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

Authors: Lele Wang; Xin Liao
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 nonconformingEQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+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.


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## Body

## 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]:(1)ut+Δ2u+αu=Δfu,X,t∈Ω×0,T,u=Δu=0,X,t∈∂Ω×0,T,uX,0=u0X,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, u0X is a given smooth function, and fu=1/2u2−2u. 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 u in L2-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 toH2Ω, and FE solution in turn shall be C1-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 inH1-norm between FE approximation uh and the interpolation of the exact solution Ihu is much better than that of u and Ihu; 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 u and auxiliary variable p in the broken H1-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 EQ1rot element (when u∈H3Ω, the consistency error is of order Oh2 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,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 Wm,pΩ 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 Lpa,b;Y with the norm ΦLp0,T;Y=∫0TΦ⋅,tYpdt1≤p<∞ and ΦL∞0,T;Y=esssup0<t<TΦ⋅,tYp=∞.


## 2. The MFE Spaces and Variational Formulation

LetΩ be a rectangular domain with edges parallel to the coordinate axes, Th be a rectangular subdivision of Ω which need not satisfy the regular condition [10]. For all K∈Th, K=xK−hxK,xK+hxK×yK−hyK,yK+hyK, assume that the barycenter of K by xK,yK, and the four vertices and four sides are zi,li=zizi+1¯mod4i=1,2,3,4, respectively. hK=maxhxK,hyK, h=maxK∈ThhK.The nonconformingEQ1rot element space [17–21, 24, 25] is defined by(2)Vh=vh:vh|K∈span1,x,y,x2,y2,∀K∈Th,∫Fvhds=0,F⊂∂K,where vh stands for the jump of vh across the boundary F and vh=vh if F⊂∂Ω.Then, we denote the norm onVh as ⋅h=∑K∈Th⋅1,K21/2.The corresponding interpolation operator is defined asIh:v∈V=H01Ω⟶Ihv∈Vh, Ih∣K=IK, satisfying(3)∫liv−IKvds=0,∫Kv−IKvdxdy=0,i=1,2,3,4,∀K∈Th.Letp=fu−Δu; then, the mixed variational formulation for (1) is find u,p∈V×V such that(4)ut,v+∇p,∇v+αu,v=0,∀v∈V,p,q−∇u,∇q=fu,q,∀q∈V,uX,0=u0X,pX,0=p0X=fu0X−Δu0X,where u,v=∫Ωuvdxdy.


