# Crank-Nicolson Fully DiscreteH1-Galerkin Mixed Finite Element Approximation of One Nonlinear Integrodifferential Model

Authors: Fengxin Chen
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider a fully discreteH1-Galerkin mixed finite element approximation of one nonlinear integrodifferential model which often arises in mathematical modeling of the process of a magnetic field penetrating into a substance. We adopt the Crank-Nicolson discretization for time derivative. Optimal order a priori error estimates for the unknown function in L2 and H1 norm and its gradient function in L2 norm are presented. A numerical example is given to verify the theoretical results.


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## Body

## 1. Introduction

The objective of this paper is to discuss a Crank-Nicolson fully discreteH1-mixed finite element scheme for the following nonlinear integrodifferential model:
(1)ut-(1+λ(t))uxx=f(x,t),(x,t)∈I×(0,T],u(0,t)=0,u(1,t)=0,0≤t≤T,u(x,0)=u0(x),x∈I,
where λ(t)=∫0t‍∫01‍(∂u/∂x)2dxds and I=[0,1]. u0(x) and f(x,t) are given functions.The above equations have been widely used to describe the process of a magnetic field penetrating into a substance, which is a generalization of the model proposed in [1–4]. The existence and uniqueness of a weak solution to the above boundary value problems were proved in [5].During the last decades, many numerical methods were developed to discretize this kind of problems. For the finite difference approximation of the above model one can refer to [6–11]. For Galerkin finite element approximation of model (1) we can refer to [11], where the authors developed error estimates for semidiscretization in the energy norm. Note that the coefficient in (1) depends on the derivative of u. When finite difference method and Galerkin method were used to solve this model, one needed to differentiate the numerical solution to determine the coefficient. This would generate an inaccurate coefficient, which then reduces the accuracy of the numerical approximation for u. In order to overcome this question an H1-Galerkin mixed finite element discrete scheme was proposed in [12]. Optimal order error estimates in L2 norm and H1 norm were presented. For more references with respect to H1-Galerkin mixed finite element method one can refer to [13–17].In [12] the backward Euler method was used to discretize the time derivative. Note that problem (1) is nonlocal due to the integration term in the coefficient. To improve the convergence order for time discretization and save the storage we construct a Crank-Nicolson H1-mixed finite element scheme for problem (1). By using elliptic projection and the boundness of the numerical solutions we prove optimal a priori error estimates for the scalar unknown function and its flux. Finally we carry out a numerical example to verify our theoretical results.The rest of this paper is organized as follows. In Section2 a Crank-Nicolson H1-mixed finite element scheme is constructed. Optimal a priori error estimates are deduced in Section 3. In Section 4 a numerical example is carried out to verify our theoretical results.Throughout the paper, we use the standard notationWm,q(Ω) for Sobolev space on Ω with a norm ∥·∥m,q and a seminorm |·|m,q. For q=2, we denote Hm(Ω)=Wm,2(Ω), ∥·∥m=∥·∥m,2, and for m=0, we denote ∥·∥=∥·∥0. Moreover, the inner products in L2(Ω) are indicated by (·,·). Let X be a Banach space and φ(t):[0,T]↦X; we set
(2)∥φ∥L2(X)2=∫0T‍∥φ(s)∥X2ds,∥φ∥L∞(X)=esssup⁡0≤t≤T∥φ∥X.In addition,C denotes a generic constant independent of the spatial mesh parameter h and time discretization parameter τ, and ɛ denotes an arbitrarily small positive constant.


## 2. Crank-Nicolson Discrete Scheme

In this section we first briefly describe the weak formulation for problem (1) and then construct a Crank-Nicolson discrete scheme for it.


### 2.1. Weak Formulation

In order to define a fully discreteH1-Galerkin mixed finite element procedure for problem (1), we firstly split (1) into a first order system. Let σ=ux; then (1) reduces to
(3)σ=ux,ut=(1+λ(t))σx+f(x,t),
where λ(t)=∫0t‍∫01‍σ2dxds.LetH01(I)={v∈H1(I);v(0)=v(1)=0}. It is natural to state the weak formulation for problem (1) in the following form:
(4)(ux,vx)=(σ,vx),v∈H01(I),(σt,w)+((1+λ(t))σx,wx)+(f,wx)=0,w∈H1(I).


