# Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

Authors: Jie Zhao; Hong Li; Zhichao Fang; Xue Bai
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimala priori error estimates for the unknown and auxiliary variables in the L2Ω norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.


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## Body

## 1. Introduction

In this article, we consider the following one-dimensional Burgers’ equation:(1)∂u∂t+αu∂u∂x−ν∂2u∂x2=0,x,t∈Ω×Γ,with initial and boundary conditions(2)ua,t=ub,t=0,t∈Γ¯,ux,0=u0x,x∈Ω¯,where Ω=a,b, Γ=0,T with 0<T<∞, α is a positive constant, ν>0 is the viscosity coefficient, and u0x is the given initial function.Burgers’ equation is a famous nonlinear evolution equation which was first derived by Bateman [1] in 1915. Burgers [2] utilized this equation to model turbulence behavior. Since its appearance, this equation has been widely concerned by researchers because of its various practical applications, such as gas dynamics, shock theory, traffic flows, viscous flow, and turbulence. The exact solutions can be expressed as a Fourier series expansion by introducing a Hopf–Cole transformation [3, 4]. Benton and Platzman [5] gave the exact solutions of Burgers’ equation in one-dimensional spatial regions for some different initial functions. In the past several decades, many numerical techniques had been constructed and tested to solve Burgers’ equation [6–13], such as finite element methods, finite difference methods, least-squares finite element methods, and spectral methods.In recent years, the mixed finite element (MFE) methods and finite volume (FV) methods have been used to solve Burgers’ equation by many researchers. Luo and Liu [14] proposed an MFE method to solve one-dimensional Burgers’ equation by introducing a flux function as an auxiliary variable and gave the existence, uniqueness, and error analyses for the discrete solutions. Chen and Jiang [15] constructed a characteristic MFE scheme to solve one-dimensional Burgers’ equation and obtained the optimal error estimates for the velocity and flux (gradient) in the L2 norm. Pany et al. [16] applied an H1-Galerkin MFE method to approximate the velocity and flux of one-dimensional Burgers’ equation and gave a priori error estimates and numerical experiments. Shi et al. [17] provided a low-order least-squares nonconforming characteristic MFE scheme to solve two-dimensional Burgers’ equation and obtained the optimal order error estimates in the L2 norm. Hu et al. [18] constructed a Crank–Nicolson time discretization MFE scheme to treat two-dimensional Burgers’ equation by using the P02−P1 pair and gave the optimal error analysis and numerical experiments. Nascimento et al. [19] applied a Fourier pseudospectral method and an FV method to solve one-dimensional Burgers’ equation and gave the numerical comparison. Guo et al. [20] proposed a fifth-order FV weighted compact scheme to solve one-dimensional Burgers’ equation and gave numerical experiments. Sheng and Zhang [21] proposed an FV method to solve two-dimensional Burgers’ equation and obtained the optimal error estimate in the H1 norm.The aim of this article is to develop mixed finite volume element (MFVE) methods to solve one-dimensional Burger’s equation by combining the MFE methods [22–25] with the finite volume element (FVE) methods [26–30]. The MFVE methods, also called mixed covolume methods, were first proposed by Russell [31] to solve the elliptic equation. Now, the methods have been applied to solve second-order elliptic equations [32–34], integrodifferential equations [35], parabolic equations [36, 37], time-fractional partial differential equations [38], and so on. In this article, we introduce a flux function as an auxiliary variable, rewrite (1) as the first-order system, and construct the semidiscrete and nonlinear backward Euler fully discrete MFVE schemes by using a transfer operator. We give the theoretical analysis for semidiscrete and fully discrete schemes in detail, including existence, uniqueness, and stability. In particular, in the analysis of the fully discrete scheme, we apply the Brouwer fixed-point theorem to prove the existence and use the Sobolev embedding theorem and the inverse inequality to prove the stability. Making use of the stability results of discrete solutions, we obtain the optimal a priori error estimates for the velocity and flux in the L2 norm. Furthermore, we give some numerical results to verify the feasibility and effectiveness of the MFVE scheme.The rest of this article is organized as follows: In Section2, we use a flux function as an auxiliary variable and give the mixed variational formulation and the semidiscrete MFVE scheme. The existence, uniqueness, stability, and convergence analyses for semidiscrete and fully discrete schemes are given in Section 3 and 4, respectively. In Section 5, a numerical example is given to verify the theoretical results. In this article, the standard definitions and notations of the Sobolev spaces as in [39] are used. Furthermore, we use the symbol C to represent a generic constant which is independent of the space and time mesh parameters h and Δt.


## 2. Semidiscrete MFVE Scheme

We introduce a flux functionpx,t=α/2fux,t−νuxx,t as an auxiliary variable, where fu=u2. Then, we can rewrite equations (1) and (2) as the following first-order system:(3)ut+px=0,x,t∈Ω×Γ,p−α2fu+νux=0,x,t∈Ω×Γ,ua,t=ub,t=0,t∈Γ¯,ux,0=u0x,x∈Ω¯.The mixed variational formulation of (3) is to find ut,pt∈H01Ω×L2Ω such that(4)ut,v−vx,p=0,∀v∈H01Ω,p,q−α2fu,q+νux,q=0,∀q∈L2Ω,ux,0=u0x,∀x∈Ω¯.Now, we construct the primal mesh for the intervalΩ¯=a,b with the nodes a=x0<x1<x2<⋯<xN=b, where N is some positive integer. Then, the primal mesh is denoted by ℑh=Ai=xi,xi+1:i=0,1,⋯,N−1, and the diameter of the primal mesh is defined by h=max0≤i≤N−1hi, where hi=xi+1−xi. We assume that the mesh satisfies the quasiuniform condition hi≥c0h0≤i≤N−1 for some constant c0>0.Next, the corresponding dual mesh is constructed by the nodesa=x0<x1/2<x3/2<⋯<xN−1/2<xN=b, where xi+1/2=xi+xi+1/2i=0,1,⋯,N−1. Denote A0∗=x0,x1/2, Ai∗=xi−1/2,xi+1/2i=1,2,⋯,N−1, and AN∗=xN−1/2,xN, then the dual mesh is defined by ℑh∗=Ai∗:i=0,1,⋯,N.We choose the mixed finite element spaceH0h×Lh as the trial function space, where(5)H0h=wh∈H01Ω:whA∈P1A,∀A∈ℑh,Lh=vh∈L2Ω:vhA∈P0A,∀A∈ℑh.Letϕi:i=1,2,⋯,N−1 and χAi:i=0,1,⋯,N−1 be the basis of the spaces H0h and Lh, respectively, where ϕi is the piecewise linear polynomial defined in [40] and χAi is the characteristic function of the set Ai.The system (3) is integrated as follows:(6)∫Ai∗utdx+∫Ai∗pxdx,t∈Γ,1≤i≤N−1,∫Aip+νuxdx=α2∫Aifudx,t∈Γ,0≤i≤N−1.Now, we define the transfer operatorγh:H0h⟶L2Ω (see [40]) as follows:(7)γhwh=∑i=1N−1whxiχAi∗,∀wh∈H0h.Thus, we haveγhϕi=χAi∗1≤i≤N−1. The range Yh of γh is used as the test function space. Then, we can rewrite (6) as(8)ut+px,γhvh=0,∀vh∈H0h,p+νux,qh=α2fu,qh,∀qh∈Lh.Note that(9)px,γhvh=∑i=1N−1∫Ai∗pxvhxidx=∑i=1N−1vhxipxi+1/2−pxi−1/2≐bγhvh,p.By a simple calculation, it is easy to have that bγhvh,qh=−vhx,qh, ∀vh∈H0h, ∀qh∈Lh. Then, we get the semidiscrete MFVE scheme to find uh,ph:0,T⟶H0h×Lh such that(10)uht,γhvh−vhx,ph=0,∀vh∈H0h,ph,qh+νuhx,qh=α2fuh,qh,∀qh∈Lh,where uh0,ph0 satisfies(11)uhx0,zh=u0x,zh,∀zh∈Lh,ph0,qh+νuhx0,qh=α2fuh0,qh,∀qh∈Lh.


