# A Godunov-Mixed Finite Element Method on Changing Meshes for the Nonlinear Sobolev Equations

Authors: Tongjun Sun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A Godunov-mixed finite element method on changing meshes is presented to simulate the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. The method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Almost optimal error estimates inL2-norm under very general changes in the mesh can be obtained. Finally, a numerical experiment is given to illustrate the efficiency of the method.


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## Body

## 1. Introduction

We consider the following nonlinear Sobolev equations:(1.1)ut-∇·(a(x,t,u)∇ut+b(x,t,u)∇u)+c(x,t,u)·∇u=f(x,t,u),x∈Ω,t∈(0,T],u(x,t)=0,x∈∂Ω,t∈[0,T],u(x,0)=u0(x),x∈Ω,
where Ω is a bounded subset of Rn (n≤3) with smooth boundary ∂Ω, u0(x) and f(x,t,u) are known functions, and the coefficients a(x,t,u),b(x,t,u),c(x,t,u)=(c1(x,t,u),c2(x,t,u))T satisfy the following condition:
(1.2)0<a0≤a(x,t,u)≤a1,|∂a(x,t,u)∂t|≤K1,0<b0≤b(x,t,u)≤b1,|c(x,t,u)|=c12(x,t,u)+c22(x,t,u)≤c0,|∂c(x,t,u)∂u|≤K2,a(x,t,u),b(x,t,u),c(x,t,u),f(x,t,u),∂c(x,t,u)∂uare  Lipschitz  continuous  aboutu,
where a0, a1, b0, b1, c0, K1, and K2 are positive constants. We assume that u(x,t) satisfy the smooth condition in the following analysis.For time-changing localized phenomena, such as sharp fronts and layers, the finite element method on changing meshes [1–3] is advantageous over fixed finite element method. The reason is that the former treats the problem with the finite element method on space domain by using different meshes and different basic functions at different time levels so that it has the capability of self-adaptive local grid modification (refinement or unrefinement) to efficiently capture propagating fronts or moving layers. The work [4] had combined this method with mixed finite element method to study parabolic problems. In [5], an upwind-mixed method on changing meshes was considered for two-phase miscible flow in porous media.Sobolev equations have important applications in many mathematical and physical problems, such as the percolation theory when the fluid flows through the cracks [6], the transfer problem of the moisture in the soil [7], and the heat conduction problem in different materials [8]. So there exists great and actual significance to research Sobolev equations. Many works had researched on numerical treatments for Sobolev equations. More attentions were paid for treating a damping term ∇·[a∇ut], which is a distinct character of Sobolev equations different from parabolic equation. For example, time stepping Galerkin method was presented for nonlinear Sobolev equations in [9, 10]. In [11, 12], nonlinear Sobolev equations with convection term were researched by using finite difference streamline-diffusion method and discontinuous Galerkin method, respectively. Two new least-squares mixed finite element procedures were formulated for solving convection-dominated Sobolev equations in [13].Methods which combined Godunov-type schemes for advection with mixed finite elements for diffusion were introduced in [14] and had been applied to flow problems in reservoir engineering, contaminant transport, and computational fluid dynamics. Applications of these types of methods to single and two-phase flow in oil reservoirs were discussed in [15, 16]; application to the Navier-Stokes equations was given in [17]. These methods had proven useful for advective flow problems because they combined element-by-element conservation of mass with numerical stability and minimal numerical diffusion. Dawson [18] researched advective flow problems in one space dimension by high-order Godunov-mixed method. In 1993, he expanded this method to multidimensions [19] and presented three variations. In these methods, advection was approximated by a Godunov-type procedure, and diffusion was approximated by a low-order-mixed finite element method.The object of this paper is to present a Godunov-mixed finite element method on changing meshes for the nonlinear Sobolev equations. The convection termc·∇u of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. This method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. We describe this method in Section 2. In Section 3, we introduce three projections and a lemma. We derive almost optimal error estimates in L2-norm under very general changes in the mesh in Section 4. In Section 5, we present results of numerical experiment, which confirm our theoretical results.Throughout the analysis, the symbolK will denote a generic constant, which is independent of mesh parameters Δt and h and not necessarily the same at different occurrences.