### 2.2. The Crank-Nicolson Discrete Scheme

First we introduce two finite element spaces. LetVh and Wh denote the finite dimensional subspaces of H01(I) and H1(I), respectively, with the following approximation properties:
(5)inf⁡ψh∈Vh{∥ψ-ψh∥0,p+h∥ψ-ψh∥1,p}≤Chk+1∥ψ∥k+1,p,llllllllllllllllllllllllllllllllllllllllllψ∈H01(I)∩Wk+1,p(I),inf⁡wh∈Wh{∥w-wh∥0,p+h∥w-wh∥1,p}≤Chr+1∥w∥r+1,p,llllllllllllllllllllllllllllllllllllllllllllllllllllllw∈Wr+1,p(I),
where 1≤p≤∞. k, r are positive integers.To define the fully discrete scheme we also need a time mesh grid. Let0=t0<t1<⋯<tN=T be a given partition of the time interval [0,T] with step length τ=T/N, for some positive integers N. Define tn=nτ and tn-(1/2)=(n-(1/2))τ. For convenience we set ϕn=ϕ(tn) and ϕn-(1/2)=(ϕn+ϕn-1)/2 for a smooth function ϕ.LetUn and Qn denote the discrete counterpart of u and σ at t=tn which satisfy the following Crank-Nicolson discrete scheme:
(6)(Uxn,vhx)=(Qn,vhx),vh∈Vh,(7)(Qn-Qn-1τ,wh)+((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)Qxn+Qxn-12,whx)+(fn-(1/2),whx)=0,wh∈Wh,
where
(8)δn,i={τ,i=1,2,…,n-1,τ2,i=n,
and U0, Q0 are to be defined later.The existence and uniqueness of the discrete solution for the above problems can be guaranteed by the theory presented in [18, page 237–239].To discretize the time integration we used the following integroformula:(9)∫0tn-(1/2)‍g(τ)ds≈∑i=1n‍δn,igi-(1/2).
Its truncation error can be estimated as follows:
(10)|∫0tn-(1/2)‍g(τ)ds-∑i=1n‍δn,ign-(1/2)|≤C(τ∫tn-1tn‍|gt|+τ2∫0tn‍|gtt|ds).


## 2.1. Weak Formulation

In order to define a fully discreteH1-Galerkin mixed finite element procedure for problem (1), we firstly split (1) into a first order system. Let σ=ux; then (1) reduces to
(3)σ=ux,ut=(1+λ(t))σx+f(x,t),
where λ(t)=∫0t‍∫01‍σ2dxds.LetH01(I)={v∈H1(I);v(0)=v(1)=0}. It is natural to state the weak formulation for problem (1) in the following form:
(4)(ux,vx)=(σ,vx),v∈H01(I),(σt,w)+((1+λ(t))σx,wx)+(f,wx)=0,w∈H1(I).


## 2.2. The Crank-Nicolson Discrete Scheme

First we introduce two finite element spaces. LetVh and Wh denote the finite dimensional subspaces of H01(I) and H1(I), respectively, with the following approximation properties:
(5)inf⁡ψh∈Vh{∥ψ-ψh∥0,p+h∥ψ-ψh∥1,p}≤Chk+1∥ψ∥k+1,p,llllllllllllllllllllllllllllllllllllllllllψ∈H01(I)∩Wk+1,p(I),inf⁡wh∈Wh{∥w-wh∥0,p+h∥w-wh∥1,p}≤Chr+1∥w∥r+1,p,llllllllllllllllllllllllllllllllllllllllllllllllllllllw∈Wr+1,p(I),
where 1≤p≤∞. k, r are positive integers.To define the fully discrete scheme we also need a time mesh grid. Let0=t0<t1<⋯<tN=T be a given partition of the time interval [0,T] with step length τ=T/N, for some positive integers N. Define tn=nτ and tn-(1/2)=(n-(1/2))τ. For convenience we set ϕn=ϕ(tn) and ϕn-(1/2)=(ϕn+ϕn-1)/2 for a smooth function ϕ.LetUn and Qn denote the discrete counterpart of u and σ at t=tn which satisfy the following Crank-Nicolson discrete scheme:
(6)(Uxn,vhx)=(Qn,vhx),vh∈Vh,(7)(Qn-Qn-1τ,wh)+((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)Qxn+Qxn-12,whx)+(fn-(1/2),whx)=0,wh∈Wh,
where
(8)δn,i={τ,i=1,2,…,n-1,τ2,i=n,
and U0, Q0 are to be defined later.The existence and uniqueness of the discrete solution for the above problems can be guaranteed by the theory presented in [18, page 237–239].To discretize the time integration we used the following integroformula:(9)∫0tn-(1/2)‍g(τ)ds≈∑i=1n‍δn,igi-(1/2).
Its truncation error can be estimated as follows:
(10)|∫0tn-(1/2)‍g(τ)ds-∑i=1n‍δn,ign-(1/2)|≤C(τ∫tn-1tn‍|gt|+τ2∫0tn‍|gtt|ds).