## 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.Lettntn=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 un=uX,tn,∂¯tun=un−un−1/Δt.The following lemma is introduced first which is important in the superclose analysis.Lemma 1 (see [18]).
For allvh∈Vh, we get(5)u−Ihu0+hu−Ihuh≤Ch2u2,u∈H2Ω,(6)∇u−Ihu,∇vhh=0,(7)vh0≤Cvhh,(8)∑K∫∂K∂u∂nvhds≤Ch2u3vhh,u∈H3Ω,where u,vh=∑K∫Kuvdxdy.Then, the B-E fully-discrete approximation scheme for (4) is find Uhn,Phn∈Vh×Vh such that(9)∂¯tUhn,vh+∇Phn,∇vhh+αUhn,vh=0,∀vh∈Vh,Phn,qh−∇Uhn,∇qhh=fUhn,qh,∀qh∈Vh,Uh0=Ihu0,Ph0=Ihp0.The existence and uniqueness of the solution for problem (9) can be found in [1].Next focus will be placed on the superclose ofUhn−Ihunh and Phn−Ihpnh.Theorem 1.
Letun,pn and Uhn,Phn be the solutions of (4) and (9), respectively, u,p∈L∞0,T;H3Ω,ut,pt∈L20,T;H3Ω,andutt∈L20,T;H2Ω; then, for n=0,1,2,…,N, we have(10)Uhn−Ihunh+Phn−Ihpnh≤Ch2+Δt.Proof.
Letun−Uhn=un−Ihun+Ihun−Uhn:=θn+ρn,pn−Phn=pn−Ihpn+Ihpn−Phn:=ξn+ηn.
The error equations can be derived from (1), (6), and (9):(11a)∂¯tρn,vh+∇ηn,∇vhh+αρn,vh=−∂¯tθn,vh−αθn,vh+∑K∫∂K∂pn∂nvhds−R1n,vh,(11b)ηn,qh−∇ρn,∇qhh=−ξn,qh+fun−fUhn,qh−∑K∫∂K∂un∂nqhds,where R1n=utn−∂¯tun=1/Δt∫tn−1tnτ−tn−1uttτdτ.
Firstly, takingvh=ηn in (11a) and qh=∂¯tρn in (11b) and then subtracting them, there holds(12)∇ρn,∇∂¯tρnh+ηnh2=−∂¯tθn,ηn−αρn,ηn−αθn,ηn+∑K∫∂K∂pn∂nηnds−R1n,ηn+ξn,∂¯tρn−fun−fUhn,∂¯tρn+∑K∫∂K∂un∂nρtds=∑i=18Ai.
It is easy to verify that(13)∇∂¯tρn,∇ρnh≥12Δtρnh2−ρn−1h2,A5=R1n,ηn≤CΔt∫tn−1tnutt02dτ+16ηnh2.
By virtue of Lemma1, we arrive at(14)A1+A2+A3=∂¯tθn,ηn+αρn,ηn+αθn,ηn≤Ch41Δt∫tn−1tnut22dτ+un22+Cρnh2+16ηnh2,(15)A4=∑K∫∂K∂pn∂nηnds≤Ch4pn32+16ηnh2.
By using the derivative transfer technique and (5), there holds(16)A6+A8=ξn,∂¯tρn+∑K∫∂K∂un∂n∂¯tρnds=∂¯t∑K∫∂K∂un∂nρnds+ξn,∂¯tρn−∑K∫∂K∂∂¯tun∂nρn−1ds≤Ch41Δt∫tn−1tnut32dτ+pn22+Cρn−1h2+ε∂¯tρn02+∂¯t∑K∫∂K∂un∂nρnds.