## 3. Theoretical Analysis for the Semidiscrete MFVE Scheme

### 3.1. Some Lemmas

For theoretical analysis, we first give some properties of the transfer operatorγh and two projection operators.Lemma 1 (see [40]).
The transfer operatorγh is bounded, that is,(12)γhwh≤3wh,∀wh∈H0h.Lemma 2 (see [40]).
The transfer operatorγh satisfies the following symmetry relation:(13)wh,γhvh=γhwh,vh,∀vh,wh∈H0h.Lemma 3 (see [40]).
The transfer operatorγh satisfies the following positivity:(14)γhwh,wh≥34wh2,∀wh∈H0h.Lemma 4 (see [40]).
LetI be an identity operator, then the transfer operator γh satisfies the following properties:(15)I−γhwh≤1212hwh1,∀wh∈H0h,vh,I−γhwh≤1212hvh1wh,∀vh,wh∈H0h.Lemma 5 (see [40]).
There exists a constantC>0 such that(16)v,I−γhwh≤Chv1wh,∀v∈H01Ω,∀wh∈H0h.Now, the elliptic projection operatorΠh:H01Ω⟶H0h is introduced below, which satisfies(17)w−Πhwx,zh=0,∀zh∈Lh,forw∈H01Ω.At the same time, theL2 orthogonal projection operator Rh:L2Ω⟶Lh is introduced to satisfy(18)φ−Rhφ,qh=0,∀qh∈Lh,forφ∈L2Ω.Referring to References [24, 25], we can know that the projection operators Πh and Rh satisfy the following estimate properties.Lemma 6.
There exists a constantC>0 such that, for i=0,1,(19)∂iw∂ti−∂iΠhw∂ti≤Ch∂iw∂ti1,∂iw∂ti∈H01Ω,∂iw∂ti−∂iΠhw∂ti1≤Chw2,∂iw∂ti∈H2Ω∩H01Ω,∂iφ∂ti−∂iRhφ∂ti≤Ch∂iφ∂ti1,∂iφ∂ti∈H1Ω.


### 3.2. Existence, Uniqueness, and Stability Analyses

Theorem 1.
There exists a unique discrete solution for the semidiscrete MFVE scheme (10).Proof.
Obviously, there exists a unique solutionuh0,ph0 for the scheme (11). Let ϕii=1N−1 and χAjj=0N−1 be the basis functions of the spaces H0h and Lh, respectively, then uh∈H0h and ph∈Lh can be expressed as follows:(20)uhx,t=∑i=1N−1uhitϕix,phx,t=∑j=0N−1phjtχAjx.
Choosingvh=ϕi1≤i≤N−1 and qh=χAj0≤j≤N−1 in (10), we rewrite the semidiscrete scheme (10) in the matrix form to find Uht,Pht such that(21)DUh′t−BTPht=0,APht+νBUht=α2FUht,where(22)Uht=uh1t,uh2t,…,uhN−1tT,Pht=ph0t,ph1t,…,phN−1tT,A=χAi,χAji,j=0,…,N−1,B=ϕix,χAji=1,…,N−1;j=0,…,N−1,D=ϕi,γhϕji,j=1,…,N−1,FUht=fUht,χAjj=0,…,N−1T.
It is easy to know that matricesA and D are symmetrically positive definite. Then, the system (21) can be rewritten as(23)DUh′t+νBTA−1BUht−α2BTA−1FUht=0,Pht=α2A−1FUht−νA−1BUht.
According to the theory of differential equation, we can see that the system (23) has a unique solution, which shows that there exists a unique discrete solution for the semidiscrete MFVE scheme (10).Theorem 2.
Letuh,ph be the discrete solution of the semidiscrete scheme (10), then there exists a constant C>0 such that(24)uht∞≤Cuht1≤Cu01+u012eCu012,pht≤Cu01+u012eCu012.Proof.
Choosingvh=uh and qh=uhx in (10), we have(25)uht,γhuh+νuhx,uhx=α2fuh,uhx.
Noting thatfuh,uhx=0, we rewrite (25) as(26)12ddtuh,γhuh+νuhx2=0.
Integrating (26) from 0 to t, we get(27)uht,γhuht+2ν∫0tuhx2dt=uh0,γhuh0.
Applying Lemma1 and Lemma 3, we obtain(28)uht2+8ν3∫0tuhx2dt≤433uh02.
Calculating the derivative of (10) with respect to t, we get(29)pht,qh+νuhxt,qh=α22uhuht,qh.
Settingvh=uht in (10) and qh=ph in (29), we have(30)uht,γhuht+1νpht,ph=ανuhuht,ph.
Applying the Sobolev embedding theorem to estimate the right-hand side of (30), we obtain(31)ανuhuht,ph≤ανuh∞⋅uht⋅ph≤38uht2+2α23ν2uh∞2ph2≤38uht2+Cuhx2ph2.
Making use of (31) and Lemma 3 in (30), we have(32)34uht2+12νddtpht2≤38uht2+Cuhx2ph2.
Integrating (32) from 0 to t, we obtain(33)pht2≤ph02+C∫0tuhx2ph2dt.
Apply the Gronwall lemma and (28) in (33), we obtain(34)pht2≤ph02eC∫0tuhx2dt≤ph02eCuh02.
Next, choosingqh=uhx in (10), we get(35)ph,uhx+νuhx2=α2fuh,uhx.
Noting thatfuh,uhx=0, we have(36)νuhxt2≤phtuhxt.
Substituting (34) into the above inequality, we have(37)uhxt≤1νpht≤1νph0eCuh02.
Thus, applying the Sobolev embedding theorem, we obtain(38)uht∞≤Cuht1≤Cuhxt≤Cph0eCuh02.
Now, we estimateuh01 and ph0. We choose zh=uhx0 in (11) to obtain(39)uh0∞≤Cuh01≤Cuhx0≤Cu0x≤Cu01.
Choosingqh=ph0 in (11), we have(40)ph02=−νuhx0,ph0+α2uh0uh0,ph0≤νuhx0ph0+α2uh0∞uh0ph0.
Making use of (39), we easily get(41)ph0≤Cu01+u012.
Thus, we obtain the proof of Theorem2.


