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 L 2 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 u ( x , t ) denotes the height from the surface of the film in epitaxial growth. The term u x x x x denotes the capillarity-driven surface diffusion, u x x denotes diffusion due to evaporation-condensation and | u x | 2 u x 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[7]).
u ( x , t ) ∈ C 0 ( [ 0 , T ] ; L 2 ( 0 , π ) ) ∩ L ∞ ( [ 0 , T ] ; H E 2 ( 0 , π ) ) ∩ L ∞ ( [ 0 , T ] ; W 1 , 4 ( 0 , 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 3, the full-discrete approximation for problem (1) is studied. In the last section, some numerical experiments which confirm our results are presented.
L p ( 0 , T ; X ) = { u ( t ) : ∥ u ∥ L p ( X ) = ( ∫ 0 T ∥ u ( t ) ∥ X p d t ) 1 p < ∞ } , 1 ≤ p ≤ ∞ .
On the other hand, the letters C, C ′ denote generic constants independent of the finite element division size and not necessarily the same at different occurrences.
S h ( k ) ⊂ H E 2 ( I ) = { u ∈ H 2 ( I ) : u x ( 0 ) = u x ( π ) = 0 } .
{ ( u h , t , v h ) + γ ( D 2 u h , D 2 v h ) + ( | D u h | 2 D u h − D u h , D v h ) = 0 , ∀ v h ∈ S h ( k ) , ( u h ( 0 ) − u ( 0 ) , v h ) = 0 , ∀ v h ∈ S h ( k ) .
∫ 0 π u h ( t , x ) d x = ∫ 0 π u h ( 0 , x ) d x , 0 ≤ t ≤ T .
where C is a positive constant depending only on γ and ∥ u h ( 0 ) ∥ 2 , 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 [ 0 , t n ) . 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).
1 2 d d t ∥ u h ( t ) ∥ 2 + γ ∥ D 2 u h ∥ 2 + ( | D u h | 2 D u h , D u h ) = ∥ D u h ∥ 2 = − ( u h , D 2 u h ) ≤ γ 2 ∥ D 2 u h ∥ 2 + 1 2 γ ∥ u h ∥ 2 .
d d t ∥ u h ( t ) ∥ 2 + γ ∥ D 2 u h ∥ 2 ≤ 1 γ ∥ u h ∥ 2 .
d d t ( e − β t ∥ u h ( t ) ∥ 2 ) ≤ 0 .
∥ u h ( t ) ∥ 2 ≤ e β t ∥ u h ( 0 ) ∥ 2 , 0 ≤ t ≤ T , β = 1 γ .
∥ u h , t ∥ 2 + γ ( D 2 u h , D 2 u h , t ) + ( | D u h | 2 D u h − D u h , D u h , t ) = 0 .
F h ( t ) = γ 2 ∥ D 2 u h ∥ 2 + 1 4 ∥ D u h ∥ L 4 4 − 1 2 ∥ D u h ∥ 2 .
d d t F h ( t ) = γ ∫ Ω D 2 u h D 2 u h , t d x + ∫ Ω | D u h | 2 D u h D u h , t d x − ∫ Ω D u h D u h , t d x = − ∥ u h , t ∥ 2 ≤ 0 .
γ 2 ∥ D 2 u h ∥ 2 + 1 4 ∥ D u h ∥ L 4 4 + 1 2 ∥ D u h ( 0 ) ∥ 2 ≤ γ 2 ∥ D 2 u h ( 0 ) ∥ 2 + 1 4 ∥ D u h ( 0 ) ∥ L 4 4 + 1 2 ∥ D u h ∥ 2 ≤ γ 2 ∥ D 2 u h ( 0 ) ∥ 2 + 1 4 ∥ D u h ( 0 ) ∥ L 4 4 + γ 4 ∥ D 2 u h ∥ 2 + 1 4 γ ∥ u h ∥ 2 ≤ γ 2 ∥ D 2 u h ( 0 ) ∥ 2 + 1 4 ∥ D u h ( 0 ) ∥ L 4 4 + γ 4 ∥ D 2 u h ∥ 2 + 1 4 γ e β T ∥ u h ( 0 ) ∥ 2 .