In order to estimateA7, the following assumption is given which will be proved later:(17)Uhn0,∞<M,n=0,1,2,…,N,where M=1+uL∞0,T;L∞Ω.
Then, we have(18)A7=fun−fUhn,∂¯tρn=−12un2−Uhn2,∂¯tρn+2un−Uhn,∂¯tρn=−12θn+ρnun+Uhn,∂¯tρn+2θn+ρn,∂¯tρn≤Cθn+ρn0un+Uhn0,∞∂¯tρn0+Cθn+ρn0∂¯tρn0≤Ch4un22+ε∂¯tρn02+Cρnh2.
Substituting (13)–(18) into (12), we get(19)12Δtρnh2−ρn−1h2+12ηnh2≤Ch4pn32+un22+1Δt∫tn−1tnut32dτ+Cρnh2+Cρn−1h2+ε∂¯tρn02+∂¯t∑K∫∂K∂un∂nρnds.
Multiplying by2Δt and then summing up the above inequality, by applying discrete Gronwall’s lemma, we can obtain(20)ρnh2+2Δt∑i=1nηih2≤Ch4+Δt2+4εΔt∑i=1n∂¯tρi02.
Secondly, taking the difference between two time levelsn and n−1 of (11b) reduces to(21)∂¯tηn,qh−∇∂¯tρn,∇qhh=−∂¯tξn,qh+∂¯tfun−fUhn,qh−∑K∫∂K∂∂¯tun∂nqhds.
Choosingvh=∂¯tρn in (11a) and qh=ηn in (21) and then adding them, we can get(22)∂¯tρn02+∂¯tηn,ηn+α∂¯tρn,ρn=−∂¯tθn,∂¯tρn−αθn,∂¯tρn+∑K∫∂K∂pn∂n∂¯tρnds−R1n,∂¯tρn−∂¯tξn,ηn+∂¯tfun−fUhn,ηn−∑K∫∂K∂∂¯tun∂nηnds=∑i=17Bi.
It is not difficult to verify that(23)∂¯tηn,ηn≥12Δtηn02−ηn−102,α∂¯tρn,ρn≥α2Δtρn02−ρn−102,B4=R1n,∂¯tρn≤CΔt∫tn−1tnutt02dτ+16∂¯tρn02.
By (5) and (17), there holds(24)B1+B2+B5=∂¯tθn,∂¯tρn+αθn,∂¯tρn+∂¯tξn,ηn≤Ch41Δt∫tn−1tnut22+pt22dτ+un22+16∂¯tρn02+Cηn02,(25)B6=∂¯tfun−fUhn,ηn=∂¯tunθn+θn−12+ρn+ρn−12+Uhn+Uhn−12∂¯tθn+∂¯tρn,ηn−2∂¯tθn+∂¯tρn,ηn≤Ch4un22+un−122+1Δt∫tn−1tnut22dτ+16∂¯tρn02+Cρn02+Cρn−1h2+Cηn02.
From derivative transfer technique and (8), it can be proved that(26)B3=∑K∫∂K∂pn∂n∂¯tρnds=∂¯t∑K∫∂K∂pn∂nρnds−∑K∫∂K∂∂¯tpn∂nρn−1ds≤Ch4Δt∫tn−1tnpt32dτ+Cρn−1h2+∂¯t∑K∫∂K∂pn∂nρnds,(27)B7=∑K∫∂K∂∂¯tun∂nηnds≤Ch4Δt∫tn−1tnut32dτ+ηnh2.
Then, substituting (23)–(27) into (22) reduces to(28)12∂¯tρn02+α2Δtρn02−ρn−102+12Δtηn02−ηn−102≤Ch41Δt∫tn−1tnut32+pt32dτ+un22+un−122+CΔt∫tn−1tnutt02dτ+∂¯t∑K∫∂K∂pn∂nρnds+Cηn02+Cρn02+ηnh2+Cρn−1h2.
Multiplying by2Δt 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 ρ0=0,η0=0, we can obtain(29)Δt∑i=1n∂¯tρi02+ρnh2+Δt∑i=1nηih2≤Ch4+Δt2.