### 3.3. A Priori Error Estimates

We first rewrite the errors as(42)u−uh=u−Πhu+Πhu−uh=ξ+η,p−ph=p−Rhp+Rhp−ph=ρ+ε,where Πh and Rh are the elliptic projection operators and the L2 orthogonal projection operator, respectively. By subtracting (10) from (4), we get that(43)ηt,γhvh−vhx,ε=−ξt,γhvh−ut,I−γhvh,∀vh∈H0h,ε,qh+νηx,qh=α2fu−fuh,qh,∀qh∈Lh,where η0 and ε0 satisfy(44)η0=0,ε0,qh+νηx0,qh=α2fu0−fuh0,qh,∀qh∈Lh.Theorem 3.
Letu,p and uh,ph be the solutions of the systems (4) and (10), respectively. Assume that the initial solution uh0,ph0 satisfies (11), then there exists a constant C>0 such that(45)ut−uht≤Chut1+∫0tut12+utt12dt1/2,pt−pht≤C1hpt1+u012+∫0tut12+utt12dt1/2,where C1=C1utL∞L∞Ω,u01 represents the function of utL∞L∞Ω and u01.Proof.
Takingvh=η and qh=ε in (43), we have(46)ηt,γhη+1νε,ε=−ξt,γhη−ut,I−γhη−α2νfuh−fu,ε.
Applying Theorem2, we have fu−fuh≤Cξ+η. We apply the Cauchy–Schwarz inequality and the Young inequality in (46) to obtain(47)12ddtη,γhη+1νε2≤12νε2+Cξt2+h2ut12+η2+ξ2.
Integrating (47) from 0 to t, and applying Lemma 3, we obtain(48)η2+83ν∫0tε2dt≤C∫0tη2+ξ2+ξt2+h2ut12dt.
Applying the Gronwall lemma in (48) yields(49)η2+83ν∫0tε2dt≤C∫0tξ2+ξt2+h2ut12dt.
Next, calculating the derivative of (43) with respect to t, we obtain(50)εt,qh+νηxt,qh=αu⋅ut−uh⋅uht,qh.
Choosingqh=ε in (50) and vh=ηt in (43), we have(51)ηt,γhηt+1νεt,ε=−ξt,γhηt−ut,I−γhηt+ανu⋅ut−uh⋅uht,ε.
Applying the stability results in Theorem2, we have(52)ανu⋅ut−uh⋅uht,ε=ανuut−uhut+uhut−uhuht,ε=ανξ+ηut+ξt+ηtuh,ε≤ανut∞ξ+ηε+uh∞ξt+ηtε≤ανut∞ξ+ηε+Cu01+u012eCu012ξt+ηtε≤316ηt2+C1ξ2+η2+ξt2+ε2,where C1=C1utL∞L∞Ω,u01 represents the function of utL∞L∞Ω and u01. Applying Lemmas 3–5 in (51), and making use of (52), we obtain(53)34ηt2+12νddtε2≤38ηt2+C1ξ2+ξt2+h2ut12+η2+ε2.
Integrating (53) from 0 to t, we get(54)∫otν4ηt2dt+εt2≤ε02+C1∫0tξ2+ξt2+h2ut12+ε2dt.
Applying the Gronwall lemma in (54), we obtain(55)εt2≤C1ε02+∫0tξ2+ξt2+h2ut12dt.
We estimateε0 by choosing qh=ε0 in (44) and noting that η0=0:(56)ε02=α2u0+uh0ξ0+η0,ε0≤Cu01ξ0ε0.
Applying Lemma6 in (56), we have(57)ε0≤Cu01ξ0≤Chu012.
Finally, we apply Lemma6 and the triangle inequality to obtain the desired conclusion.


## 3.1. Some Lemmas

For theoretical analysis, we first give some properties of the transfer operatorγh and two projection operators.Lemma 1 (see [40]).
The transfer operatorγh is bounded, that is,(12)γhwh≤3wh,∀wh∈H0h.Lemma 2 (see [40]).
The transfer operatorγh satisfies the following symmetry relation:(13)wh,γhvh=γhwh,vh,∀vh,wh∈H0h.Lemma 3 (see [40]).
The transfer operatorγh satisfies the following positivity:(14)γhwh,wh≥34wh2,∀wh∈H0h.Lemma 4 (see [40]).
LetI be an identity operator, then the transfer operator γh satisfies the following properties:(15)I−γhwh≤1212hwh1,∀wh∈H0h,vh,I−γhwh≤1212hvh1wh,∀vh,wh∈H0h.Lemma 5 (see [40]).
There exists a constantC>0 such that(16)v,I−γhwh≤Chv1wh,∀v∈H01Ω,∀wh∈H0h.Now, the elliptic projection operatorΠh:H01Ω⟶H0h is introduced below, which satisfies(17)w−Πhwx,zh=0,∀zh∈Lh,forw∈H01Ω.At the same time, theL2 orthogonal projection operator Rh:L2Ω⟶Lh is introduced to satisfy(18)φ−Rhφ,qh=0,∀qh∈Lh,forφ∈L2Ω.Referring to References [24, 25], we can know that the projection operators Πh and Rh satisfy the following estimate properties.Lemma 6.
There exists a constantC>0 such that, for i=0,1,(19)∂iw∂ti−∂iΠhw∂ti≤Ch∂iw∂ti1,∂iw∂ti∈H01Ω,∂iw∂ti−∂iΠhw∂ti1≤Chw2,∂iw∂ti∈H2Ω∩H01Ω,∂iφ∂ti−∂iRhφ∂ti≤Ch∂iφ∂ti1,∂iφ∂ti∈H1Ω.


