Finite element analysis of a nonlinear parabolic equation modeling epitaxial thin-film growth
- PDF / 701,703 Bytes
- 13 Pages / 595.276 x 793.701 pts Page_size
- 6 Downloads / 211 Views
RESEARCH
Open Access
Finite element analysis of a nonlinear parabolic equation modeling epitaxial thin-film growth Fengnan Liu1* , Xiaopeng Zhao2 and Bo Liu1 *
Correspondence: [email protected] College of Mathematics, Jilin University, Changchun, 130012, China Full list of author information is available at the end of the article 1
Abstract In this paper, we consider a nonlinear model describing crystal surface growth. For the equation, the finite element method is presented and a nice error estimate is derived in the L2 norm by means of a finite element biharmonic projection approximation.
1 Introduction The finite element method is essentially a discretization method for the approximate solution of partial differential equations. It has the natural advantage of keeping the physical properties of the primitive problems. There are many papers that have already been published to study the finite element method for a fourth-order nonlinear parabolic equation (see [–]). In this paper, we consider the finite element analysis for the following problem: ⎧ ∂u ⎪ ⎪ ⎨ ∂t + γ uxxxx – (|ux | ux – ux )x = , (x, t) ∈ (, π) × (, T), ux (x, t) = uxxx (x, t) = , x = , π, ⎪ ⎪ ⎩ u(x, ) = u (x), in (, π),
()
where γ is a positive constant. Problem () arises in epitaxial growth of nanoscale thin films [, ], where u(x, t) denotes the height from the surface of the film in epitaxial growth. The term uxxxx denotes the capillarity-driven surface diffusion, uxx denotes diffusion due to evaporationcondensation and |ux | ux corresponds to the upward hopping of atoms. During the past years, many authors have paid much attention to problem (), for example [, –]. Here, we give the existence and uniqueness of a global solution for problem () (see[]). Theorem . Suppose that HE (, π) = {u ∈ H (, π) : ux () = ux (π) = }, and u ∈ HE (, π) ∩ W , (, π), then there exists a unique global solution u(x, t) for problem (), such that u(x, t) ∈ C [, T]; L (, π) ∩ L∞ [, T]; HE (, π) ∩ L∞ [, T]; W , (, π) . The outline of this paper is as follows. In the next section, we establish a semi-discrete approximation and derive its error bound. In Section , the full-discrete approximation ©2014 Liu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Liu et al. Boundary Value Problems 2014, 2014:46 http://www.boundaryvalueproblems.com/content/2014/1/46
Page 2 of 13
for problem () is studied. In the last section, some numerical experiments which confirm our results are presented. Throughout this paper, we denote the L , Lp , L∞ , H k norms in (, ) simply by · , · Lp , | · |∞ , and · k . Define the inner product of L space as (·, ·), we have the space
Lp (, T; X) = u(t) : uLp (X) =
u(t) p dt X
T
p
Data Loading...