∥ D 2 u h ∥ 2 ≤ C .
∥ D u h ∥ 2 = − ( u h , D 2 u h ) ≤ 1 2 ∥ u h ∥ 2 + 1 2 ∥ D 2 u h ∥ 2 .
C 1 ′ ( ∥ D 2 u h ∥ 2 + ∥ D u h ∥ L 4 4 ) ≤ C 2 ′ ( ∥ D 2 u h ( 0 ) ∥ 2 + ∥ D u h ( 0 ) ∥ L 4 4 + ∥ u h ( 0 ) ∥ 2 ) .
where C 1 is a positive constant dependent only on γ, independent of u h ( 0 ) and h.
a ( u − R h u , v h ) ≡ γ ( D 2 ( u − R h u ) , D 2 v h ) + ( u − R h u , v h ) = 0 , ∀ v h ∈ S h ( k ) .
where c 0 is a positive constant depends only on γ and μ. Hence, a ( u , v ) is a symmetrical positive determined bilinear form, and there exists a unique solution u h ∈ S h ( k ) for problem (12).
∥ u − R h u ∥ + h ∥ u − R h u ∥ 1 + h 2 ∥ u − R h u ∥ 2 ≤ C h r + 1 ∥ u ∥ r + 1 , 2 ≤ r ≤ k .
u − u h = u − R h u + R h u − u h = η ( t ) + θ ( t ) .
( η t + θ t , v h ) + γ ( D 2 η + D 2 θ , D 2 v h ) + ( | D u | 2 D u − | D u h | 2 D u h , D v h ) − ( D η + D θ , D v h ) = 0 .
( θ t , v h ) + γ ( D 2 θ , D 2 v h ) = ( η − η t , v h ) − ( | D u | 2 D u − | D u h | 2 D u h , D v h ) + ( D θ , D v h ) .
− ( | D u | 2 D u − | D u h | 2 D u h , D θ ) ≤ C ( ∥ θ ∥ 2 + ∥ η ∥ 2 ) + γ 4 ∥ D 2 θ ∥ 2 .
∥ η ∥ 2 = ∥ u − R h u ∥ 2 ≤ ∥ u ∥ 2 + C ∥ u ∥ 2 ≤ C .
| D u | 2 D u − | D u h | 2 D u h = [ ( D u ) 2 + D u D u h + ( D u h ) 2 ] ( D u − D u h ) .
| D u | ∞ ≤ C ′ , | D u h | ∞ ≤ C ′ , | D θ | ∞ ≤ C ′ ∥ θ ∥ 2 .
− ( | D u | 2 D u − | D u h | 2 D u h , D θ ) = − ( [ ( D u ) 2 + D u D u h + ( D u h ) 2 ] ( D u − D u h ) , D θ ) = ( [ ( D u ) 2 + D u D u h + ( D u h ) 2 ] ( θ + η ) , D 2 θ ) + ( θ + η , D θ [ 2 D u D 2 u + D u D 2 u h + D 2 u D u h + 2 D u h D 2 u h ] ) ≤ ∥ θ + η ∥ ∥ D 2 θ ∥ + | D θ | ∞ × ∥ θ + η ∥ ( 2 | D u | ∞ ∥ D 2 u ∥ + | D u | ∞ ∥ D 2 u h ∥ + | D u h | ∞ ∥ D 2 u ∥ + 2 | D u h | ∞ ∥ D 2 u h ∥ ) ≤ ∥ θ + η ∥ ∥ D 2 θ ∥ + C ′ ∥ θ ∥ 2 ∥ θ + η ∥ ≤ C ( ∥ θ ∥ 2 + ∥ η ∥ 2 ) + ε ∥ D 2 θ ∥ 2 .