## 3.2. Existence, Uniqueness, and Stability Analyses

Theorem 1.
There exists a unique discrete solution for the semidiscrete MFVE scheme (10).Proof.
Obviously, there exists a unique solutionuh0,ph0 for the scheme (11). Let ϕii=1N−1 and χAjj=0N−1 be the basis functions of the spaces H0h and Lh, respectively, then uh∈H0h and ph∈Lh can be expressed as follows:(20)uhx,t=∑i=1N−1uhitϕix,phx,t=∑j=0N−1phjtχAjx.
Choosingvh=ϕi1≤i≤N−1 and qh=χAj0≤j≤N−1 in (10), we rewrite the semidiscrete scheme (10) in the matrix form to find Uht,Pht such that(21)DUh′t−BTPht=0,APht+νBUht=α2FUht,where(22)Uht=uh1t,uh2t,…,uhN−1tT,Pht=ph0t,ph1t,…,phN−1tT,A=χAi,χAji,j=0,…,N−1,B=ϕix,χAji=1,…,N−1;j=0,…,N−1,D=ϕi,γhϕji,j=1,…,N−1,FUht=fUht,χAjj=0,…,N−1T.
It is easy to know that matricesA and D are symmetrically positive definite. Then, the system (21) can be rewritten as(23)DUh′t+νBTA−1BUht−α2BTA−1FUht=0,Pht=α2A−1FUht−νA−1BUht.
According to the theory of differential equation, we can see that the system (23) has a unique solution, which shows that there exists a unique discrete solution for the semidiscrete MFVE scheme (10).Theorem 2.
Letuh,ph be the discrete solution of the semidiscrete scheme (10), then there exists a constant C>0 such that(24)uht∞≤Cuht1≤Cu01+u012eCu012,pht≤Cu01+u012eCu012.Proof.
Choosingvh=uh and qh=uhx in (10), we have(25)uht,γhuh+νuhx,uhx=α2fuh,uhx.
Noting thatfuh,uhx=0, we rewrite (25) as(26)12ddtuh,γhuh+νuhx2=0.
Integrating (26) from 0 to t, we get(27)uht,γhuht+2ν∫0tuhx2dt=uh0,γhuh0.
Applying Lemma1 and Lemma 3, we obtain(28)uht2+8ν3∫0tuhx2dt≤433uh02.
Calculating the derivative of (10) with respect to t, we get(29)pht,qh+νuhxt,qh=α22uhuht,qh.
Settingvh=uht in (10) and qh=ph in (29), we have(30)uht,γhuht+1νpht,ph=ανuhuht,ph.
Applying the Sobolev embedding theorem to estimate the right-hand side of (30), we obtain(31)ανuhuht,ph≤ανuh∞⋅uht⋅ph≤38uht2+2α23ν2uh∞2ph2≤38uht2+Cuhx2ph2.
Making use of (31) and Lemma 3 in (30), we have(32)34uht2+12νddtpht2≤38uht2+Cuhx2ph2.
Integrating (32) from 0 to t, we obtain(33)pht2≤ph02+C∫0tuhx2ph2dt.
Apply the Gronwall lemma and (28) in (33), we obtain(34)pht2≤ph02eC∫0tuhx2dt≤ph02eCuh02.
Next, choosingqh=uhx in (10), we get(35)ph,uhx+νuhx2=α2fuh,uhx.
Noting thatfuh,uhx=0, we have(36)νuhxt2≤phtuhxt.
Substituting (34) into the above inequality, we have(37)uhxt≤1νpht≤1νph0eCuh02.
Thus, applying the Sobolev embedding theorem, we obtain(38)uht∞≤Cuht1≤Cuhxt≤Cph0eCuh02.
Now, we estimateuh01 and ph0. We choose zh=uhx0 in (11) to obtain(39)uh0∞≤Cuh01≤Cuhx0≤Cu0x≤Cu01.
Choosingqh=ph0 in (11), we have(40)ph02=−νuhx0,ph0+α2uh0uh0,ph0≤νuhx0ph0+α2uh0∞uh0ph0.
Making use of (39), we easily get(41)ph0≤Cu01+u012.
Thus, we obtain the proof of Theorem2.


## 3.3. A Priori Error Estimates

We first rewrite the errors as(42)u−uh=u−Πhu+Πhu−uh=ξ+η,p−ph=p−Rhp+Rhp−ph=ρ+ε,where Πh and Rh are the elliptic projection operators and the L2 orthogonal projection operator, respectively. By subtracting (10) from (4), we get that(43)ηt,γhvh−vhx,ε=−ξt,γhvh−ut,I−γhvh,∀vh∈H0h,ε,qh+νηx,qh=α2fu−fuh,qh,∀qh∈Lh,where η0 and ε0 satisfy(44)η0=0,ε0,qh+νηx0,qh=α2fu0−fuh0,qh,∀qh∈Lh.Theorem 3.
Letu,p and uh,ph be the solutions of the systems (4) and (10), respectively. Assume that the initial solution uh0,ph0 satisfies (11), then there exists a constant C>0 such that(45)ut−uht≤Chut1+∫0tut12+utt12dt1/2,pt−pht≤C1hpt1+u012+∫0tut12+utt12dt1/2,where C1=C1utL∞L∞Ω,u01 represents the function of utL∞L∞Ω and u01.Proof.
Takingvh=η and qh=ε in (43), we have(46)ηt,γhη+1νε,ε=−ξt,γhη−ut,I−γhη−α2νfuh−fu,ε.
Applying Theorem2, we have fu−fuh≤Cξ+η. We apply the Cauchy–Schwarz inequality and the Young inequality in (46) to obtain(47)12ddtη,γhη+1νε2≤12νε2+Cξt2+h2ut12+η2+ξ2.
Integrating (47) from 0 to t, and applying Lemma 3, we obtain(48)η2+83ν∫0tε2dt≤C∫0tη2+ξ2+ξt2+h2ut12dt.
Applying the Gronwall lemma in (48) yields(49)η2+83ν∫0tε2dt≤C∫0tξ2+ξt2+h2ut12dt.
Next, calculating the derivative of (43) with respect to t, we obtain(50)εt,qh+νηxt,qh=αu⋅ut−uh⋅uht,qh.
Choosingqh=ε in (50) and vh=ηt in (43), we have(51)ηt,γhηt+1νεt,ε=−ξt,γhηt−ut,I−γhηt+ανu⋅ut−uh⋅uht,ε.
Applying the stability results in Theorem2, we have(52)ανu⋅ut−uh⋅uht,ε=ανuut−uhut+uhut−uhuht,ε=ανξ+ηut+ξt+ηtuh,ε≤ανut∞ξ+ηε+uh∞ξt+ηtε≤ανut∞ξ+ηε+Cu01+u012eCu012ξt+ηtε≤316ηt2+C1ξ2+η2+ξt2+ε2,where C1=C1utL∞L∞Ω,u01 represents the function of utL∞L∞Ω and u01. Applying Lemmas 3–5 in (51), and making use of (52), we obtain(53)34ηt2+12νddtε2≤38ηt2+C1ξ2+ξt2+h2ut12+η2+ε2.
Integrating (53) from 0 to t, we get(54)∫otν4ηt2dt+εt2≤ε02+C1∫0tξ2+ξt2+h2ut12+ε2dt.
Applying the Gronwall lemma in (54), we obtain(55)εt2≤C1ε02+∫0tξ2+ξt2+h2ut12dt.
We estimateε0 by choosing qh=ε0 in (44) and noting that η0=0:(56)ε02=α2u0+uh0ξ0+η0,ε0≤Cu01ξ0ε0.
Applying Lemma6 in (56), we have(57)ε0≤Cu01ξ0≤Chu012.
Finally, we apply Lemma6 and the triangle inequality to obtain the desired conclusion.


## 4. Fully Discrete MFVE Scheme and Its Theoretical Analysis

### 4.1. Fully Discrete MFVE Scheme

Let0=t0<t1<⋯<tM=T be an equidistant partition of the time interval 0,T, where M is a positive integer. Denote Δt=T/M to represent the step length and tn=nΔt,n=0,1,…,M. And denote ψn=ψtn and ∂tψn=ψn−ψn−1/Δt for a function ψ.Letuhn and phn be the fully discrete solutions of u and p at t=tn, respectively. We can obtain the following nonlinear backward Euler MFVE scheme to find uhn,phn∈H0h×Lhn=1,2,…,M:(58)∂tuhn,γhvh−vhx,phn=0,∀vh∈H0h,phn,qh+νuhxn,qh=α2fuhn,qh,∀qh∈Lh,where the initial value uh0,ph0 satisfies the following equations:(59)uhx0,zh=u0x,zh,∀zh∈Lh,ph0,qh+νuhx0,qh=α2fuh0,qh,∀qh∈Lh.Remark 1.
The fully discrete MFVE scheme (58) is implicit in time. In the actual calculation of uhn,phn1≤n≤M, we need to make predictions first by using the linear backward Euler MFVE scheme defined as follows:(60)∂tuhn,γhvh−vhx,phn=0,∀vh∈H0h,n≥1,phn,qh+νuhxn,qh=α2fuhn−1,qh,∀qh∈Lh,n≥1.
Similar to the proof process of Theorem 4 in Reference [40], we can also obtain that there exists a unique discrete solution for the linear MFVE scheme (60).