At last, choosingvh=∂¯tηn in (11a) and qh=∂¯tρn in (21) and then substituting them, it yields(30)∇∂¯tηn,∇ηnh+∂¯tρnh2=−∂¯tθn,∂¯tηn−αθn,∂¯tηn−αρn,∂¯tηn+∑K∫∂K∂pn∂n∂¯tηnds−R1n,∂¯tηn+∂¯tξn,∂¯tρn−∂¯tfun−fUhn,∂¯tρn+∑K∫∂K∂∂¯tun∂n∂¯tρnds=∑i=18Di.
Similar to the estimates of (20) and (29), we have(31)∇∂¯tηn,∇ηnh≥12Δtηnh2−ηn−1h2,(32)D1+D2=∂¯tθn,∂¯tηn+αθn,∂¯tηn=∂¯t∂¯tθn,ηn+α∂¯tθn,ηn−∂¯t∂¯tθn,ηn−1−α∂¯tθn,ηn−1≤∂¯t∂¯tθn,ηn+αθn,ηn+Ch4Δt∫tn−2tnutt22dτ+∫tn−1tnut22dτ+Cηn−102,(33)D3+D4+D5=−αρn,∂¯tηn+∑K∫∂K∂pn∂n∂¯tηnds−R1n,∂¯tηn≤∂¯t−αρn,ηn+∑K∫∂K∂pn∂nηnds−R1n,ηn+C∂¯tρn02+C∂¯tR1n02+Ch4Δt∫tn−1tnpt32dτ+Cηn−1h2,(34)D6+D7=∂¯tξn,∂¯tρn−∂¯tfun−fUhn,∂¯tρn≤Ch41Δt∫tn−1tnpt22+ut22dτ+un22+un−122+C∂¯tρn02+Cρn02+Cρn−102,(35)D8=∑K∫∂K∂∂¯tun∂n∂¯tρnds≤Ch4Δt∫tn−1tnut32dτ+12∂¯tρnh2.
Substituting (31)–(35) into (30), we have(36)12Δtηnh2−ηn−1h2≤Ch41Δt∫tn−1tnpt32+ut32+utt22dτ+un22+un−122+∂¯t∂¯tθn,ηn+αθn,ηn−αρn,ηn+∑K∫∂K∂pn∂nηnds−R1n,ηn+C∂¯tρn02+Cρn02+Cρn−102.
Multiplying by2Δt and summing up the above inequality, by discrete Gronwall’s lemma and (29), there holds(37)ηnh2+Δt∑i=1n∂¯tρih2≤Ch4+Δt2.
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].
Letμn:=un−Uhn. Initially, when n=0, we have μ00,∞=u0−Ihu00,∞≤Ch<1, and the assumption is true.
Furthermore, we assume that whenn=k−1, there holds μk−10,∞<1. Then, by Theorem 1, we have Uhk−1−Ihuk−1h≤Ch2+Δt.
Additionally, we consider the situation atn=k. We know that μt0,∞ is continuous function about time t, so there exists δ>0, for ∀ϵ>0; when tk−1−tk=Δt<δ, there holds(38)μk−10,∞−μk0,∞<ϵ.
Takingϵ=Δt in (38), we have(39)μk0,∞≤μk−10,∞+Δt≤Uhk−1−Ihuk−10,∞+uk−1−Ihuk−10,∞+Δt≤Ch−1Uhk−1−Ihuk−10+Chuk−11,∞+Δt≤Ch−1Δt+h+Ch+Δt<1.
In the last step of (39), we need the time step Δt and space step h satisfy the condition Δt=Oh1+α0<α≤1. Above all, choose appropriate h1 to make that Ch1α<1, which ends the proof.


