High Accuracy Analysis of Nonconforming Mixed Finite Element Method for the Nonlinear Sivashinsky Equation

*e 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 EQrot 1 element (see Lemma 1), the superclose estimates of order O(h2 + Δt) in the brokenH1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (BE for short) fully-discrete scheme. Secondly, the global superconvergence results of order O(h2 + Δt) 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.


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. e 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]: u(X, 0) � u 0 (X), X ∈ Ω, where Ω is the interior of the rectangle [0, a] × [0, b], X � (x, y), T > 0, a > 0, b > 0, α > 0 are fixed constants, u 0 (X) is a given smooth function, and f(u) � (1/2)u 2 − 2u. Due to the nonlinearity of this equation, it is very difficult to find out the true solution. us, 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 u in L 2 -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 H 2 (Ω), and FE solution in turn shall be C 1 -continuous.
is 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). e 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. e nonconforming MFEM brings down the smoothness requirement on FE solution compared to the conforming case. Readers with more interests may refer [11][12][13][14][15] and the references listed. For problem (1), Omrani [1] developed the convergence analysis of the corresponding variables in the semidiscrete and fullydiscrete 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 H 1 -norm between FE approximation u h and the interpolation of the exact solution I h u is much better than that of u and I h u; this fascinating characteristic is called superclose.
e 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][16][17][18][19][20][21][22][23]. As far as our knowledge is concerned, research on superconvergence for Sivashinsky equation is yet to be found. e 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 u and auxiliary variable p in the broken H 1 -norm are obtained for the B-E fully-discrete scheme. e 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 EQ rot 1 element (when u ∈ H 3 (Ω), the consistency error is of order O(h 2 ) 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. roughout this article, C denotes a positive constant that may take different values at different places but remains independent of the subdivision parameter h and time step Δt. Meanwhile, we use the notations as in [10] for the Sobolev spaces W m,p (Ω) with norm ‖·‖ m,p and seminorm |·| m,p , where m and p are nonnegative integer numbers. Especially, for p � 2, p will be omitted in the above norms and seminorms. Furthermore, we define the space

The MFE Spaces and Variational Formulation
Let Ω be a rectangular domain with edges parallel to the coordinate axes, T h be a rectangular subdivision of Ω which need not satisfy the regular condition [10].
assume that the barycenter of K by (x K , y K ), and the four vertices and four sides are e nonconforming EQ rot 1 element space [17-21, 24, 25] is defined by where [v h ] stands for the jump of v h across the boundary F and en, we denote the norm on V h as ‖·‖ h � ( K∈T h |·| 2 1,K ) 1/2 . e corresponding interpolation operator is defined as Let p � f(u) − Δu; then, the mixed variational formulation for (1) is find (u, p) ∈ V × V such that