### 4.2. Existence, Uniqueness, and Stability Analyses

For the existence analysis, we first give the Brouwer fixed-point theorem [41, 42].Lemma 7.
LetH,⋅,⋅H be a finite-dimensional inner product space with a norm ⋅H. Let g:H⟶H be a continuous operator. Assume that there exists β>0 such that gz,z≥0 for z∈H with zH=β. Then, there exists z∗∈H such that gz∗=0 and z∗≤β.Theorem 4.
Suppose thatuhi,phii=0n−1 has been given, then there exists a unique fully discrete solution uhn,phn for the nonlinear backward Euler MFVE scheme (58).Proof.
Choosingqh=vhx in (58), we have(61)∂tuhn,γhvh+νuhxn,vhx=α2fuhn,vhx.
The operatorg:H0h⟶H0h is defined as follows:(62)gV,vhH1=V,γhvh+ΔtνVx,vhx−uhn−1,γhvh−αΔt2fV,vhx.
It is easy to know that the operatorg is continuous. Setting vh=V in (62), we have(63)gV,VH1=V,γhV+ΔtνVx,Vx−uhn−1,γhV−αΔt2fV,Vx≥34V2+ΔtνVx2−3uhn−1V=34VV−433uhn−1+ΔtνVx2.
SelectingV∈H0h to satisfy V=1+43/3uhn−1, we have gV,VH1>0. Making use of Lemma 7, we know that there exists V∗∈H0h such that gV∗=0. Thus, we choose uhn=V∗ to satisfy (61). Furthermore, selecting uhn=V∗ in (58), we get(64)phn,qh=−νVx∗,qh+α2fV∗,qh.
Thus, there exists a solutionphn∈Lh which satisfies (64). Then, it is obviously known that uhn,phn satisfies the scheme (58), which proves the existence of the fully discrete solutions.
Next, we give the uniqueness for the fully discrete scheme (58). Let Vn,Qn∈H0h×Lhn=1,2,…,M be another solution of the scheme (58) with the initial value V0=uh0, then we have(65)∂tVn,γhvh+νVxn,vhx=α2fVn,vhx.
LetEn=uhn−Vn, then we have(66)∂tEn,γhvh+νExn,vhx=α2uhn⋅uhn−Vn⋅Vn,vhx=α2uhn⋅uhn−uhnVn+uhnVn−Vn⋅Vn,vhx=α2uhnEn+VnEn,vhx.
The following proof is based on the mathematical induction. First,E0=uh0−V0=0; next assume En−1=0 and then choose vh=En in (66) to obtain(67)∂tEn,γhEn+νExn2=α2uhnEn+VnEn,Exn.
Applying the stability results in Theorem5 (because the existence of the discrete solutions has been proved), we have(68)1ΔtEn,γhEn+νExn2≤ν2Exn2+Cuhn∞2+Vn∞2En2≤ν2Exn2+Cph02eCuh012En2.
Multiplying (68) by Δt, and making use of Lemma 3, we obtain(69)34En2+ν2ΔtExn2≤CΔtph02eCuh012En2.
SelectingΔt to satisfy CΔtph02eCuh012<3/4, we have En2=0 and Exn2=0, which indicate that Vn=uhn, and then Qn=phn is known from (58). Thus, we complete the proof of Theorem 4.Theorem 5.
Letuhn,phnn=1M and uh0,ph0 be the discrete solutions of the systems (58) and (59), respectively. Then, there exists a constant C>0 such that(70)phn≤Cph0⋅eCuh012,uhn∞≤Cuhn1≤Cph0eCuh012.
Furthermore, the initial solutionuh0,ph0 satisfies(71)ph0≤Cu01+u012,uh0∞≤Cuh01≤Cuhx0≤Cu01.Proof.
Similarly to the proof process of Theorem2, we can obtain the estimate of the initial solution uh0,ph0. We choose vh=uhn and qh=uhxn in (58) to obtain(72)∂tuhn,γhuhn+νuhxn,uhxn=α2fuhn,uhxn.
Taking note offuhn,uhxn=0 and ∂tuhn,γhuhn≥1/2Δtuhn,γhuhn−uhn−1,γhuhn−1 in (72), we obtain(73)12Δtuhn,γhuhn−uhn−1,γhuhn−1+νuhxn2≤0.
Multiplying (73) by 2Δt, summing from 1 to n, and applying Lemma 1 and Lemma 3, we have(74)34uhn2+2νΔt∑k=1nuhxk2≤3uh02.
Next, we make use of (58) and (59) to obtain(75)∂tphn,qh+ν∂tuhxn,qh=α2uhn⋅uhn−uhn−1⋅uhn−1Δt,qh.
Choosingvh=∂tuhn in (58) and qh=phn in (75), we get(76)∂tuhn,γh∂tuhn+1ν∂tphn,phn=α2νuhn⋅uhn−uhn−1⋅uhn−1Δt,phn=α2νuhn⋅∂tuhn+uhn−1∂tuhn,phn.
Noting that(77)α2νuhn∂tuhn+uhn−1∂tuhn,phn≤α2νuhn∞∂tuhnphn+uhn−1∞∂tuhnphn≤38∂tuhn2+α23ν2uhn∞2+uhn−1∞2phn2,and applying Lemma 3 in (76), we have(78)34∂tuhn2+12νΔtphn2−phn−12≤38∂tuhn2+α23ν2uhn∞2+uhn−1∞2phn2.
Summing (78) from 1 to n and applying the inverse inequality and the Sobolev embedding theorem, we obtain(79)phn2≤ph02+2α23ν2Δt∑k=1nuhk∞2+uhk−1∞2phk2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+2α23ν2Δtuhn∞2+uhn−1∞2phn2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+CΔthuhn2+uhn−12phn2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+CΔthuh02phn2.
SelectingΔt and h in (79) to satisfy Cuh02⋅Δt/h<1/2, making use of (74), and applying the discrete Gronwall lemma, we have(80)phn2≤Cph02⋅eCΔt∑k=1n−1uhxk2+uhxk−12≤Cph02⋅eCuh012.
Choosingqh=uhxn in (58), we get(81)phn,uhxn+νuhxn2=α2fuhn,uhxn=0.
Applying the Cauchy–Schwarz inequality, we obtain(82)νuhxn2≤phnuhxn.
Making use of (80) in (82), and applying the Sobolev embedding theorem, we obtain(83)uhn∞≤Cuhn1≤Cph0eCuh012.
Thus, we complete the proof of Theorem5.