## 2. The Godunov-Mixed Method on Changing Meshes

At first we give some notation and basic assumptions. The usual Sobolev spaces and norms are adopted onΩ. The inner product on L2(Ω) is denoted by (f,g)=∫Ω‍fgdx. Define the following spaces and norms:
(2.1)Hm(div,Ω)={f=(fx,fy);fx,fy,∇·f∈Hm(Ω)},m≥0,∥f∥Hm(div)2=∥fx∥m2+∥fy∥m2+∥∇·f∥m2,H(div,Ω)=H0(div,Ω),W=L2(Ω){φ≡constantonΩ},V={v∈H(div;Ω)∣v·n=0on∂Ω,nis  the  unit  outward  norm  to∂Ω}.LetΔtn>0  (n=1,2,…,N*) denote different time steps, tn=∑k=1nΔtk, T=∑n=1N*Δtn,  Δt=maxnΔtn. Assume that the time steps Δtn do not change too rapidly; that is, we assume there exist positive constants t* and t* which are independent of n and Δt such that
(2.2)t*≤ΔtnΔtn-1≤t*.
For a given function g(x,t), let gn=g(x,tn).AssumeΩ=(0,1)×(0,1). At each time level tn, we construct a quasiuniform rectangular partition Khn={ein} of Ω:
(2.3)δxn:0=x1/2n<x3/2n<⋯<xi+1/2n<⋯<xIn+1/2n=1,δyn:0=y1/2n<y3/2n<⋯<yj+1/2n<⋯<yJn+1/2n=1.
Let hi,xn=xi+1/2n-xi-1/2n, hi,yn=yj+1/2n-yj-1/2n, hn=maxi,j{hi,xn,hi,yn}, and h=maxnhn. And xin=(xi+1/2n+xi-1/2n)/2 is the midpoint of [xi-1/2n,xi+1/2n], hi+1/2,xn=xi+1n-xin, Δtn=O(hn). Let yjn,hj+1/2,yn be defined analogously.Let(2.4)M-1k(δxn)={f:f∣Bi,xn∈Pk(Bi,xn),i=1,…,In},M0k(δxn)=M-1k(δxn)∩C0([0,1]),
where Bi,xn=[xi-1/2n,xi+1/2n] and Pk(Bi,xn) is the set of all polynomials of degree less than or equal to k defined on Bi,xn. Similar definitions are given to M-10(δyn) and M01(δyn).Then the lowest order Raviart-Thomas spacesWhn and Vhn are given by
(2.5)Whn=M-10(δxn)⊗M-10(δyn),Vhn=Vh,xn×Vh,yn,Vh,xn=M01(δxn)⊗M-10(δyn),Vh,yn=M-10(δxn)⊗M01(δyn).
That is, the space Whn is the space of functions which are constant on each element ein∈Khn, and Vhn is the space of vector valued functions whose components are continuous and linear on each element ein∈Khn. The degrees of freedom of a function vn∈Vhn correspond to the values of vn·γ at the midpoints of the sides of ein, where γ is the unit outward normal to ∂ein. It is easy to see that Whn×Vhn⊂W×V and divVhn=Whn.By introducing variablesz-=-∇u,  z=bz-+az-t,  g=cu=(c1u,c2u)T=(g1,g2)T, and c-(x,t,u)=(∂c1/∂u,∂c2/∂u)T, we modify the first equation in (1.1) as
(2.6)ut+∇·z+(∇·g+uc-(u)·z-)=f(u).Here we are using the so-called “expanded” mixed finite element method, proposed by Arbogast et al. [20], which gives a gradient approximation z- as well as an approximation to the diffusion term z.The weak form of (2.6) is
(2.7)(ut,w)+(∇·z,w)+(∇·g,w)+(uc-(u)·z-,w)=(f(u),w),
so that
(2.8)(ut,w)+(∇·z,w)+(∇·g,w)+(uc-(u)·z-,w)=(f(u),w),∀w∈W,(z-,v)=(u,∇·v),∀v∈V,(z,v)=(b(u)z-+a(u)z-t,v),∀v∈V.The Godunov-mixed method on changing meshes is presented as follows: at each time leveln, find Un∈Whn, Zn∈Vhn such that
(2.9)(Un-RnUn-1Δtn,wn)+(∇·Zn,wn)+(Unc-(Un)·Z-n,wn)=(f(Un),wn)-(∇·Gn-1,wn),∀wn∈Whn,(RnUn-1-Un-1,wn)=0,∀wn∈Whn,(Z-n,vn)=(Un,∇·vn),∀vn∈Vhn,(Zn,vn)=(b(Un)Z-n+a(Un)Z-n-RnZ-n-1Δtn,vn),∀vn∈Vhn,(a(Un)(RnZ~n-1-Z~n-1),vn)=0,∀vn∈Vhn.
