Finite element analysis of a nonlinear parabolic equation modeling epitaxial thin-film growth
© Liu et al.; licensee Springer. 2014
Received: 11 October 2013
Accepted: 7 February 2014
Published: 24 February 2014
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 norm by means of a finite element biharmonic projection approximation.
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 [1–6]).
where γ is a positive constant.
Problem (1) arises in epitaxial growth of nanoscale thin films [7, 8], where denotes the height from the surface of the film in epitaxial growth. The term denotes the capillarity-driven surface diffusion, denotes diffusion due to evaporation-condensation and corresponds to the upward hopping of atoms. During the past years, many authors have paid much attention to problem (1), for example [7, 9–12].
Here, we give the existence and uniqueness of a global solution for problem (1) (see).
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 3, the full-discrete approximation for problem (1) is studied. In the last section, some numerical experiments which confirm our results are presented.
On the other hand, the letters C, denote generic constants independent of the finite element division size and not necessarily the same at different occurrences.
2 Semi-discrete approximation
where C is a positive constant depending only on γ and , independent of h.
Proof The equation of problem (3) is an ordinary differential equation and according to ODE theory, there exists a unique local solution to problem (3) in the interval . If we have (4), then according to the extension theorem, we can also obtain the existence of unique global solution. So, we only need to prove (4).
By (7), (10), and (11), we complete the proof of Theorem 2.1. □
where is a positive constant dependent only on γ, independent of and h.
where is a positive constant depends only on γ and μ. Hence, is a symmetrical positive determined bilinear form, and there exists a unique solution for problem (12).
Then the proof of Lemma 2.1 is completed. □
Remark 2.2 We use the integration by parts for the term in the proof of Lemma 2.1. Then a better convergency is obtained.
where the constant .
Combing (20) and (14) (noticing that ), and using the triangle inequality, we complete the proof of Theorem 2.2. □
where the constant . We will prove it in the next step.
3 Full-discrete approximation
We will prove the correctness of (26) in the end of this proof.
Using the discrete Gronwall inequality, (12), and the triangle inequality, we obtain (25).
where is the Hermite type interpolation approximation of the function u. Hence, (26) is correct for . Then, using the inductive method, the correctness of (26) is proved, and the proof of Theorem 3.1 is completed. □
We will prove it in the next step.
4 Numerical approximation
In addition, we consider the change of error when the time . Since there is no exact solution to problem (1) to the best of our knowledge, we make a comparison between the solution of (22) on coarse meshes and the fine mesh.
The error for difference time step at
In Table 1, it is easy to see that the third column is monotone decreasing along with the time step’s waning and the fourth column is not monotone decreasing along with the time step’s waning. Then the order of convergence for time is of and . It is easy to see that the result of the numerical analysis on time is better than the theoretical result. The reason may be the existence of a nonlinear term or the limit of the theoretical proof tool.
The error for difference h at
which means the order of the error estimates is .
The error for the differences h and Δ t at
which means that the order of error estimates is of .
The authors would like to express their deep thanks for the referee’s valuable suggestions about the revision and improvement of the manuscript.
- Barrett JW, Blowey JF, Garcke H: Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 1999, 37: 286-318. 10.1137/S0036142997331669MathSciNetView ArticleGoogle Scholar
- Choo SM, Kim YH: Finite element scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient. J. Appl. Math. Comput. 2005, 19: 385-395. 10.1007/BF02935813MathSciNetView ArticleGoogle Scholar
- Elliott CM, French DA: A nonconforming finite element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 1989, 26: 884-903. 10.1137/0726049MathSciNetView ArticleGoogle Scholar
- Feng X, Wu H: A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow. J. Comput. Math. 2008, 26: 767-796.MathSciNetGoogle Scholar
- Kovács M, Larsson S, Mesforush A: Finite element approximation of the Cahn-Hilliard-Cook equation. SIAM J. Numer. Anal. 2011, 49: 2407-2429. 10.1137/110828150MathSciNetView ArticleGoogle Scholar
- Zhang T: Finite element analysis for Cahn-Hilliard equation. Math. Numer. Sin. 2006, 28: 281-292.MathSciNetGoogle Scholar
- King BB, Stein O, Winkler M: A fourth order parabolic equation modeling epitaxial thin film growth. J. Math. Anal. Appl. 2003, 286: 459-490. 10.1016/S0022-247X(03)00474-8MathSciNetView ArticleGoogle Scholar
- Zangwill A: Some causes and a consequence of epitaxial roughening. J. Cryst. Growth 1996, 163: 8-21. 10.1016/0022-0248(95)01048-3View ArticleGoogle Scholar
- Kohn RV, Yan X: Upper bound on the coarsening rate for an epitaxial growth model. Commun. Pure Appl. Math. 2003, 56: 1549-1564. 10.1002/cpa.10103MathSciNetView ArticleGoogle Scholar
- Liu C: Regularity of solutions for a fourth order parabolic equation. Bull. Belg. Math. Soc. Simon Stevin 2006, 13(3):527-535.MathSciNetGoogle Scholar
- Zhao X, Liu C: The existence of global attractor for a fourth-order parabolic equation. Appl. Anal. 2013, 92: 44-59. 10.1080/00036811.2011.590476MathSciNetView ArticleGoogle Scholar
- Zhao X, Liu C: Optimal control of a fourth-order parabolic equation modeling epitaxial thin-film growth. Bull. Belg. Math. Soc. Simon Stevin 2013, 20(3):547-557.MathSciNetGoogle Scholar
- Ciarlet P: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam; 1978.Google Scholar
This article is published under license to BioMed Central Ltd. 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 credited.