experiments are provided for showing that some results on the convergence rate and gradient fields are consistent with theoretical conclusions.