## 3. Error Analysis

### 3.1. Preliminaries

We begin by recalling some preliminary knowledge that will be used in the following convergence analysis.We define the following elliptic projections:u~h(t)∈Vh, σ~h(t)∈Wh, which satisfy
(11)(ux-u~hx,vhx)=0,∀vh∈Vh,(σx-σ~hx,whx)+α(σ-σ~h,wh)=0,∀wh∈Wh.
Here α is chosen to guarantee the H1-coercivity of the bilinear form in the second equations. Moreover, it is easy to check that the bilinear form is bounded.Letη=u-u~h, ρ=σ-σ~h; then η and ρ satisfy the following estimates from [19]:
(12)∥η(t)∥j+∥ηt(t)∥j≤Chk+1-j(∥u∥k+1+∥ut∥k+1),j=0,1,∥ρ(t)∥j+∥ρt(t)∥j≤Chr+1-j(∥q∥r+1+∥qt∥r+1),j=0,1.To derive the error estimates we also need the following discrete Gronwall inequality.Lemma 1 (discrete Gronwall inequality; see [20]).
Letτ,B1,C1>0 and let an, bn, cn, and dn be sequences of nonnegative numbers satisfying
(13)∀n≥0,an+τ∑i=0n‍bi≤B1+C1τ∑i=0n‍ai+τ∑i=0n‍ci.
Then, if C1τ<1,
(14)∀n≥0,an+τ∑i=0n‍bi≤eC1(n+1)τ(B1+τ∑i=0n‍ci).


