Abstract
The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming element (see Lemma 1), the superclose estimates of order in the broken -norm for the original variable and intermediate variable are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.
1. Introduction
Under certain conditions, the dilute binary alloy will solidify, at which point the solid-liquid interface is unstable and has a cellular structure. When the solute rejection coefficient is close to unity, near the stability threshold, the characteristic cell size may significantly be beyond the diffusion width of the solidification zone. The Sivashinsky equation describes the dynamic of the onset and stabilization of the cellular structure, which is considered as the following fourth-order nonlinear equation [1, 2]:where is the interior of the rectangle , , are fixed constants, is a given smooth function, and . Due to the nonlinearity of this equation, it is very difficult to find out the true solution. Thus, a lot of numerical simulation methods have been considered for (1), such as the finite difference method, finite element method (FEM for short), and region decomposition method. For one-dimensional case, Benammou and Omrani [3] studied the FEM and obtained the convergence analysis of the original variable in -norm; Momani [4] presented a numerical scheme based on the region decomposition method; and Omrani and Reza and Kenan [5, 6] provided two kinds of finite difference schemes and proved the uniqueness and convergence, respectively. For two-dimensional case, Denet [7] gave the stability of the solution under the rectangular region; Rouis and Omrani [8] proposed a linearized three-level difference scheme; and Ilati and Dehghan [9] derived an error analysis by a meshless method based on radial point interpolation technique.
As it is known to all that in regard to the fourth-order problem, the conforming Galerkin finite element (FE for short) approximation space belongs to , and FE solution in turn shall be -continuous. This leads to the higher degree of piecewise polynomials, and the related computation is complicated and difficult (both triangular Bell element and rectangular Bogner–Fox–Schmit element [10] are typical examples). The MFEM is an optimal choice to overcome the above deficiencies, which transforms a fourth-order problem into 2 coupled second-order problems by introducing an intermediate variable; thus, the low-order elements can be used to solve. The nonconforming MFEM brings down the smoothness requirement on FE solution compared to the conforming case. Readers with more interests may refer [11–15] and the references listed. For problem (1), Omrani [1] developed the convergence analysis of the corresponding variables in the semidiscrete and fully-discrete schemes by using conforming MFEM; however, situation involving nonconforming MFEM was not available till now.
It is also well known that the superconvergence analysis is an important approach to improve the precision of FE solution. More precisely, based on the so-called integral identity technique, the order of error in -norm between FE approximation and the interpolation of the exact solution is much better than that of and ; this fascinating characteristic is called superclose. The global superconvergence will then be investigated by adding a simple postprocessing without changing the existing FE program. Meanwhile, superconvergence is critical in practical engineering numerical calculation and has always been a research hotspot. To find out more applications, readers may refer [12, 15–23]. As far as our knowledge is concerned, research on superconvergence for Sivashinsky equation is yet to be found.
The main purpose of this article is to develop a nonconforming MFE scheme for problem (1), and the superclose and superconvergence results of the original variable and auxiliary variable in the broken -norm are obtained for the B-E fully-discrete scheme. The outline is organized as follows: in Section 2, the MFE spaces and variational formulation are introduced. In Section 3, based on the special property of the nonconforming element (when , the consistency error is of order which is one order higher than the interpolation error), the superclose results for the above two variables are deduced. In Section 4, the global superconvergence properties are derived with the help of interpolation postprocessing technique. In Section 5, a numerical example is given to verify the theoretical analysis. In the last section, a brief conclusion is drawn.
Throughout this article, denotes a positive constant that may take different values at different places but remains independent of the subdivision parameter and time step . Meanwhile, we use the notations as in [10] for the Sobolev spaces with norm and seminorm , where and are nonnegative integer numbers. Especially, for , will be omitted in the above norms and seminorms. Furthermore, we define the space with the norm and .
2. The MFE Spaces and Variational Formulation
Let be a rectangular domain with edges parallel to the coordinate axes, be a rectangular subdivision of which need not satisfy the regular condition [10]. For all , , assume that the barycenter of by , and the four vertices and four sides are , respectively. , .
The nonconforming element space [17–21, 24, 25] is defined bywhere stands for the jump of across the boundary and if .
Then, we denote the norm on as .
The corresponding interpolation operator is defined as , , satisfying
Let ; then, the mixed variational formulation for (1) is find such thatwhere .
3. Superclose Analysis for the Fully-Discrete Approximation Scheme
In this section, the superclose analysis for the B-E fully-discrete scheme will be studied.
Let be a uniform partition of with the time step . For a given continuous function on , we define that .
The following lemma is introduced first which is important in the superclose analysis.
Lemma 1 (see [18]). For all , we getwhere .