## 4. Superconvergence Analysis for the Fully-Discrete Approximation Scheme

To obtain the superconvergence results, we combine the adjacent four elementsK1, K2, K3, and K4 into a big element K˜, i.e., K˜=∪i=14Ki (see Figure 1). The corresponding subdivision is defined by T2h.Figure 1
The big elementK˜.Then, construct the interpolation postprocessing operatorI2h on K˜ as in [27–29] which satisfies(40)I2hu∣K˜∈P2K˜,∫LiI2hu−uds=0,i=1,2,3,4,∀K˜∈T2h,∫K1∪K3I2hu−udxdy=0,∫K2∪K4I2hu−udxdy=0,where P2K˜ denotes the space of polynomial on K˜ with degree less than or equal to 2 and Lii=1,2,3,4 are the four sides of K˜.From [27], the interpolation postprocessing operator I2h satisfies(41)I2hIhu=I2hu,I2hu−uh≤Ch2u3,I2hvhh≤Cvhh,∀vh∈Vh.Theorem 2.
Under the assumption of Theorem1, there holds(42)un−I2hUhnh+pn−I2hPhnh≤Ch2+Δt.Proof.
From triangle inequality, (10), and (41), there holds(43)un−I2hUhnh=un−I2hIhun+I2hIhun−I2hUhnh≤un−I2hunh+CIhun−Uhnh≤Ch2+Δt.
The superconvergence result ofp can be obtained similarly, which completes the proof.Remark 1.
(i)
From the analysis of Theorems1 and 2, the superclose and superconvergence results for the two variables u and p in the broken H1-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 theQ1rot element [30], rectangular constrained Q1rot 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 Oh; for nonconforming P1 element [39] and p1mod element [40], though the consistency errors reach to Oh3 order, the interpolation error term can be estimated as ∇u−Ihu,∇vhh≤Chu2vhh. 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]:(44)ut+Δ2u+αu=Δfu+gX,X,t∈Ω×0,T,uX,0=u0X,X∈Ω,where Ω=0,1×0,1,α=0.5,T=1.Letp=fu−Δu; then, the mixed variational formulation for (44) is find u,p∈V×V such that(45)ut,v+∇p,∇v+αu,v=gX,v,∀v∈V,p,q−∇u,∇q=fu,q,∀q∈V,uX,0=u0X,pX,0=p0X=fu0X−Δu0X.The exact solutions areux,y,t=1+t6sinπxsinπy and px,y,t=1/21+t62sinπx2sinπy2+2π2−11+t6sinπxsinπy.The fully-discrete approximation scheme for (45) is find Uhn,Phn∈Vh×Vh such that(46)∂¯tUhn,vh+∇Phn,∇vhh+αUhn,vh=gX,vh,∀vh∈Vh,Phn,qh−∇Uhn,∇qhh=fUhn,qh,∀qh∈Vh,Uh0=Ihu0,Ph0=Ihp0.We divide the domainΩ into m×n rectangular meshes (see Figure 2). Then, the FE solutions Uhn,Phn can be calculated according to (46); here, we choose Δt=1/m2+1/n2.Figure 2
The rectangular mesh.For simplicity and concreteness, we just plot the exact solutionsu,p and the numerical solutions Uh,Ph on 16 × 32 meshes at t=0.5 (see Figures 3 and 4), respectively.Figure 3
Exact solutionu (a) and numerical solution Uh (b) on 16 × 32 meshes at t=0.5.
(a)
(b)Figure 4
Exact solutionp (a) and numerical solution Ph (b) on 16 × 32 meshes at t=0.5.
(a)
(b)Then, the convergence, superclose, and superconvergence results ofu and p in the broken H1-norm at time t=0.1 and 0.5 are listed in Tables 1–4, respectively.Table 1
Numerical results ofu at t=0.1.
m
×
n
u
n
−
U
h
n
h
Order
U
h
n
−
I
h
u
n
h
Order
u
n
−
I
2
h
U
h
n
h
Order
4
×
8
4.2013e − 01
—
1.4228e − 01
—
2.6226e − 01
—
8
×
16
2.0201e − 01
1.0564
3.6301e − 02
1.9707
6.7246e − 02
1.9634
16
×
32
9.9915e − 02
1.0156
9.1169e − 03
1.9933
1.6911e − 02
1.9914
32
×
64
4.9818e − 02
1.0040
2.2818e − 03
1.9983
4.2340e − 03
1.9978Table 2
Numerical results ofu at t=0.5.
m
×
n
u
n
−
U
h
n
h
Order
U
h
n
−
I
h
u
n
h
Order
u
n
−
I
2
h
U
h
n
h
Order
4
×
8
4.2408e − 01
—
1.3655e − 01
—
2.6207e − 01
—
8
×
16
2.0484e − 01
1.0497
3.5024e − 02
1.9630
6.7315e − 02
1.9609
16
×
32
1.0147e − 01
1.0134
9.2067e − 03
1.9275
1.7147e − 02
1.9729
32
×
64
5.0596e − 02
1.0039
2.3042e − 03
1.9983
4.2931e − 03
1.9978Table 3
Numerical results ofp at t=0.1.
m
×
n
p
n
−
P
h
n
h
Order
P
h
n
−
I
h
p
n
h
Order
p
n
−
I
2
h
P
h
n
h
Order
4
×
8
7.2813
—
1.2981
—
4.1865
—
8
×
16
3.6128
1.0110
3.2303e − 01
2.0066
1.0770
1.9586
16
×
32
1.8031
1.0026
8.0602e − 02
2.0027
2.7098e − 01
1.9908
32
×
64
9.0114e − 01
1.0006
2.0140e − 02
2.0007
6.7852e − 02
1.9977Table 4
Numerical results ofp at t=0.5.
m
×
n
p
n
−
P
h
n
h
Order
P
h
n
−
I
h
p
n
h
Order
p
n
−
I
2
h
P
h
n
h
Order
4
×
8
7.3758
—
1.1901
—
4.2129
—
8
×
16
3.66791
1.0078
2.9887e − 01
1.9935
1.0854
1.9565
16
×
32
1.8318
1.0016
8.2041e − 02
1.8651
2.7536e − 01
1.9788
32
×
64
9.1551e − 01
1.0006
2.0500e − 02
2.0006
6.8948e − 02
1.9977From Tables1 and 2, we can see that un−Uhnh are convergent at order Oh and Uhn−Ihunh and un−I2hUhnh are convergent at order Oh2, which coincide with the theoretical analysis. Meanwhile, the results of un−I2hUhnh are better than un−Uhnh, which indicate the superiority of the superconvergence algorithm. The results of p in Tables 3 and 4 are consistent with those of u 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 brokenH1-norm are obtained, which are one order higher than those of convergence. Furthermore, a numerical example demonstrates the efficiency of the theoretical analysis.



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*Source: 8416898-2020-04-30.xml*