When different finite element spaces are used at time levels tn and tn-1, the second and fifth equations of (2.9) give the L2-projection {RnUn-1,RnZ~n-1} of the previous approximate solution {Un-1,Z~n-1} into the current finite element space Whn×Vhn. This projection is used in the first and third equations of (2.9) as initial value to calculate {Un,Z~n}. If the finite element spaces are the same at time levels tn and tn-1, we know that RnUn-1=Un-1, RnZ~n-1=Z~n-1.LetGn in (2.9) be constructed by the Godunov method. We adopt the Godunov flux [19] Hw(wL,wR) for a given flux function w(s):
(2.10)Hw(wL,wR)={minwL≤s≤wRw(s),ifwL≤wR,maxwR≤s≤wLw(s),otherwise,
where  wL,wR are the left and right states, respectively. It is easily seen that Hw is Lipschitz in its arguments if w is Lipschitz in s and Hw is consistent, that is, Hw(s,s)=w(s).LetG=(g-1,g-2), gi=ciu, Gi+1/2,j+1/2n-1=(g-1,i+1/2,jn-1,g-2,i,j+1/2n-1). Now we give the calculation of G by Hw by the following three steps.First Step
We construct a piecewise linear functionRUn-1 on each element ein-1:
(2.11)RUn-1∣ein-1=Ui,jn-1+(x-xi)δxUi,jn-1+(y-yj)δyUi,jn-1,
where δxUi,jn-1 and δyUi,jn-1 are x,y slopes. The x slope calculation can be performed as
(2.12)δxUi,jn-1={▵+xUi,jn-1,if|▵+xUi,jn-1|≤|▵-xUi,jn-1|,▵-xUi,jn-1,otherwise,
where ▵+x and ▵-x are forward and backward difference operators in the x direction, respectively:
(2.13)▵+xUi,jn-1=Ui+1,jn-1-Ui,jn-1hi+1/2,xn-1,▵-xUi,jn-1=Ui,jn-1-Ui-1,jn-1hi-1/2,xn-1,
and δx satisfy
(2.14)hi,xn-12|δxUi,jn-1|≤C∑j-L≤j≤j+L‍∑i-R≤i≤i+R‍|Ui,jn-1|,δxui,jn-1=ux(xi,yj,tn-1)+O(h),
where L and R are positive integers independent of h. The y slope is defined analogously.Second Step
By the Taylor expansion, foru with smooth second derivatives, we have
(2.15)u(xi+1/2,yj,tn-1/2)=u+hi,xn2ux+Δtn2ut+O(h2+Δt2).
By (2.6), we see that
(2.16)ut=f-uc-(u)·z--g1uux-g2y-∇·z,(2.17)ui+1/2,yn-1/2=u+(hi,xn2-Δtn2g1u)ux-Δtn2g2y+σ,
where
(2.18)σ=Δtn2(f-uc-(u)·z--∇·z)+O(h2+Δt2)=O(h2+Δt).
Based on (2.17) and a similar expansion about (xi+1,yj,tn-1), we define UL,UR:
(2.19)Ui+1/2,jL,n-1≜Ui,jn-1+(hi,xn-12-Δtn-12g1u(Ui,jn-1))δxUi,jn-1-Δtn-12hj,yn-1[Γi,j+1/2n-1-Γi,j-1/2n-1],Ui+1/2,jR,n-1≜Ui+1,jn-1+(hi,xn-12-Δtn-12g1u(Ui+1,jn-1))δxUi+1,jn-1-Δtn-12hj,yn-1[Γi+1,j+1/2n-1-Γi+1,j-1/2n-1],
where
(2.20)Γi,j+1/2n-1=Hg(Ui,j+1/2L,n-1,Ui,j+1/2R,n-1),Ui,j+1/2L,n-1=Ui,jn-1+hj,yn-12δyUi,jn-1,Ui,j+1/2R,n-1=Ui,j+1n-1+hj+1,yn-12δyUi,j+1n-1.Third Step
With the above definitions,(g-1,i+1/2,jn-1,g-2,i,j+1/2n-1) is calculated as follows.
Ifc1(Ui+1/2,jn-1)>0,  define
(2.21)g-1,i+1/2,jn-1≜Hg1(Ui+1/2,jL,n-1,Ui+1/2,jL,n-1)≜c1(Ui+1/2,jL,n-1)Ui+1/2,jL,n-1.