### 3.2. Error Analysis

To estimate the errors, we firstly decompose the errors into(15)u(tn)-Un=u(tn)-u~h(tn)+u~h(tn)-Un=ηn+ζn,σ(tn)-Qn=σ(tn)-σ~h(tn)+σ~h(tn)-Qn=ρn+ξn.
Note that the estimates of ηn and ρn can be found out easily from (12) at t=tn. Therefore it remains to estimate ζn and ξn.Settingt=tn-(1/2) in (4) and combining (6) and (7) with auxiliary projections, we deduce the following error equations with respect to ζn and ξn:
(16)(ζxn,vhx)=(ρn,vhx)+(ξn,vhx),(17)(ξn-ξn-1τ,wh)+(ξxn+ξxn-12,whx)=(σn-σn-1τ-σtn-(1/2),wh)-(ρn-ρn-1τ,wh)+α(ρn+ρn-12,wh)-(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)vvvvvvvvv×Qxn+Qxn-12,whx(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))+(σxn+σxn-12-σxn-(1/2),whx).Theorem 2.
Suppose thatU0=u~h(0), Q0=σ~h(0), and 1≤J≤N. Then there exists a positive constant C independent of h and τ such that for sufficiently small τ(18)∥uJ-UJ∥+h∥uJ-UJ∥1+∥σJ-QJ∥≤C(hmin⁡{k+1,r+1}+τ2).
Here k,r≥1 are positive integers.Proof.
Choosingvh=ζn in (16) yields
(19)(ζxn,ζxn)=(ρn,ζxn)+(ξn,ζxn),
which implies
(20)∥ζxn∥≤∥ρn∥+∥ξn∥.
Setting wh=(ξn+ξn-1)/2 in (17) gives
(21)(ξn-ξn-1τ,ξn+ξn-12)+(ξxn+ξxn-12,ξxn+ξxn-12)+(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)vvvvvvvvv×Qxn+Qxn-12,ξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))=(σn-σn-1τ-σtn-(1/2),ξn+ξn-12)-(ρn-ρn-1τ,ξn+ξn-12)+α(ρn+ρn-12,ξn+ξn-12)+(σxn+σxn-12-σxn-(1/2),ξxn+ξxn-12).
For the third term on the left side we have(22)(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×Qxn+Qxn-12,ξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))=((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ξxn+ξxn-12,ξxn+ξxn-12)+((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvvvvvvξxn+ξxn-12(λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2))+((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn-(1/2)-σxn+σxn-12),vvvvvvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn-(1/2)-σxn+σxn-12))-α((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12,ξn+ξn-12).
Then we obtain
(23)12τ(∥ξn∥2-∥ξn-1∥2)+(1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×∥ξxn+ξxn-12∥2=-((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvvvvvξxn+ξxn-12(λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2))+((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)),vvvvvvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)))+(σn-σn-1τ-σtn-(1/2),ξn+ξn-12)-(ρn-ρn-1τ,ξn+ξn-12)+α((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12,vvvvvvvvvvvξn+ξn-12(1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12)+(σxn+σxn-12-σhxn-(1/2),ξxn+ξxn-12).
Now it remains to estimate the terms on the right side of (23). It is easy to prove that
(24)∥ρn-ρn-1τ∥2≤C1τ∫tn-1tn‍∥ρt∥2ds,∥σn-σn-1τ-σtn-(1/2)∥2≤Cτ3∫tn-1tn‍∥σttt∥2ds,∥σxn+σxn-12-σxn-(1/2)∥2≤Cτ3∫tn-1tn‍∥σxttt∥2ds.
Here the Taylor formula with integral remainder was used.
Note that(25)|λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)|=|∫01‍(∫0tn-(1/2)‍σ2ds-∑i=1n‍δn,i(Qi)2+(Qi-1)22)dx|≤|∫01‍∫0tn-(1/2)‍(σ2-σ~h2)dsdx|+|∫01‍(∫0tn-(1/2)‍σ~h2ds-∑i=1n‍δn,i(σ~hi)2+(σ~hi-1)22)dx|+|∫01‍∑i=1n‍δn,i((σ~hi)2+(σ~hi-1)22-(Qi)2+(Qi-1)22)dx|∶=I1+I2+I3.
By Hölder inequality and the bound of the projection σ~h we have
(26)I1=|∫01‍∫0tn-(1/2)‍(σ2-σ~h2)dsdx|≤∥σ-σ~h∥L2(0,tn;L2(I))(∥σ∥L2(0,tn;L2(I))+∥σ~h∥L2(0,tn;L2(I)))≤C∥ρ∥L2(0,tn;L2(I)).
Using the error formula (10) and the bound of the projection σ~h we derive
(27)I2=|∫01‍(∫0tn-(1/2)‍σ~h2ds-∑i=1n‍δn,i(σ~hi-(1/2))2)dx|≤C|(∫01‍τ∫tn-1tn‍ddt(σ~h2)ds+τ2∫0tn‍d2dt2(σ~h2)ds)dx|=C|∫01‍(τ∫tn-1tn‍2σ~hσ~htds+τ2∫0tn‍(2σ~ht2+2σ~htσ~htt)ds)dx|≤Cτ∫tn-1tn‍∥σ~h∥∥σ~ht∥dsvvvvvvvvv+Cτ2(∥σ~ht∥L2(0,tn;L2(I))vvvvvvvvvvvvvvvvv+∥σ~h∥L2(0,tn;L2(I))∥σ~htt∥L2(0,tn;L2(I)))≤C(τ3/2(∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds)1/2+τ2).
For I3 we have
(28)I3=12|∫01‍∑i=1n‍δn,i((σ~hi)2-(Qi)2+(σ~hi-1)2-(Qi-1)2)dx|=12|∫01‍∑i=1n‍δn,i((σ~hi-Qi)(σ~hi+Qi)vvvvvvvvvvvvvv+(σ~hi-1-Qi-1)(σ~hi-1+Qi-1))dx12|∫01‍∑i=1n‍δn,i((σ~hi-Qi)(σ~hi+Qi)|≤12∑i=1n‍δn,i(∥ξi∥(∥σ~hi∥+∥Qi∥)+∥ξi-1∥(∥σ~hi-1∥+∥Qi-1∥)).
To bound I3 we need to derive the boundness of Qi. From (7) we can deduce
(29)((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)Qxn+Qxn-12,Qxn+Qxn-12)+(Qn-Qn-1τ,Qn+Qn-12)+(fn-(1/2),Qxn+Qxn-12)=0.
By ɛ inequality we have
(30)12τ(∥Qn∥2-∥Qn-1∥2)+(12+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×∥Qxn+Qxn-12∥2≤∥fn-(1/2)∥2.
Then, multiplying by 2τ and summing from 1 to l we conclude
(31)∥Ql∥2≤τ∑i=1l‍∥fn-(1/2)∥2+∥Q0∥2,
which implies Ql is bounded.
Combining (31) with the above estimate of I3 and using the boundness of σ~h, we can get
(32)I3≤Cτ∑i=1n‍∥ξi∥,
where ξ0=0 was used.
Collecting the above estimates forI1~I3 and using ɛ inequality, we obtain
(33)|((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvξxn+ξxn-12((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),)|≤C(∥ρ∥L2(0,tn;L2(I))+τ3/2(∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds)1/2+τ2)×∥ξxn+ξxn-12∥+τ∑i=1n‍∥ξi∥∥ξxn+ξxn-12∥≤C(∥ρ∥L2(0,tn;L2(I))2+τ3∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds+τ4)+τ∑i=1n‍∥ξi∥2+ɛ∥ξxn+ξxn-12∥2≤C(∥ρ∥L2(0,tn;L2(I))2+τ3∫tn-1tn‍∥σ~h∥2∥σ~ht∥2dsvvvvvv+τ4+τ∑i=1n‍∥ξi∥2)+ɛ∥ξxn+ξxn-12∥2,|((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)),vvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))|≤C∥σxn+σxn-12-σxn-(1/2)∥∥ξxn+ξxn-12∥≤C∥σxn+σxn-12-σxn-(1/2)∥2+ɛ∥ξxn+ξxn-12∥2≤Cτ3∫tn-1tn‍∥σxttt∥2ds+ɛ∥ξxn+ξxn-12∥2.
Inserting (24) and (33) into (23) leads to
(34)12τ(∥ξn∥2-∥ξn-1∥2)≤C(∑i=1n‍∥ξi∥2∥ρ∥L2(0,tn;L2(I))2+∥ρn∥2+∥ρn-1∥2vvvvvvvv+τ3∫tn-1tn‍∥σ~h∥2ds+τ4+τ∑i=1n‍∥ξi∥2)+C1τ∫tn-1tn‍∥ρt∥2ds+Cτ3∫tn-1tn‍∥σxttt∥2ds+Cτ3∫tn-1tn‍∥σttt∥2ds+C∥ξn+ξn-12∥2.
Multiplying by 2τ and summing from 1 to J leads to
(35)∥ξJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+Cτ4∫0tJ‍∥σ~h∥2ds+Cτ4+C∫0tJ‍∥ρt∥2ds+Cτ4∫0tJ‍∥σxttt∥2ds+Cτ4∫0tJ‍∥σttt∥2ds+Cτ2∑n=1J‍∑i=1n‍∥ξi∥2+Cτ∑n=1J‍∥ξn+ξn-12∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ∑n=1J‍∥ξn∥2+Cτ4(∫0tJ‍∥σ~h∥2ds+∫0tJ‍∥σxttt∥2dsvvvvvvvvvvv+∫0tJ‍∥σttt∥2ds+1)≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ∑n=1J‍∥ξn∥2+Cτ4.
Taking τ1, let 0<τ≤τ1, such that 1-Cτ>0; then by discrete Gronwall’s lemma we obtain that
(36)∥ξJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ4.
Substituting (36) into (20) yields
(37)∥ζxJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ4
Combining (36), (37), and the estimates of ηn, ρn and using the triangle inequality, we can complete the proof.