Superclose Analysis for the Fully-Discrete Approximation Scheme
In this section, the superclose analysis for the B-E fullydiscrete scheme will be studied.
Let t n | t n � nΔt; n � 0, 1, 2, . . . , N be a uniform partition of [0, T] with the time step Δt � (T/N). For a given continuous function u on [0, T], we define that u n � u(X, t n ), z t u n � u n − u n− 1 /Δt. e following lemma is introduced first which is important in the superclose analysis.
Lemma 1 (see [18]). For all v h ∈ V h , we get K zK en, the B-E fully-discrete approximation scheme for e existence and uniqueness of the solution for problem (9) can be found in [1].
Next focus will be placed on the superclose of ‖U n h − I h u n ‖ h and ‖P n h − I h p n ‖ h . Theorem 1. Let (u n , p n ) and (U n h , P n h ) be the solutions of (4) and (9), respectively, u, p ∈ L ∞ (0,T; H 3 (Ω)),u t , p t ∈ L 2 (0, T; H 3 (Ω)), and u tt ∈ L 2 (0,T; H 2 (Ω)); then, for n � 0, 1, 2,... , N, we have Proof. Let : � ξ n + η n . e error equations can be derived from (1), (6), and (9): Firstly, taking v h � η n in (11a) and q h � z t ρ n in (11b) and then subtracting them, there holds It is easy to verify that By virtue of Lemma 1, we arrive at Mathematical Problems in Engineering By using the derivative transfer technique and (5), there holds In order to estimate A 7 , the following assumption is given which will be proved later: where M � 1 + ‖u‖ L ∞ (0,T;L ∞ (Ω)) . en, we have Substituting (13)- (18) into (12), we get Multiplying by 2Δt and then summing up the above inequality, by applying discrete Gronwall's lemma, we can obtain Secondly, taking the difference between two time levels n and n − 1 of (11b) reduces to Choosing v h � z t ρ n in (11a) and q h � η n in (21) and then adding them, we can get It is not difficult to verify that Mathematical Problems in Engineering By (5) and (17), there holds From derivative transfer technique and (8), it can be proved that en, substituting (23)-(27) into (22) reduces to Multiplying by 2Δt and summing up the above inequality and then plugging (28) into (20), by discrete Gronwall's lemma again, choosing appropriate Δt and ε > 0 such that 1 − CεΔt > 0, by applying (8) and noting that ρ 0 � 0, η 0 � 0, we can obtain At last, choosing v h � z t η n in (11a) and q h � z t ρ n in (21) and then substituting them, it yields Similar to the estimates of (20) and (29), we have Mathematical Problems in Engineering Multiplying by 2Δt and summing up the above inequality, by discrete Gronwall's lemma and (29), there holds With (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].
en, by eorem 1, we have Additionally, we consider the situation at n � k. We know that ‖μ(t)‖ 0,∞ is continuous function about time t, so there exists δ > 0, for ∀ϵ > 0; when |t k− 1 − t k | � Δt < δ, there holds Taking ϵ � Δt in (38), we have 6 Mathematical Problems in Engineering In the last step of (39), we need the time step Δt and space step h satisfy the condition Δt � O(h 1+α )(0 < α ≤ 1). Above all, choose appropriate h 1 to make that Ch α 1 < 1, which ends the proof.

Superconvergence Analysis for the Fully-Discrete Approximation Scheme
To obtain the superconvergence results, we combine the adjacent four elements K 1 , K 2 , K 3 , and K 4 into a big element K, i.e., K � ∪ 4 i�1 K i (see Figure 1). e corresponding subdivision is defined by T 2h . en, construct the interpolation postprocessing operator I 2h on K as in [27][28][29] which satisfies where P 2 (K) denotes the space of polynomial on K with degree less than or equal to 2 and L i (i � 1, 2, 3, 4) are the four sides of K. From [27], the interpolation postprocessing operator I 2h satisfies

Theorem 2. Under the assumption of eorem 1, there holds
Proof. From triangle inequality, (10), and (41), there holds e superconvergence result of p can be obtained similarly, which completes the proof. (i) From the analysis of eorems 1 and 2, the superclose and superconvergence results for the two variables u and p in the broken H 1 -norm are derived, which are one order higher than the convergence results in [1].
(ii) e conclusions of this paper are applicable to other nonconforming elements such as the Q rot 1 element [30], rectangular constrained Q rot 1 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 O(h); for nonconforming P 1 element [39] and p mod 1 element [40], though the consistency errors reach to O(h 3 ) order, the interpolation error term can be estimated as (∇(u − I h u), erefore, the superconvergence results are unable to get.

Numerical Example
Numerical simulation results are presented in this section.
e Newton iterative algorithm is used to solve the nonlinear system.
Consider the following problem [1]: Let p � f(u) − Δu; then, the mixed variational formulation for (44) is find (u, p) ∈ V × V such that Figure 1: e big element K.

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(1/m 2 ) + (1/n 2 ). For simplicity and concreteness, we just plot the exact solutions u, p and the numerical solutions U h , P h on 16 × 32 meshes at t � 0.5 (see Figures 3 and 4), respectively. en, the convergence, superclose, and superconvergence results of u and p in the broken H 1 -norm at time t � 0.1 and 0.5 are listed in Tables 1-4     better than ‖u n − U n h ‖ h , which indicate the superiority of the superconvergence algorithm. e results of p in Tables 3 and 4 are consistent with those of u in Tables 1 and 2.

Conclusions
In this work, we study the nonconforming MFEM for fourth-order nonlinear Sivashinsky equation. e superconvergence results of the relevant variables in the broken H 1 -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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest regarding the publication of this paper.