Ifc1(Ui+1/2,jn-1)≤0,  define
(2.22)g-1,i+1/2,jn-1≜Hg1(Ui+1/2,jR,n-1,Ui+1/2,jR,n-1)≜c1(Ui+1/2,jR,n-1)Ui+1/2,jR,n-1.
For g-2,i,j+1/2n-1, the definition is similar.
The equations (2.19)–(2.22) hold for elements at least one element away from the boundary. At the left and right boundaries, we can set
(2.23)U1/2,jL,n-1=UIn-1+1/2,jR,n-1=0,j=1,2,…,Jn-1,n=1,2,…,N,
and in the slope calculation procedure, we define δxU1,jn-1 and δxUIn-1,jn-1 using (2.12) with
(2.24)▵-xU1,jn-1=2U1,jn-1h1,xn-1,▵+xUIn-1,jn-1=-2UIn-1,jn-1hIn-1,xn-1.
On the bottom and top boundaries, we set
(2.25)Γi,1/2n-1=Γi,Jn-1+1/2n-1=0,i=1,2,…,In-1,
in (2.19).


## 3. Projections and Lemma

We introduce three projections and a lemma to obtain error estimates. Let{u~n,z~n}:J→Whn×Vhn be the projection of {u,z-} in mixed finite element space such that
(3.1)(z~n,vn)-(u~n,∇·vn)=0,∀vn∈Vhn,
and define πzn as the π-projection of zn such that
(3.2)(∇·(zn-πzn),wn)=0,∀wn∈Whn.These projections satisfy ([21, 22]):
(3.3)∥un-u~n∥+∥z-n-z~n∥H(div)≤Khn,∥∂∂t(un-u~n)∥+∥∂∂t(z-n-z~n)∥H(div)≤Khn,
and it is easy to see that ∥u~n∥L∞≤K.Letg~∈Vhn satisfies that the value of g~·γ at the midpoint of the boundaries is equal to the average value of g·γ on boundaries, that is,
(3.4)(∇·(g~n-gn),wn)=0,∀wn∈Whn.
Let g~=(g~1,g~2). From (3.4), we can define
(3.5)g~1(xi+1/2,yj,tn)≜1hj,yn∫yj-1/2yj+1/2‍c1(xi+1/2,yj,tn,u)u(xi+1/2,y,tn)dy≜1hj,yn∫yj-1/2yj+1/2‍g1(u(xi+1/2,y,tn))dy.
Then, g~2(xi,yj+1/2,tn) can be defined similarly.Lemma 3.1.
Letξ=U-u~ and assume that u is sufficiently smooth. Then
(3.6)∥Gn-1(·,Un-1)-g~n-1∥≤K(∥ξn-1∥+h+Δt).Proof.
Firstly, constructu~i+1/2,jL,n-1,u~i+1/2,jR,n-1 from u~ given in (3.1) and (3.2) by
(3.7)u~i+1/2,jL,n-1=u~i,jn-1+(hi,xn-12-Δtn2g1u(u~i,jn-1))δxu~i,jn-1-Δtn2hj,yn-1[Γi,j+1/2n-1(u~)-Γi,j-1/2n-1(u~)],
at interior edges, where
(3.8)δxu~i,jn-1={▵+xu~i,jn-1,ifδxUi,jn-1=▵+xUi,jn-1,▵-xu~i,jn-1,otherwise,Γi,j+1/2n-1(u~)=Hg(u~i,j+1/2L,n-1,u~i,j+1/2R,n-1),u~i,j+1/2L,n-1=u~i,jn-1+hj,yn-12δyu~i,jn-1,u~i,j+1/2R,n-1=u~i,j+1n-1+hj+1,yn-12δyu~i,j+1n-1.
And define u~i+1/2,jR,n-1 similarly.
Then, assumingg1 is twice differentiable and Lipschitz continuous, and using the Lipschitz continuity of the Godunov flux and (2.14), it can be shown that
(3.9)|Ui+1/2,jL,n-1-u~i+1/2,jL,n-1|≤K∑l=i-1i+2‍∑m=j-1j+2‍|Ul,mn-1-u~l,mn-1|.
A similar bound holds for |Ui+1/2,jR,n-1-u~i+1/2,jR,n-1|.
At the boundaries, we follow (2.23) and (2.25) and define
(3.10)u~1/2,jL,n-1=u~In-1+1/2,jR,n-1=0,j=1,2,…,Jn-1,Γi,1/2n-1(u~)=Γi,Jn-1+1/2n-1(u~)=0,i=1,2,…,In-1.