### 4.3. A Priori Error Estimates

First, the errors are expressed as(84)utn−uhn=utn−Πhutn+Πhutn−uhn=ξn+ηn,ptn−phn=ptn−Rhptn+Rhptn−phn=ρn+εn.By subtracting (58) from (4), we can obtain the error equations about ηn and εn as follows:(85)∂tηn,γhvh−vhx,εn=−utn,I−γhvh−θn,γhvh−∂tξn,γhvh,∀vh∈H0h,εn,qh+νηxn,qh=α2fun−fuhn,qh,∀qh∈Lh,where θn=utn−∂tun and η0 and ε0 satisfy(86)η0=0,ε0,qh+νηx0,qh=α2fu0−fuh0,qh,∀qh∈Lh.Theorem 6.
Letuhn,phn be the discrete solution of the fully discrete scheme (58), and the exact solution u,p of the mixed variational formulation (4) satisfies the following regularity properties:(87)u,ut∈L∞H01Ω,utt∈L2L2Ω,p∈L∞H1Ω.
Then, there exists a constantC>0 such that(88)max1≤m≤Mutm−uhm+ptm−phm≤Ch+Δt.Proof.
Choosingvh=ηn and qh=ηxn in (85), we have(89)∂tηn,γhηn+νηxn2=−utn,I−γhηn−θn,γhηn−∂tξn,γhηn−α2fuhn−fun,ηxn.
Applying Theorem5, we get fuhn−fun2≤Cξn2+ηn2. Noting that(90)∂tηn,γhηn≥12Δtηn,γhηn−ηn−1,γhηn−1,we easily get that(91)12Δtηn,γhηn−ηn−1,γhηn−1+νηxn2≤Ch2utn12+θn2+ξn2+∂tξn2+Cηn2+ν2ηxn2.
Multiplying (91) by 2Δt, and summing from 1 to m, we have(92)ηm,γhηm+Δtν∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2+ηn2.
Applying Lemma 3 in (92), we obtain(93)34ηm2+νΔt∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2+CΔt∑n=1m−1ηn2+CΔtηm2.
SelectingΔt to satisfy CΔt<3/8, and applying the discrete Gronwall lemma, we obtain(94)ηm2+8ν3Δt∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2.
According to the integral remainder formula of Taylor formula, we get(95)∂tξn2≤CΔt∫tn−1tnξt2dt,θn2≤CΔt∫tn−1tnutt2dt,n≥1.
Applying the above inequalities and Lemma6, we obtain(96)ηm≤ChuL∞H1Ω+utL∞H1Ω+utL2H1Ω+CuttL2L2ΩΔt.
Next, we estimateptm−phm. Making use of (85) and (86), we have(97)∂tεn,qh+ν∂tηxn,qh=α2unun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,qh.
The right-hand side of (97) can be estimated by Theorem 5 as follows:(98)unun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,qh=un+uhn−un−1+uhn−1∂tξn+∂tηn+2un−1−uhn−1∂tun,qh≤un∞+uhn∞+un−1∞+uhn−1∞∂tξn+∂tηnqh+2utL∞L∞Ωξn−1+ηn−1qh≤K1∂tξn+∂tηnqh+K2ξn−1+ηn−1qh,where K1=CuL∞L∞Ω+u01+u012eCu012 and K2=2utL∞L∞Ω. For the convenience of description, we use C2=C2K1,K2 to denote the function of K1 and K2.
Now, choosingvh=∂tηn in (85) and qh=εn in (97), we have(99)∂tηn,γh∂tηn+1ν∂tεn,εn=α2νunun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,εn−utn,I−γh∂tηn−θn,γh∂tηn−∂tξn,γh∂tηn.
Applying (98) and Lemmas 3–5, we obtain(100)34∂tηn2+12νΔtεn2−εn−12≤38∂tηn2+CC2h2utn12+∂tξn2+θn2+ξn−12+ηn−12+εn2.
Multiplying (100) by 2νΔt, and summing from 1 to m, we get(101)3ν4Δt∑n=1m∂tηn2+εm2≤ε02+CC2Δt∑n=1m−1εn2+CΔtεm2+CC2Δt∑n=1mh2utn12+∂tξn2+θn2+ξn−12+ηn−12.
SelectingΔt to satisfy C2Δt<1/2, and applying the discrete Gronwall lemma, we have(102)εm2≤Cε02+C2Δt∑n=1mh2utn12+∂tξn2+θn2+ξn−12+ηn−12.
We estimateε0 by choosing qh=ε0 in (86):(103)ε02=α2fu0−fuh0,ε0=α2u0+uh0ξ0+η0,ε0≤Cu01ξ0ε0.
Applying Lemma6, we have(104)ε0≤Chu012.
Taking note ofη0=0 in (102), and applying (95), (96), and Lemma 6, we obtain(105)εm≤CC2hu0H1Ω2+uL∞H1Ω+utL∞H1Ω+utL2H1Ω+CC2uttL2L2ΩΔt.
Finally, we apply the triangular inequality and Lemma6 to complete the proof of Theorem 6.


## 4.1. Fully Discrete MFVE Scheme

Let0=t0<t1<⋯<tM=T be an equidistant partition of the time interval 0,T, where M is a positive integer. Denote Δt=T/M to represent the step length and tn=nΔt,n=0,1,…,M. And denote ψn=ψtn and ∂tψn=ψn−ψn−1/Δt for a function ψ.Letuhn and phn be the fully discrete solutions of u and p at t=tn, respectively. We can obtain the following nonlinear backward Euler MFVE scheme to find uhn,phn∈H0h×Lhn=1,2,…,M:(58)∂tuhn,γhvh−vhx,phn=0,∀vh∈H0h,phn,qh+νuhxn,qh=α2fuhn,qh,∀qh∈Lh,where the initial value uh0,ph0 satisfies the following equations:(59)uhx0,zh=u0x,zh,∀zh∈Lh,ph0,qh+νuhx0,qh=α2fuh0,qh,∀qh∈Lh.Remark 1.
The fully discrete MFVE scheme (58) is implicit in time. In the actual calculation of uhn,phn1≤n≤M, we need to make predictions first by using the linear backward Euler MFVE scheme defined as follows:(60)∂tuhn,γhvh−vhx,phn=0,∀vh∈H0h,n≥1,phn,qh+νuhxn,qh=α2fuhn−1,qh,∀qh∈Lh,n≥1.
Similar to the proof process of Theorem 4 in Reference [40], we can also obtain that there exists a unique discrete solution for the linear MFVE scheme (60).