## 3.1. Preliminaries

We begin by recalling some preliminary knowledge that will be used in the following convergence analysis.We define the following elliptic projections:u~h(t)∈Vh, σ~h(t)∈Wh, which satisfy
(11)(ux-u~hx,vhx)=0,∀vh∈Vh,(σx-σ~hx,whx)+α(σ-σ~h,wh)=0,∀wh∈Wh.
Here α is chosen to guarantee the H1-coercivity of the bilinear form in the second equations. Moreover, it is easy to check that the bilinear form is bounded.Letη=u-u~h, ρ=σ-σ~h; then η and ρ satisfy the following estimates from [19]:
(12)∥η(t)∥j+∥ηt(t)∥j≤Chk+1-j(∥u∥k+1+∥ut∥k+1),j=0,1,∥ρ(t)∥j+∥ρt(t)∥j≤Chr+1-j(∥q∥r+1+∥qt∥r+1),j=0,1.To derive the error estimates we also need the following discrete Gronwall inequality.Lemma 1 (discrete Gronwall inequality; see [20]).
Letτ,B1,C1>0 and let an, bn, cn, and dn be sequences of nonnegative numbers satisfying
(13)∀n≥0,an+τ∑i=0n‍bi≤B1+C1τ∑i=0n‍ai+τ∑i=0n‍ci.
Then, if C1τ<1,
(14)∀n≥0,an+τ∑i=0n‍bi≤eC1(n+1)τ(B1+τ∑i=0n‍ci).