Define ui+1/2,jL,n-1,ui+1/2,jR,n-1 analogous to u~i+1/2,jL,n-1,u~i+1/2,jR,n-1.
Ifc1(Ui+1/2,jn-1)>0, we consider
(3.11)g-1,i+1/2,jn-1-g~1(xi+1/2,yj,tn-1)=Hg1(Ui+1/2,jL,n-1,Ui+1/2,jL,n-1)-1hj,yn-1∫yj-1/2yj+1/2‍g1(u(xi+1/2,y,tn-1))dy=[Hg1(Ui+1/2,jL,n-1,Ui+1/2,jL,n-1)-Hg1(u~i+1/2,jL,n-1,u~i+1/2,jL,n-1)]+[Hg1(u~i+1/2,jL,n-1,u~i+1/2,jL,n-1)-Hg1(ui+1/2,jL,n-1,ui+1/2,jL,n-1)]+[Hg1(ui+1/2,jL,n-1,ui+1/2,jL,n-1)-g1(ui+1/2,jn-1/2)]+[g1(ui+1/2,jn-1/2)-1hj,yn-1∫yj-1/2yj+1/2‍g1(u(xi+1/2,y,tn-1))dy]=R1+R2+R3+R4.
By the Lipschitz continuity ofHg1 and (3.9), we have
(3.12)|R1|≤K{|Ui+1/2,jL,n-1-u~i+1/2,jL,n-1|+|Ui+1/2,jR,n-1-u~i+1/2,jR,n-1|}≤K∑l=i-1i+2‍∑m=j-1j+2‍|Ul,mn-1-u~l,mn-1|.
The similar arguments are applied to R2 to get
(3.13)|R2|≤K∑l=i-1i+2‍∑m=j-1j+2‍|u~l,mn-1-ui+1/2,jn-1|.
By the consistency of Hg1, we see
(3.14)|R3|≤K|ui+1/2,jL,n-1-ui+1/2,jn-1/2|=O(Δt+h2).
For R4, it is easy to have
(3.15)|R4|≤K(Δt+h2).
Using (3.3), (3.12)–(3.15), and equivalence of norms, we have
(3.16)∥g-1n-1-g~1n-1∥≤K∑j‍∑i|g-1,i+1/2,jn-1-g~1,i+1/2,jn-1|‍2hi+1/2,xn-1hj,yn-1≤K(∥ξn-1∥+Δt+h).
The similar approach can be used whenc1(Ui+1/2,jn-1)≤0. Defining g-2 and g~2 similarly and following analogous arguments yields
(3.17)∥g-2n-1-g~2n-1∥≤C(∥ξn-1∥+Δt+h).
Thus, Lemma 3.1 holds.


## 4. Error Estimates

At time leveltn, for all wn∈Whn, vn∈Vhn, the exact solutions satisfy
(4.1)(un-un-1Δtn,wn)+(∇·zn,wn)+(unc-(un)·z-n,wn)=(f(un),wn)-(∇·gn,wn)-(ρn,wn),(z-,vn)=(un,∇·vn),(zn,vn)=(b(un)z-n+a(un)z-n-z-n-1Δtn,vn)+(rn,vn),
where ρn=utn-(un-un-1)/Δtn, rn=a(un)(z-tn-(z-n-z-n-1)/Δtn).Let(4.2)ξu=U-u~,βu=u-u~,ξ-z=Z--z~,β-z=z--z~,ξz=Z-πz,βz=z-πz.Using projections (3.1) and (3.2), we subtract (2.9) from (4.1) to get
(4.3)(ξun-ξun-1Δtn,wn)+(∇·ξzn,wn)+(Unc-(Un)·Z-n-unc-(un)·z-n,wn)=(f(Un)-f(un),wn)+(ρn,wn)+(∇·(g~n-Gn-1),wn)+(βun-βun-1Δtn,wn),(ξ-zn,vn)=(ξun,∇·vn),(ξzn,vn)=([b(Un)Z-n-b(un)z-n],vn)-(rn,vn)-(βzn,vn)+(a(Un)Z-n-Z-n-1Δtn-a(un)z-n-z-n-1Δtn,vn),
where the last term of the first equation in (4.3) is related to the changing meshes. If the meshes do not change, this term will be zero.Takingw=ξun, v=ξzn in the second equation and v=ξ-zn in the third, then adding them together, we have
(4.4)(ξun-ξun-1Δtn,ξun)+(b(Un)Z-n-b(un)z-n,ξ-zn)+(a(Un)Z-n-Z-n-1Δtn,ξ-zn)-(a(un)z-n-z-n-1Δtn,ξ-zn)=(f(Un)-f(un),ξun)+(ρn,ξun)+(∇·(g~n-Gn-1),ξun)-(Unc-(Un)·Z-n-unc-(un)·z-n,ξun)+(rn,ξ-zn)-(βzn,ξ-zn)+(βun-βun-1Δtn,ξun).