## 4.2. Existence, Uniqueness, and Stability Analyses

For the existence analysis, we first give the Brouwer fixed-point theorem [41, 42].Lemma 7.
LetH,⋅,⋅H be a finite-dimensional inner product space with a norm ⋅H. Let g:H⟶H be a continuous operator. Assume that there exists β>0 such that gz,z≥0 for z∈H with zH=β. Then, there exists z∗∈H such that gz∗=0 and z∗≤β.Theorem 4.
Suppose thatuhi,phii=0n−1 has been given, then there exists a unique fully discrete solution uhn,phn for the nonlinear backward Euler MFVE scheme (58).Proof.
Choosingqh=vhx in (58), we have(61)∂tuhn,γhvh+νuhxn,vhx=α2fuhn,vhx.
The operatorg:H0h⟶H0h is defined as follows:(62)gV,vhH1=V,γhvh+ΔtνVx,vhx−uhn−1,γhvh−αΔt2fV,vhx.
It is easy to know that the operatorg is continuous. Setting vh=V in (62), we have(63)gV,VH1=V,γhV+ΔtνVx,Vx−uhn−1,γhV−αΔt2fV,Vx≥34V2+ΔtνVx2−3uhn−1V=34VV−433uhn−1+ΔtνVx2.
SelectingV∈H0h to satisfy V=1+43/3uhn−1, we have gV,VH1>0. Making use of Lemma 7, we know that there exists V∗∈H0h such that gV∗=0. Thus, we choose uhn=V∗ to satisfy (61). Furthermore, selecting uhn=V∗ in (58), we get(64)phn,qh=−νVx∗,qh+α2fV∗,qh.
Thus, there exists a solutionphn∈Lh which satisfies (64). Then, it is obviously known that uhn,phn satisfies the scheme (58), which proves the existence of the fully discrete solutions.
Next, we give the uniqueness for the fully discrete scheme (58). Let Vn,Qn∈H0h×Lhn=1,2,…,M be another solution of the scheme (58) with the initial value V0=uh0, then we have(65)∂tVn,γhvh+νVxn,vhx=α2fVn,vhx.
LetEn=uhn−Vn, then we have(66)∂tEn,γhvh+νExn,vhx=α2uhn⋅uhn−Vn⋅Vn,vhx=α2uhn⋅uhn−uhnVn+uhnVn−Vn⋅Vn,vhx=α2uhnEn+VnEn,vhx.
The following proof is based on the mathematical induction. First,E0=uh0−V0=0; next assume En−1=0 and then choose vh=En in (66) to obtain(67)∂tEn,γhEn+νExn2=α2uhnEn+VnEn,Exn.
Applying the stability results in Theorem5 (because the existence of the discrete solutions has been proved), we have(68)1ΔtEn,γhEn+νExn2≤ν2Exn2+Cuhn∞2+Vn∞2En2≤ν2Exn2+Cph02eCuh012En2.
Multiplying (68) by Δt, and making use of Lemma 3, we obtain(69)34En2+ν2ΔtExn2≤CΔtph02eCuh012En2.
SelectingΔt to satisfy CΔtph02eCuh012<3/4, we have En2=0 and Exn2=0, which indicate that Vn=uhn, and then Qn=phn is known from (58). Thus, we complete the proof of Theorem 4.Theorem 5.
Letuhn,phnn=1M and uh0,ph0 be the discrete solutions of the systems (58) and (59), respectively. Then, there exists a constant C>0 such that(70)phn≤Cph0⋅eCuh012,uhn∞≤Cuhn1≤Cph0eCuh012.
Furthermore, the initial solutionuh0,ph0 satisfies(71)ph0≤Cu01+u012,uh0∞≤Cuh01≤Cuhx0≤Cu01.Proof.
Similarly to the proof process of Theorem2, we can obtain the estimate of the initial solution uh0,ph0. We choose vh=uhn and qh=uhxn in (58) to obtain(72)∂tuhn,γhuhn+νuhxn,uhxn=α2fuhn,uhxn.
Taking note offuhn,uhxn=0 and ∂tuhn,γhuhn≥1/2Δtuhn,γhuhn−uhn−1,γhuhn−1 in (72), we obtain(73)12Δtuhn,γhuhn−uhn−1,γhuhn−1+νuhxn2≤0.
Multiplying (73) by 2Δt, summing from 1 to n, and applying Lemma 1 and Lemma 3, we have(74)34uhn2+2νΔt∑k=1nuhxk2≤3uh02.
Next, we make use of (58) and (59) to obtain(75)∂tphn,qh+ν∂tuhxn,qh=α2uhn⋅uhn−uhn−1⋅uhn−1Δt,qh.
Choosingvh=∂tuhn in (58) and qh=phn in (75), we get(76)∂tuhn,γh∂tuhn+1ν∂tphn,phn=α2νuhn⋅uhn−uhn−1⋅uhn−1Δt,phn=α2νuhn⋅∂tuhn+uhn−1∂tuhn,phn.
Noting that(77)α2νuhn∂tuhn+uhn−1∂tuhn,phn≤α2νuhn∞∂tuhnphn+uhn−1∞∂tuhnphn≤38∂tuhn2+α23ν2uhn∞2+uhn−1∞2phn2,and applying Lemma 3 in (76), we have(78)34∂tuhn2+12νΔtphn2−phn−12≤38∂tuhn2+α23ν2uhn∞2+uhn−1∞2phn2.
Summing (78) from 1 to n and applying the inverse inequality and the Sobolev embedding theorem, we obtain(79)phn2≤ph02+2α23ν2Δt∑k=1nuhk∞2+uhk−1∞2phk2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+2α23ν2Δtuhn∞2+uhn−1∞2phn2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+CΔthuhn2+uhn−12phn2≤ph02+CΔt∑k=1n−1uhxk2+uhxk−12phk2+CΔthuh02phn2.
SelectingΔt and h in (79) to satisfy Cuh02⋅Δt/h<1/2, making use of (74), and applying the discrete Gronwall lemma, we have(80)phn2≤Cph02⋅eCΔt∑k=1n−1uhxk2+uhxk−12≤Cph02⋅eCuh012.
Choosingqh=uhxn in (58), we get(81)phn,uhxn+νuhxn2=α2fuhn,uhxn=0.
Applying the Cauchy–Schwarz inequality, we obtain(82)νuhxn2≤phnuhxn.
Making use of (80) in (82), and applying the Sobolev embedding theorem, we obtain(83)uhn∞≤Cuhn1≤Cph0eCuh012.
Thus, we complete the proof of Theorem5.