Then, the B-E fully-discrete approximation scheme for (4) is find such that
The existence and uniqueness of the solution for problem (9) can be found in [1].
Next focus will be placed on the superclose of and .
Theorem 1. Let and be the solutions of (4) and (9), respectively, ,; then, for , we have
Proof. Let .
The error equations can be derived from (1), (6), and (9):where .
Firstly, taking in (11a) and in (11b) and then subtracting them, there holdsIt is easy to verify thatBy virtue of Lemma 1, we arrive atBy using the derivative transfer technique and (5), there holdsIn order to estimate , the following assumption is given which will be proved later:where .
Then, we haveSubstituting (13)–(18) into (12), we getMultiplying by and then summing up the above inequality, by applying discrete Gronwall’s lemma, we can obtainSecondly, taking the difference between two time levels and of (11b) reduces toChoosing in (11a) and in (21) and then adding them, we can getIt is not difficult to verify thatBy (5) and (17), there holdsFrom derivative transfer technique and (8), it can be proved thatThen, substituting (23)–(27) into (22) reduces toMultiplying by and summing up the above inequality and then plugging (28) into (20), by discrete Gronwall’s lemma again, choosing appropriate and such that , by applying (8) and noting that , we can obtainAt last, choosing in (11a) and in (21) and then substituting them, it yieldsSimilar to the estimates of (20) and (29), we haveSubstituting (31)–(35) into (30), we haveMultiplying by and summing up the above inequality, by discrete Gronwall’s lemma and (29), there holdsWith (29) and (37), the proof is completed.
Finally, we use mathematical induction to verify assumption (15) which is similar to the technique used in [22, 23, 26].
Let . Initially, when , we have , and the assumption is true.
Furthermore, we assume that when , there holds . Then, by Theorem 1, we have .
Additionally, we consider the situation at . We know that is continuous function about time , so there exists , for ; when , there holdsTaking in (38), we haveIn the last step of (39), we need the time step and space step satisfy the condition . Above all, choose appropriate to make that , which ends the proof.
4. Superconvergence Analysis for the Fully-Discrete Approximation Scheme
To obtain the superconvergence results, we combine the adjacent four elements , , , and into a big element , i.e., (see Figure 1). The corresponding subdivision is defined by .
Then, construct the interpolation postprocessing operator on as in [27–29] which satisfieswhere denotes the space of polynomial on with degree less than or equal to 2 and are the four sides of .
From [27], the interpolation postprocessing operator satisfies
Theorem 2. Under the assumption of Theorem 1, there holds
Proof. From triangle inequality, (10), and (41), there holdsThe superconvergence result of can be obtained similarly, which completes the proof.
Remark 1. (i)From the analysis of Theorems 1 and 2, the superclose and superconvergence results for the two variables and in the broken -norm are derived, which are one order higher than the convergence results in [1].(ii)The conclusions of this paper are applicable to other nonconforming elements such as the element [30], rectangular constrained element [31], quasi-Wilson element [32, 33], modified quasi-Wilson element [34], and quasi-Carey element [35].(iii)For nonconforming linear triangle Crouzeix–Raviart element [36], Carey element [37], and Wilson element [38], the consistency errors are only of order ; for nonconforming element [39] and element [40], though the consistency errors reach to order, the interpolation error term can be estimated as . Therefore, the superconvergence results are unable to get.
5. Numerical Example
Numerical simulation results are presented in this section. The Newton iterative algorithm is used to solve the nonlinear system.
Consider the following problem [1]:where .
Let ; then, the mixed variational formulation for (44) is find such that
The exact solutions are and .
The fully-discrete approximation scheme for (45) is find such that
We divide the domain into rectangular meshes (see Figure 2). Then, the FE solutions can be calculated according to (46); here, we choose .
For simplicity and concreteness, we just plot the exact solutions and the numerical solutions on 16 32 meshes at (see Figures 3 and 4), respectively.
(a)
(b)
(a)
(b)
Then, the convergence, superclose, and superconvergence results of and in the broken -norm at time and 0.5 are listed in Tables 1–4, respectively.
From Tables 1 and 2, we can see that are convergent at order and and are convergent at order , which coincide with the theoretical analysis. Meanwhile, the results of are better than , which indicate the superiority of the superconvergence algorithm. The results of in Tables 3 and 4 are consistent with those of in Tables 1 and 2.
6. Conclusions
In this work, we study the nonconforming MFEM for fourth-order nonlinear Sivashinsky equation. The superconvergence results of the relevant variables in the broken -norm are obtained, which are one order higher than those of convergence. Furthermore, a numerical example demonstrates the efficiency of the theoretical analysis.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant no. 11671369) and support program for Key Scientific Research Project of Universities in Henan Province (18A110033).