## 3.2. Error Analysis

To estimate the errors, we firstly decompose the errors into(15)u(tn)-Un=u(tn)-u~h(tn)+u~h(tn)-Un=ηn+ζn,σ(tn)-Qn=σ(tn)-σ~h(tn)+σ~h(tn)-Qn=ρn+ξn.
Note that the estimates of ηn and ρn can be found out easily from (12) at t=tn. Therefore it remains to estimate ζn and ξn.Settingt=tn-(1/2) in (4) and combining (6) and (7) with auxiliary projections, we deduce the following error equations with respect to ζn and ξn:
(16)(ζxn,vhx)=(ρn,vhx)+(ξn,vhx),(17)(ξn-ξn-1τ,wh)+(ξxn+ξxn-12,whx)=(σn-σn-1τ-σtn-(1/2),wh)-(ρn-ρn-1τ,wh)+α(ρn+ρn-12,wh)-(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)vvvvvvvvv×Qxn+Qxn-12,whx(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))+(σxn+σxn-12-σxn-(1/2),whx).Theorem 2.
Suppose thatU0=u~h(0), Q0=σ~h(0), and 1≤J≤N. Then there exists a positive constant C independent of h and τ such that for sufficiently small τ(18)∥uJ-UJ∥+h∥uJ-UJ∥1+∥σJ-QJ∥≤C(hmin⁡{k+1,r+1}+τ2).
Here k,r≥1 are positive integers.Proof.
Choosingvh=ζn in (16) yields
(19)(ζxn,ζxn)=(ρn,ζxn)+(ξn,ζxn),
which implies
(20)∥ζxn∥≤∥ρn∥+∥ξn∥.
Setting wh=(ξn+ξn-1)/2 in (17) gives
(21)(ξn-ξn-1τ,ξn+ξn-12)+(ξxn+ξxn-12,ξxn+ξxn-12)+(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)vvvvvvvvv×Qxn+Qxn-12,ξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))=(σn-σn-1τ-σtn-(1/2),ξn+ξn-12)-(ρn-ρn-1τ,ξn+ξn-12)+α(ρn+ρn-12,ξn+ξn-12)+(σxn+σxn-12-σxn-(1/2),ξxn+ξxn-12).
For the third term on the left side we have(22)(λn-(1/2)σxn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×Qxn+Qxn-12,ξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))=((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ξxn+ξxn-12,ξxn+ξxn-12)+((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvvvvvvξxn+ξxn-12(λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2))+((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn-(1/2)-σxn+σxn-12),vvvvvvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn-(1/2)-σxn+σxn-12))-α((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12,ξn+ξn-12).
Then we obtain
(23)12τ(∥ξn∥2-∥ξn-1∥2)+(1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×∥ξxn+ξxn-12∥2=-((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvvvvvξxn+ξxn-12(λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2))+((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)),vvvvvvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)))+(σn-σn-1τ-σtn-(1/2),ξn+ξn-12)-(ρn-ρn-1τ,ξn+ξn-12)+α((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12,vvvvvvvvvvvξn+ξn-12(1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)ρn+ρn-12)+(σxn+σxn-12-σhxn-(1/2),ξxn+ξxn-12).
Now it remains to estimate the terms on the right side of (23). It is easy to prove that
(24)∥ρn-ρn-1τ∥2≤C1τ∫tn-1tn‍∥ρt∥2ds,∥σn-σn-1τ-σtn-(1/2)∥2≤Cτ3∫tn-1tn‍∥σttt∥2ds,∥σxn+σxn-12-σxn-(1/2)∥2≤Cτ3∫tn-1tn‍∥σxttt∥2ds.
Here the Taylor formula with integral remainder was used.
Note that(25)|λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)|=|∫01‍(∫0tn-(1/2)‍σ2ds-∑i=1n‍δn,i(Qi)2+(Qi-1)22)dx|≤|∫01‍∫0tn-(1/2)‍(σ2-σ~h2)dsdx|+|∫01‍(∫0tn-(1/2)‍σ~h2ds-∑i=1n‍δn,i(σ~hi)2+(σ~hi-1)22)dx|+|∫01‍∑i=1n‍δn,i((σ~hi)2+(σ~hi-1)22-(Qi)2+(Qi-1)22)dx|∶=I1+I2+I3.
By Hölder inequality and the bound of the projection σ~h we have