Remark 2.2 We use the integration by parts for the term − ( | D u | 2 D u − | D u h | 2 D u h , D θ ) in the proof of Lemma 2.1. Then a better convergency is obtained.
∥ u ( 0 ) − u h ( 0 ) ∥ ≤ C h k + 1 ∥ u ( 0 ) ∥ k + 1 .
where the constant C ′ = C ′ ( u , u h ( 0 ) , γ ) .
1 2 d d t ∥ θ ∥ 2 + γ ∥ D 2 θ ∥ 2 ≤ ( ∥ η ∥ + ∥ η t ∥ ) ∥ θ ∥ − ( | D u | 2 D u − | D u h | 2 D u h , D θ ) + ∥ D 2 θ ∥ ∥ θ ∥ ≤ 1 2 ( ∥ η ∥ + ∥ η t ∥ ) 2 + C ( ∥ θ ∥ 2 + ∥ η ∥ 2 ) + γ 4 ∥ D 2 θ ∥ 2 + 1 γ ∥ θ ∥ 2 + γ 4 ∥ D 2 θ ∥ 2 ≤ ( C + 1 2 ) ( ∥ η ∥ + ∥ η t ∥ ) 2 + γ 2 ∥ D 2 θ ∥ 2 + ( C + 1 γ ) ∥ θ ∥ 2 .
d d t ∥ θ ∥ 2 + γ ∥ D 2 θ ∥ 2 ≤ ( 2 C + 1 ) ( ∥ η ∥ + ∥ η t ∥ ) 2 + ( 2 C + 2 γ ) ∥ θ ∥ 2 .
∥ θ ∥ 2 ≤ C ′ ( ∥ θ ( 0 ) ∥ 2 + ∫ 0 t ( ∥ η ( τ ) ∥ 2 + ∥ η t ( τ ) ∥ 2 ) d τ ) .
where the constant C ′ = C ′ ( u , u h ( 0 ) , γ ) . We will prove it in the next step.
where u n = u ( t n ) .
{ ( δ t U n , v h ) + γ ( D 2 U n , D 2 v h ) − ( D U n , D v h ) + ( ( D U n − 1 ) 2 D U n , D v h ) = 0 , ∀ v h ∈ S h ( k ) , ( U 0 − u ( 0 ) , v h ) = 0 , ∀ v h ∈ S h ( k ) .
u n − U n = u n − R h u n + R h u n − U n = η n + θ n , θ n ∈ S h ( k ) .
( δ t R h u n , v h ) + γ ( D 2 R h u n , D 2 v h ) = ( η n − ε n − δ t η n , v h ) + ( D R h u n , D v h ) − ( | D u n | 2 D u n , D v h ) .
( δ t θ n , v h ) + γ ( D 2 θ n , D 2 v h ) = ( η n − ε n − δ t η n , v h ) + ( D θ n , D v h ) − ( | D u n | 2 D u n − | D U n − 1 | 2 D U n , D v h ) .
∥ u n − U n ∥ ≤ C ( Δ t + h k ) , n = 0 , 1 , … , M − 1 .
∥ D u m − D U m ∥ ∞ ≤ C , m = 1 , 2 , … , n − 1 .
We will prove the correctness of (26) in the end of this proof.
( δ t θ n , θ n ) + γ ∥ D 2 θ n ∥ 2 ≤ ∥ η n − ε n − δ t η n ∥ ∥ θ n ∥ + ∥ D θ n ∥ 2 − ∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ ∥ D θ n ∥ ≤ 1 2 ∥ η n − ε n − δ t η n ∥ 2 + 1 2 ∥ θ n ∥ 2 + 2 ∥ D θ n ∥ 2 + 1 4 ∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ 2 ≤ 1 2 ∥ η n − ε n − δ t η n ∥ 2 + ( 1 2 + 2 γ ) ∥ θ n ∥ 2 + γ 2 ∥ D 2 θ n ∥ 2 + 1 4 ∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ 2 .