## 4.3. A Priori Error Estimates

First, the errors are expressed as(84)utn−uhn=utn−Πhutn+Πhutn−uhn=ξn+ηn,ptn−phn=ptn−Rhptn+Rhptn−phn=ρn+εn.By subtracting (58) from (4), we can obtain the error equations about ηn and εn as follows:(85)∂tηn,γhvh−vhx,εn=−utn,I−γhvh−θn,γhvh−∂tξn,γhvh,∀vh∈H0h,εn,qh+νηxn,qh=α2fun−fuhn,qh,∀qh∈Lh,where θn=utn−∂tun and η0 and ε0 satisfy(86)η0=0,ε0,qh+νηx0,qh=α2fu0−fuh0,qh,∀qh∈Lh.Theorem 6.
Letuhn,phn be the discrete solution of the fully discrete scheme (58), and the exact solution u,p of the mixed variational formulation (4) satisfies the following regularity properties:(87)u,ut∈L∞H01Ω,utt∈L2L2Ω,p∈L∞H1Ω.
Then, there exists a constantC>0 such that(88)max1≤m≤Mutm−uhm+ptm−phm≤Ch+Δt.Proof.
Choosingvh=ηn and qh=ηxn in (85), we have(89)∂tηn,γhηn+νηxn2=−utn,I−γhηn−θn,γhηn−∂tξn,γhηn−α2fuhn−fun,ηxn.
Applying Theorem5, we get fuhn−fun2≤Cξn2+ηn2. Noting that(90)∂tηn,γhηn≥12Δtηn,γhηn−ηn−1,γhηn−1,we easily get that(91)12Δtηn,γhηn−ηn−1,γhηn−1+νηxn2≤Ch2utn12+θn2+ξn2+∂tξn2+Cηn2+ν2ηxn2.
Multiplying (91) by 2Δt, and summing from 1 to m, we have(92)ηm,γhηm+Δtν∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2+ηn2.
Applying Lemma 3 in (92), we obtain(93)34ηm2+νΔt∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2+CΔt∑n=1m−1ηn2+CΔtηm2.
SelectingΔt to satisfy CΔt<3/8, and applying the discrete Gronwall lemma, we obtain(94)ηm2+8ν3Δt∑n=1mηxn2≤CΔt∑n=1mξn2+h2utn12+θn2+∂tξn2.
According to the integral remainder formula of Taylor formula, we get(95)∂tξn2≤CΔt∫tn−1tnξt2dt,θn2≤CΔt∫tn−1tnutt2dt,n≥1.
Applying the above inequalities and Lemma6, we obtain(96)ηm≤ChuL∞H1Ω+utL∞H1Ω+utL2H1Ω+CuttL2L2ΩΔt.
Next, we estimateptm−phm. Making use of (85) and (86), we have(97)∂tεn,qh+ν∂tηxn,qh=α2unun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,qh.
The right-hand side of (97) can be estimated by Theorem 5 as follows:(98)unun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,qh=un+uhn−un−1+uhn−1∂tξn+∂tηn+2un−1−uhn−1∂tun,qh≤un∞+uhn∞+un−1∞+uhn−1∞∂tξn+∂tηnqh+2utL∞L∞Ωξn−1+ηn−1qh≤K1∂tξn+∂tηnqh+K2ξn−1+ηn−1qh,where K1=CuL∞L∞Ω+u01+u012eCu012 and K2=2utL∞L∞Ω. For the convenience of description, we use C2=C2K1,K2 to denote the function of K1 and K2.
Now, choosingvh=∂tηn in (85) and qh=εn in (97), we have(99)∂tηn,γh∂tηn+1ν∂tεn,εn=α2νunun−uhnuhn−un−1un−1+uhn−1uhn−1Δt,εn−utn,I−γh∂tηn−θn,γh∂tηn−∂tξn,γh∂tηn.
Applying (98) and Lemmas 3–5, we obtain(100)34∂tηn2+12νΔtεn2−εn−12≤38∂tηn2+CC2h2utn12+∂tξn2+θn2+ξn−12+ηn−12+εn2.
Multiplying (100) by 2νΔt, and summing from 1 to m, we get(101)3ν4Δt∑n=1m∂tηn2+εm2≤ε02+CC2Δt∑n=1m−1εn2+CΔtεm2+CC2Δt∑n=1mh2utn12+∂tξn2+θn2+ξn−12+ηn−12.
SelectingΔt to satisfy C2Δt<1/2, and applying the discrete Gronwall lemma, we have(102)εm2≤Cε02+C2Δt∑n=1mh2utn12+∂tξn2+θn2+ξn−12+ηn−12.
We estimateε0 by choosing qh=ε0 in (86):(103)ε02=α2fu0−fuh0,ε0=α2u0+uh0ξ0+η0,ε0≤Cu01ξ0ε0.
Applying Lemma6, we have(104)ε0≤Chu012.
Taking note ofη0=0 in (102), and applying (95), (96), and Lemma 6, we obtain(105)εm≤CC2hu0H1Ω2+uL∞H1Ω+utL∞H1Ω+utL2H1Ω+CC2uttL2L2ΩΔt.
Finally, we apply the triangular inequality and Lemma6 to complete the proof of Theorem 6.


## 5. Numerical Example

Now, we give a numerical example to verify the feasibility and effectiveness of the MFVE scheme. We consider Burgers’ equation as follows:(106)ut+uux−νuxx=0,x,t∈0,1×0,2,u0,t=u1,t=0,t∈0,2,ux,0=sinπx,x∈0,1,where ν=1/48. Following Reference [4], we can obtain the exact solution (Fourier series solution) as follows:(107)ux,t=2πν∑n=1∞anexp−n2π2νtnsinnπxa0+∑n=1∞anexp−n2π2νtcosnπx,where(108)a0=∫01exp−2πν−11−cosπxdx,an=2∫01exp−2πν−11−cosπxcosnπxdx,n=1,2,3,…,.The auxiliary variablepx,t=1/2u2x,t−νuxx,t is determined by the above data.In practical calculation, we use the equidistant grids of spatial regions and truncate the series of the exact solutions with∑n=1500 when calculating error results in the L∞L2Ω norm of u and p. We give the error results and convergence orders in Table 1 with mesh sizes h=Δt=1/20,1/40,1/80,1/160. We can see that the convergence orders are approximate to 1, and these results are consistent with Theorem 6. The space-time graphs of the exact solutions of u and p at t=2 with h=1/40 are shown in Figures 1 and 2, respectively. The space-time graphs of the nonlinear backward Euler fully discrete solutions uh and ph at t=2 with h=Δt=1/20 are shown in Figures 3 and 4. The figures show that the numerical solutions of u and p have the same numerical behavior as their exact solutions. The numerical results and figures verify that the MFVE method for Burgers’ equation is feasible and efficient.Table 1
Error results and convergence orders.
h
,
Δ
t
u
−
u
h
L
∞
L
2
Ω
Order
p
−
p
h
L
∞
L
2
Ω
Order
1/20, 1/20
2.6767e − 02
2.1361e − 02
1/40, 1/40
1.4241e − 02
0.9104
1.1424e − 02
0.9029
1/80, 1/80
7.4366e − 03
0.9374
5.9368e − 03
0.9444
1/160, 1/160
3.8548e − 03
0.9480
3.0320e − 03
0.9694Figure 1
Graph of the exact solution ofux,t at t=2 with h=1/40.Figure 2
Graph of the exact solution ofpx,t at t=2 with h=1/40.Figure 3
Graph of the numerical solution ofuhx,t at t=2 with h=Δt=1/20.Figure 4
Graph of the numerical solution ofphx,t at t=2 with h=Δt=1/20.


## 6. Conclusions

In this article, our main aim is to construct the MFVE scheme for solving Burgers’ equation by introducing the auxiliary variablepx,t and the transfer operator γh. We give the detailed theoretical results on existence, uniqueness, and stability for the semidiscrete and fully discrete schemes and obtain the optimal a priori error estimates in the L2Ω norm for the velocity u and flux p by using the stability results in Theorems 2 and 5. Moreover, we give a numerical example to verify that the proposed scheme for Burgers’ equation is feasible and efficient. In the future, we will apply the MFVE methods to solve some other nonlinear partial differential equations.



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