For (4.4), we see(I)(4.5)(b(Un)Z-n-b(un)z-n,ξ-zn)=(b(Un)ξ-zn,ξ-zn)-(b(Un)β-zn,ξ-zn)+([b(Un)-b(un)]z-n,ξ-zn),(II)(4.6)(a(Un)Z-n-Z-n-1Δtn,ξ-zn)-(a(un)z-n-z-n-1Δtn,ξ-zn)=(a(Un)ξ-zn-ξ-zn-1Δtn,ξ-zn)-(a(Un)β-zn-β-zn-1Δtn,ξ-zn)+([a(Un)-a(un)]z-n-z-n-1Δtn,ξ-zn),(III)(4.7)(Unc-(Un)·Z-n-unc-(un)·z-n,ξun)=([ξun-βun]c-(Un)·Z-n,ξun)+(unc-(Un)·[ξ-zn-β-zn],ξun)+(un[c-(Un)-c-(un)]·z-n,ξun).
The last second term in (4.6) is related to the changing meshes. If the meshes do not change, this term will be zero.Substitute (4.5)–(4.7) into (4.4) to get
(4.8)(ξun-ξun-1Δtn,ξun)+(b(Un)ξ-zn,ξ-zn)+(a(Un)ξ-zn-ξ-zn-1Δtn,ξ-zn)=(f(Un)-f(un),ξun)+(ρn,ξun)+(∇·(g~n-Gn-1),ξun)+(rn,ξ-zn)-(βzn,ξ-zn)-([ξun-βun]c-(Un)·Z-n,ξun)-(unc-(Un)·[ξ-zn-β-zn],ξun)-(un[c-(Un)-c-(un)]·z-n,ξun)+(βun-βun-1Δtn,ξun)+(b(Un)β-zn,ξ-zn)+([b(un)-b(Un)]z-n,ξ-zn)+([a(un)-a(Un)]z-n-z-n-1Δtn,ξ-zn)+(a(Un)β-zn-β-zn-1Δtn,ξ-zn)=T1+T2+⋯+T13.
Furthermore, for the first and third terms on the left-hand side of (4.8), we have
(4.9)(ξun-ξun-1Δtn,ξun)=12Δtn[(ξun,ξun)-(ξun-1,ξun-1)]+12Δtn∥ξun-ξun-1∥2,(a(Un)ξ-zn-ξ-zn-1Δtn,ξ-zn)≥12Δtn[(a(Un)ξ-zn,ξ-zn)-(a(Un-1)ξ-zn-1,ξ-zn-1)]-a22∥ξ-zn-1∥2+a02Δtn∥ξ-zn-ξ-zn-1∥2.
By (4.9), we modify (4.8) as
(4.10)12Δtn[(ξun,ξun)-(ξun-1,ξun-1)]+12Δtn∥ξun-ξun-1∥2+(b(Un)ξ-zn,ξ-zn)+12Δtn[(a(Un)ξ-zn,ξ-zn)-(a(Un-1)ξ-zn-1,ξ-zn-1)]+a02Δtn∥ξ-zn-ξ-zn-1∥2≤a22∥ξ-zn-1∥2+T1+T2+⋯+T13.By the Lipschitz continuity off(u), a(u), b(u), ∂c/∂u, and (3.3), the following terms on the right-hand side of (4.10) can be obtained:
(4.11)T1≤K∥Un-un∥2+K∥ξun∥2≤K∥ξun∥2+Kh2,T2≤KΔtn∥∂2u∂t2∥L2(tn-1,tn;H1)2+K∥ξun∥2,T4≤KΔtn∥∂2u∂t2∥L2(tn-1,tn;H1)2+K∥ξ-zn∥2,T5≤K∥ξ-zn∥2+Kh2,T6≤K∥ξun∥2+Kh2,T7≤K∥ξun∥2+K∥ξ-zn∥2+Kh2,T8≤K∥Un-un∥2+K∥ξun∥2≤K∥ξun∥2+Kh2,T10≤K(b1)h2+K∥ξ-zn∥2,T11≤K∥ξ-zn∥2+K∥ξun∥2+Kh2,T12≤K∥ξ-zn∥2+K∥Un-un∥2+K∥z-n-z-n-1Δtn∥2≤K∥ξ-zn∥2+K∥ξun∥2+K(Δtn)-1∥∂u∂t∥L2(tn-1,tn;H1)2+Kh2.We turn to considerT3. Letting v=g~n-Gn-1 in the second equation of (4.10), then we have
(4.12)(ξ-zn,g~n-Gn-1)=(ξun,∇·(g~n-Gn-1)).