∥ θ n ∥ 2 + γ ∥ D 2 θ n ∥ 2 ≤ ∥ θ n − 1 ∥ 2 + 2 Δ t { 1 2 ∥ η n − ε n − δ t η n ∥ 2 + ( 1 2 + 2 γ ) ∥ θ n ∥ 2 + 1 4 ∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ 2 } .
∥ η n ∥ ≤ C h k + 1 ∥ u ( t n ) ∥ k + 1 , ∥ ε n ∥ ≤ ∫ t n − 1 t n ∥ u t t ( τ ) ∥ d τ ≤ ( Δ t ) 1 2 ( ∫ t n − 1 t n ∥ u t t ( τ ) ∥ 2 d τ ) 1 2 , ∥ δ t η n ∥ = ∥ 1 Δ t ∫ t n − 1 t n η t ( τ ) d τ ∥ ≤ C 1 Δ t h k + 1 ∫ t n − 1 t n ∥ u t ( τ ) ∥ k + 1 d τ ≤ C 1 ( Δ t ) 1 2 h k + 1 ( ∫ t n − 1 t n ∥ u t ( τ ) ∥ k + 1 2 d τ ) 1 2 .
∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ = ∥ | D u n | 2 D u n − | D u n − 1 | 2 D u n + | D u n − 1 | 2 D u n − | D U n − 1 | 2 D u n + | D U n − 1 | 2 D u n − | D U n − 1 | 2 D U n ∥ ≤ ∥ | D u n | 2 D u n − | D u n − 1 | 2 D u n ∥ + ∥ | D u n − 1 | 2 D u n − | D U n − 1 | 2 D u n ∥ + ∥ | D U n − 1 | 2 D u n − | D U n − 1 | 2 D U n ∥ ≤ | D u n | ∞ | D u n + D u n − 1 | ∞ ∥ D u n − D u n − 1 ∥ + | D u n | ∞ 2 | D u n − 1 + D U n − 1 | ∞ ∥ D u n − 1 − D U n − 1 ∥ + | D U n − 1 | ∞ 2 ∥ D u n − D U n ∥ .
| D U n − 1 | ∞ ≤ | D u n − 1 − D U n − 1 | ∞ + | D u n − 1 | ∞ ≤ C + | D u n − 1 | ∞ , | D u n − 1 + D U n − 1 | ∞ ≤ | D u n − 1 − D U n − 1 | ∞ + 2 | D u n − 1 | ∞ ≤ C + | D u n − 1 | ∞ .
∥ | D u n | 2 D u n − | D U n − 1 | 2 D U n ∥ ≤ C ( ∥ D u n − D u n − 1 ∥ + ∥ D u n − 1 − D U n − 1 ∥ + ∥ D u n − D U n ∥ ) ≤ C ( ∥ D u n − D u n − 1 ∥ + ∥ D θ n − 1 + D η n − 1 ∥ + ∥ D θ n + D η n ∥ ) ≤ C ( ∫ t n − 1 t n ∥ D u t ( τ ) ∥ d τ + ∥ D η n − 1 ∥ + ∥ D θ n − 1 ∥ + ∥ D η n ∥ + ∥ D θ n ∥ ) ≤ C [ ( Δ t ) 1 2 ( ∫ t n − 1 t n ∥ D u t ( τ ) ∥ 2 d τ ) 1 2 + h k ( ∥ u ( t n ) ∥ k + 1 + ∥ u ( t n − 1 ) ∥ k + 1 ) + ∥ D θ n − 1 ∥ + ∥ D θ n ∥ ] .
∥ θ n ∥ 2 − ∥ θ n − 1 ∥ 2 + γ ∥ D 2 θ n ∥ 2 ≤ C Δ t ( ∥ D θ n ∥ 2 + ∥ D θ n − 1 ∥ 2 + h 2 k ∥ u ( t n ) ∥ k + 1 2 ) + C [ h 2 k + ( Δ t ) 2 ] ( ∫ t n − 1 t n ( ∥ u t ( τ ) ∥ k + 1 2 + ∥ u t t ∥ 2 ) d τ ) .