(26)I1=|∫01‍∫0tn-(1/2)‍(σ2-σ~h2)dsdx|≤∥σ-σ~h∥L2(0,tn;L2(I))(∥σ∥L2(0,tn;L2(I))+∥σ~h∥L2(0,tn;L2(I)))≤C∥ρ∥L2(0,tn;L2(I)).
Using the error formula (10) and the bound of the projection σ~h we derive
(27)I2=|∫01‍(∫0tn-(1/2)‍σ~h2ds-∑i=1n‍δn,i(σ~hi-(1/2))2)dx|≤C|(∫01‍τ∫tn-1tn‍ddt(σ~h2)ds+τ2∫0tn‍d2dt2(σ~h2)ds)dx|=C|∫01‍(τ∫tn-1tn‍2σ~hσ~htds+τ2∫0tn‍(2σ~ht2+2σ~htσ~htt)ds)dx|≤Cτ∫tn-1tn‍∥σ~h∥∥σ~ht∥dsvvvvvvvvv+Cτ2(∥σ~ht∥L2(0,tn;L2(I))vvvvvvvvvvvvvvvvv+∥σ~h∥L2(0,tn;L2(I))∥σ~htt∥L2(0,tn;L2(I)))≤C(τ3/2(∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds)1/2+τ2).
For I3 we have
(28)I3=12|∫01‍∑i=1n‍δn,i((σ~hi)2-(Qi)2+(σ~hi-1)2-(Qi-1)2)dx|=12|∫01‍∑i=1n‍δn,i((σ~hi-Qi)(σ~hi+Qi)vvvvvvvvvvvvvv+(σ~hi-1-Qi-1)(σ~hi-1+Qi-1))dx12|∫01‍∑i=1n‍δn,i((σ~hi-Qi)(σ~hi+Qi)|≤12∑i=1n‍δn,i(∥ξi∥(∥σ~hi∥+∥Qi∥)+∥ξi-1∥(∥σ~hi-1∥+∥Qi-1∥)).
To bound I3 we need to derive the boundness of Qi. From (7) we can deduce
(29)((1+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)Qxn+Qxn-12,Qxn+Qxn-12)+(Qn-Qn-1τ,Qn+Qn-12)+(fn-(1/2),Qxn+Qxn-12)=0.
By ɛ inequality we have
(30)12τ(∥Qn∥2-∥Qn-1∥2)+(12+∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)×∥Qxn+Qxn-12∥2≤∥fn-(1/2)∥2.
Then, multiplying by 2τ and summing from 1 to l we conclude
(31)∥Ql∥2≤τ∑i=1l‍∥fn-(1/2)∥2+∥Q0∥2,
which implies Ql is bounded.
Combining (31) with the above estimate of I3 and using the boundness of σ~h, we can get
(32)I3≤Cτ∑i=1n‍∥ξi∥,
where ξ0=0 was used.
Collecting the above estimates forI1~I3 and using ɛ inequality, we obtain
(33)|((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),vvvvvξxn+ξxn-12((λn-(1/2)-(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))σxn-(1/2),)|≤C(∥ρ∥L2(0,tn;L2(I))+τ3/2(∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds)1/2+τ2)×∥ξxn+ξxn-12∥+τ∑i=1n‍∥ξi∥∥ξxn+ξxn-12∥≤C(∥ρ∥L2(0,tn;L2(I))2+τ3∫tn-1tn‍∥σ~h∥2∥σ~ht∥2ds+τ4)+τ∑i=1n‍∥ξi∥2+ɛ∥ξxn+ξxn-12∥2≤C(∥ρ∥L2(0,tn;L2(I))2+τ3∫tn-1tn‍∥σ~h∥2∥σ~ht∥2dsvvvvvv+τ4+τ∑i=1n‍∥ξi∥2)+ɛ∥ξxn+ξxn-12∥2,|((∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22)(σxn+σxn-12-σxn-(1/2)),vvvvξxn+ξxn-12(∑i=1n‍δn,i∥Qi∥2+∥Qi-1∥22))|≤C∥σxn+σxn-12-σxn-(1/2)∥∥ξxn+ξxn-12∥≤C∥σxn+σxn-12-σxn-(1/2)∥2+ɛ∥ξxn+ξxn-12∥2≤Cτ3∫tn-1tn‍∥σxttt∥2ds+ɛ∥ξxn+ξxn-12∥2.
Inserting (24) and (33) into (23) leads to
(34)12τ(∥ξn∥2-∥ξn-1∥2)≤C(∑i=1n‍∥ξi∥2∥ρ∥L2(0,tn;L2(I))2+∥ρn∥2+∥ρn-1∥2vvvvvvvv+τ3∫tn-1tn‍∥σ~h∥2ds+τ4+τ∑i=1n‍∥ξi∥2)+C1τ∫tn-1tn‍∥ρt∥2ds+Cτ3∫tn-1tn‍∥σxttt∥2ds+Cτ3∫tn-1tn‍∥σttt∥2ds+C∥ξn+ξn-12∥2.
Multiplying by 2τ and summing from 1 to J leads to
(35)∥ξJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+Cτ4∫0tJ‍∥σ~h∥2ds+Cτ4+C∫0tJ‍∥ρt∥2ds+Cτ4∫0tJ‍∥σxttt∥2ds+Cτ4∫0tJ‍∥σttt∥2ds+Cτ2∑n=1J‍∑i=1n‍∥ξi∥2+Cτ∑n=1J‍∥ξn+ξn-12∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ∑n=1J‍∥ξn∥2+Cτ4(∫0tJ‍∥σ~h∥2ds+∫0tJ‍∥σxttt∥2dsvvvvvvvvvvv+∫0tJ‍∥σttt∥2ds+1)≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ∑n=1J‍∥ξn∥2+Cτ4.
Taking τ1, let 0<τ≤τ1, such that 1-Cτ>0; then by discrete Gronwall’s lemma we obtain that
(36)∥ξJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ4.
Substituting (36) into (20) yields
(37)∥ζxJ∥2≤C∥ρ∥L∞(0,tJ;L2(I))2+C∫0tJ‍∥ρt∥2ds+Cτ4
Combining (36), (37), and the estimates of ηn, ρn and using the triangle inequality, we can complete the proof.