By Lemma 3.1, we have
(4.13)T3≤12(b(Un)ξ-zn,ξ-zn)+K(b0-1)∥g~n-Gn-1∥2≤12(b(Un)ξ-zn,ξ-zn)+K(b0-1)∥g~n-g~n-1∥2+∥g~n-1-Gn-1∥2≤12(b(Un)ξ-zn,ξ-zn)+K(b0-1)(∥ξun-1∥2+h2+Δt2).Substituting (4.11) and (4.13) into (4.10), we have
(4.14)12Δtn[(ξun,ξun)-(ξun-1,ξun-1)]+12Δtn∥ξun-ξun-1∥2+12(b(Un)ξ-zn,ξ-zn)+a02Δtn∥ξ-zn-ξ-zn-1∥2+12Δtn[(anξ-zn,ξ-zn)-(an-1ξ-zn-1,ξ-zn-1)]≤K(∥ξ-zn∥2+∥ξ-zn-1∥2)+K(∥ξun∥2+∥ξun-1∥2+Δt2+h2)+K(Δtn)(∥∂u∂t∥L2(tn-1,tn;H1)2+∥∂2u∂t2∥L2(tn-1,tn;H1)2)+T9+T13.LetN be the time step at which ∥ξun∥ is maximum, that is,
(4.15)∥ξuN∥2=max1≤n≤N*∥ξun∥2.
Multiplying (4.14) by 2Δtn and summing on n from 1 to N, we obtain
(4.16)∥ξuN∥2+∑n=1N‍∥ξun-ξun-1∥2+∑n=1N‍(b(Un)ξ-zn,ξ-zn)Δtn+a0∑n=1N‍∥ξ-zn-ξ-zn-1∥2+a0∥ξ-zN∥2≤K(Δt2+h2)+K∑n=1N‍∥ξun∥2Δtn+∥ξu0∥2+a1∥ξ-z0∥2+K∑n=1N‍∥ξ-zn∥2Δtn+∑n=1N‍(βun-βun-1Δtn,ξun)Δtn+∑n=1N‍(a(Un)β-zn-β-zn-1Δtn,ξ-zn)Δtn.Assuming that the mesh is modified at mostM times, and M≤M*, where M* is independent of h and Δt, we get [3]:
(4.17)∑n=1N‍(βun-βun-1Δtn,ξun)Δtn=∑n=1N‍[(βun,ξun)-(βun-1,ξun)]≤K(M*h)2+14∥ξuN∥2,∑n=1N‍(a(Un)β-zn-β-zn-1Δtn,ξ-zn)Δtn≤Kh2+ε∑n=1N‍∥ξ-zn∥2Δtn,
where ε is a sufficiently small positive constant.Substituting (4.17) into (4.16), we find
(4.18)∥ξuN∥2+∑n=1N‍∥ξun-ξun-1∥2+∑n=1N‍(b(Un)ξ-zn,ξ-zn)Δtn+a0∑n=1N‍∥ξ-zn-ξ-zn-1∥2+a0∥ξ-zN∥2≤K(Δt2+h2+(M*h)2)+∥ξu0∥2+a1∥ξ-z0∥2+K∑n=1N‍∥ξun∥2Δtn+K∑n=1N‍∥ξ-zn∥2Δtn+ε∑n=1N‍∥ξ-zn∥2Δtn+14∥ξuN∥2.
Using the discrete Gronwall's lemma, we have
(4.19)maxn∥ξun∥2≤K(h2+Δt2+(M*h)2).
Similarly as (4.14), we can derive
(4.20)maxn∥ξ-zn∥2≤K(h2+Δt2+(M*h)2).By the triangle inequality and (3.3), we can obtain the following result.Theorem 4.1.