∥ θ n ∥ 2 + γ ∑ i = 1 n ∥ D 2 θ i ∥ 2 ≤ C Δ t [ ∥ D θ 0 ∥ 2 + ∑ i = 1 n ∥ D θ i ∥ 2 ] + C [ ( Δ t ) 2 + h 2 k ] ( ∫ 0 t n ( ∥ u t ( τ ) ∥ k + 1 2 + ∥ u t t ∥ 2 ) d τ ) ≤ C Δ t [ ∥ D θ 0 ∥ 2 + 1 2 ∑ i = 1 n ( ∥ D 2 θ i ∥ 2 + ∥ θ i ∥ 2 ) ] + C [ ( Δ t ) 2 + h 2 k ] ( ∫ 0 t n ( ∥ u t ( τ ) ∥ k + 1 2 + ∥ u t t ∥ 2 ) d τ ) .
∥ θ n ∥ 2 ≤ C Δ t ∑ i = 1 n − 1 ∥ θ i ∥ 2 + C [ ( Δ t ) 2 + h 2 k ] .
Using the discrete Gronwall inequality, (12), and the triangle inequality, we obtain (25).
∥ u n − U n ∥ 2 ≤ C ( Δ t + h k − 2 ) , n = 0 , 1 , … , M − 1 .
We will prove it in the next step.
In this section, using the full-discrete form (22), we approximate the solution of problem (1). Let u 0 = cos x , γ = 1 , T = 1 , h = π 400 , Δ t = 1 100 . We get the solution which evolves from t = 0 to t = 1 (cf. Figure 1).
The approximation solution of the full-discrete form.
In addition, we consider the change of error when the time t = 0.5 . 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.
err ( 0.5 , Δ t ) = ( ∫ 0 1 [ u N k ( x , 0.5 ) − u N min ( x , 0.5 ) ] 2 d x ) 1 2 , k = 1 , 2 , 3 , 4 .
Then the error is shown in Table 1.
In Table 1, it is easy to see that the third column err ( 0.5 , Δ t ) Δ t is monotone decreasing along with the time step’s waning and the fourth column err ( 0.5 , Δ t ) ( Δ t ) 2 is not monotone decreasing along with the time step’s waning. Then the order of convergence for time is of O ( Δ t ) and O ( ( Δ t ) 2 ) . 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.
Now, we consider the error for the difference h at t = 0.5 . We choose h = 0.1517 , 0.0785 , 0.0628 , 0.0524 , respectively, to solve (22). Set u N min ( x , 0.5 ) as the solution for h = π 400 , Δ t = 1 100 , and use the discrete L 2 norm to obtain the error err ( 0.5 , h ) for t = 0.5 . Then the error is shown in Table 2.
which means the order of the error estimates is O ( h ) 3 .
On the other hand, we consider the error for difference h and Δt at t = 0.5 . We choose ( h , Δ t ) = ( π 8 , 1 10 ) , ( π 10 , 1 20 ) , ( π 16 , 1 40 ) , ( π 20 , 1 60 ) , ( π 25 , 1 80 ) , respectively, to solve (22), set u N min ( x , 0.5 ) as the solution for h = π 400 , Δ t = 1 100 , use the discrete L 2 norm to obtain the error err ( 0.5 , h , Δ t ) for t = 0.5 , which is shown in Table 3.
which means that the order of error estimates is of O ( h 3 + Δ t ) .
The authors would like to express their deep thanks for the referee’s valuable suggestions about the revision and improvement of the manuscript.
Below are the links to the authors’ original submitted files for images.
The authors declare that they have no competing interests.
XZ and FL wrote the first draft, FL made the figure of numerical solution and results on errors, BL and XZ corrected and improved the final version. All authors read and approved the final draft.
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.
By using this website, you agree to our Terms and Conditions, Privacy statement and Cookies policy. Manage the cookies we use in the preference centre.
© 2019 BioMed Central Ltd unless otherwise stated. Part of Springer Nature.