## 4. Numerical Example

In this section a numerical example is carried out to verify the theorems presented in this paper.Example 1.
Let us consider the initial and boundary problem (1) with the initial value u(x,0)=x(1-x)sin(x) and the right hand term
(38)f=x(1-x)cos⁡(x+t)-(1+1160t-18cos⁡(t)sin(t)vvvv-18cos⁡(t+1)sin(t+1)+18cos⁡(1)sin(1))×(-2sin(x+t)+2(1-x)cos⁡(x+t)vvvv-2xcos⁡(x+t)-x(1-x)sin(x+t)).
This example is taken from [11].In this example we use piecewise linear finite element spaces to approximate the unknown functionsu and σ, respectively. The Crank-Nicolson method is used to approximate the time derivative. Then the corresponding error estimates reduce to
(39)∥uJ-UJ∥+h∥uJ-UJ∥1+∥σJ-QJ∥≤C(h2+τ2).
In the numerical implementation we choose h=τ. The errors and the corresponding rate of convergence for uJ-UJ and σJ-QJ are displayed in Tables 1, 2, and 3, respectively.Table 1
The errors of∥uJ-UJ∥ at different time.
h
=
τ
Time
t
=
0.2
t
=
0.4
t
=
0.6
t
=
0.8
Error
Order
Error
Order
Error
Order
Error
Order
1
/
20
1.0174
e
-
004
∖
1.1819
e
-
004
∖
1.2690
e
-
004
∖
1.2445
e
-
004
∖
1
/
40
2.5472
e
-
005
1.9979
2.9553
e
-
005
1.9997
3.1723
e
-
005
2.0001
3.1112
e
-
005
2.0000
1
/
80
6.3711
e
-
006
1.9993
7.3900
e
-
006
1.9997
7.9329
e
-
006
1.9996
7.7813
e
-
006
1.9994Table 2
The errors of∥uxJ-UxJ∥ at different time.
h
=
τ
Time
t
=
0.2
t
=
0.4
t
=
0.6
t
=
0.8
Error
Order
Error
Order
Error
Order
Error
Order
1
/
20
0.0020
∖
0.0024
∖
0.0025
∖
0.0025
∖
1
/
40
0.0010
1.0000
0.0012
1.0000
0.0013
0.9434
0.0012
1.0589
1
/
80
5.0969
e
-
04
0.9723
5.9120
e
-
04
1.0213
6.3463
e
-
04
1.0345
6.2251
e
-
04
0.9469Table 3
The errors of∥σJ-QJ∥ at different time.
h
=
τ
Time
t
=
0.2
t
=
0.4
t
=
0.6
t
=
0.8
Error
Order
Error
Order
Error
Order
Error
Order
1
/
20
1.9532
e
-
004
∖
3.5339
e
-
004
∖
5.4745
e
-
004
∖
7.6700
e
-
004
∖
1
/
40
4.7504
e
-
005
2.0397
8.6496
e
-
005
2.0306
1.3441
e
-
004
2.0261
1.8884
e
-
004
2.0221
1
/
80
1.1699
e
-
005
2.0217
2.1381
e
-
005
2.0163
3.3284
e
-
005
2.0137
4.6818
e
-
005
2.0120We can observe that the numerical results are in agreement with our theoretical results proposed in Section3.



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