Assuming that the coefficients satisfy condition (1.2), the mesh is modified at most M times, M≤M*, and u∈L2(H1), ut∈L2(H1), utt∈L2(H1), then we have
(4.21)maxn∥un-Un∥≤K(h+Δt)+K(M*)h,maxn∥z-n-Z-n∥≤K(h+Δt)+K(M*)h,
where K and M* are positive constants independent of h and Δt.


## 5. Numerical Example

In this section, we give numerical experiment of the following nonlinear equations to illustrate the efficiency of the Godunov-mixed finite element method on changing meshes:(5.1)ut=(a(x,t)uxt+b(x,t,u)ux)x-c(x,t)ux+f(x,t,u),(x,t)∈(0,1)×(0,1],u(0,t)=u(1,t)=0,t∈[0,1],u(·,0)=sinπx,x∈[0,1],
where (x,t)∈[0,1]×[0,1], a(x,t)=t2(x+0.25)+0.25, b(x,t,u)=0.005ut2(x+0.05), c(x,t)=0.05t(x+1)+1, and f(x,t,u) is chosen properly so that the exact solution is u=e-tsinπx.In the numerical example,h and Δt change as the following four cases.Case 1.
t
∈
[
0,0.4
], set h=Δt=0.1; t∈(0.4,0.6], set h=Δt=0.05; t∈(0.6,1.0], set h=Δt=0.1 and calculate 40 steps at every time interval.Case 2.
t
∈
[
0,0.4
], set h=Δt=0.05; t∈(0.4,0.6], set h=Δt=0.025; t∈(0.6,1.0], set h=Δt=0.05 and calculate 80 steps at every time interval.Case 3.
t
∈
[
0,0.4
], set h=Δt=0.025; t∈(0.4,0.6], set h=Δt=0.0125; t∈(0.6,1.0], set h=Δt=0.025 and calculate 160 steps at every time interval.Case 4.
t
∈
[
0,0.4
], set h=Δt=0.0125; t∈(0.4,0.6], set h=Δt=0.00625; t∈(0.6,1.0], set h=Δt=0.0125 and calculate 320 steps at every time interval.
The numerical solutionsUn, Z-n are calculated and the L2 error estimates of Un-un,Z-n-z-n are obtained; see Tables 1 and 2.Table 1
L
2-norm estimates of Un-un.
Time
Case1
Case2
Case3
Case4
t
=
0.4
0.0454
0.0232
0.0117
0.0057
t
=
0.6
0.0253
0.0135
0.0064
0.0032
t
=
1.0
0.0288
0.0160
0.0080
0.0039Table 2
L
2-norm estimates of  Z-n-z-n.
Time
Case1
Case2
Case3
Case4
t
=
0.4
0.0537
0.0307
0.0161
0.0069
t
=
0.6
0.0646
0.0353
0.0164
0.0080
t
=
1.0
0.0643
0.0395
0.0198
0.0099In Table3, convergence rates of Un-un,Z-n-z-n are given. The figures of convergence rate are shown by Figure 1. The convergence rate of Un-un in Figure 1(a) is one order, and in Figure 1(b) the convergence rate of Z-n-z-n is a little smaller than one order at the beginning. But the convergence rate Z-n-z-n will be close to one order when h minishes, which is consistent with the analysis in this paper.Table 3
Convergence rate ofL2-norm.
Time
Convergence rates of∥un-Un∥
Convergence rates of∥z-n-Z-n∥
Case1/Case 2
Case2/Case 3
Case3/Case 4
Case1/Case 2
Case2/Case 3
Case3/Case 4
t
=
0.4
0.9670
0.9852
1.0392
0.8052
0.9271
1.2167
t
=
0.6
0.9023
1.0724
0.9974
0.8692
1.1121
1.0290
t
=
1.0
0.8464
1.0075
1.0242
0.7009
0.9963
1.0007Convergence rate figures.
(a)
Convergence rates ofu
(b)
Convergence rates ofz~For Case4, we compare the exact solution {u,z~} with the approximate solution {U,Z~} at time t=0.25, 0.5, and 0.75, respectively; see Figure 2. From Figures 2(a) and 2(b), we can see that the approximate solutions are very close to the exact solutions.Compare figures for Case4 at time t=0.25, 0.5, and 0.75.
(a)
Comparingu with U
(b)
Comparingz~ with Z~



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*Source: 413718-2012-12-17.xml*