Split least-squares finite element methods for linear and nonlinear parabolic problems

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Journal of Computational and Applied

Mathematics

journal homepage:www.elsevier.com/locate/cam

Split least-squares finite element methods for linear and nonlinear

parabolic problems

I

Hongxing Rui

a,∗

_{, Sang Dong Kim}

b

_{, Seokchan Kim}

c

a_{School of Mathematics, Shandong University, Jinan, Shandong, 250100, China}

b_{Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea}

c_{Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea}

a r t i c l e i n f o

Article history:

Received 9 June 2007

Received in revised form 5 March 2008

MSC: 65M12 65M15 65M60 Keywords: Split Least squares Finite element Error estimates Parabolic problem a b s t r a c t

In this paper, we propose some least-squares finite element procedures for linear and nonlinear parabolic equations based on first-order systems. By selecting the least-squares functional properly each proposed procedure can be split into two independent symmetric positive definite sub-procedures, one of which is for the primary unknown variableuand the other is for the expanded flux unknown variable_σ. Optimal order error estimates are developed. Finally we give some numerical examples which are in good agreement with the theoretical analysis.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The purpose of this paper is to consider the least-squares finite element procedures for linear and nonlinear parabolic problems written as first-order systems. It is well known that, compared to mixed element methods, the least-squares finite element method has two typical advantages as follows: it is not subject to the Ladyzhenkaya–Babuska–Brezzi [13, 1,4] consistency condition, so the choice of approximation spaces becomes flexible, and it results in a symmetric positive definite system.

Least-squares finite element methods for elliptic problems, based on first-order systems, were introduced by [12] where a least-squares residual minimization is introduced for the mixed system in primary unknown variableuand expanded unknown flux σ. Then an elegant theory for least-squares finite element approximation for general elliptic boundary value problems was established, see, for example, [12,10,11,16,5,6] and the references therein. Concerning the parabolic problems, [14] and [15] introduced the least-squares finite element procedure with semi-discretization in time and fully discrete scheme. They also established the a posterior error estimates and constructed adaptive algorithms.

In this paper we consider the least-squares finite element procedures for linear and nonlinear parabolic problems. Like [14,15] we define the least-squares functionals using weight-factors. By selecting different weight-factors we get different

I_{The work was supported by the Korea Research Foundation under contract number KRF-2005-070-C00017, the National Natural Science Foundation} of China Grant No. 10771124 and the Research Fund for Doctoral Program of High Education by the State Education Ministry of China No. 20060422006.

∗_{Corresponding author.}

E-mail addresses:[email protected](H. Rui),[email protected](S.D. Kim),[email protected](S. Kim). 0377-0427/$ – see front matter©2008 Elsevier B.V. All rights reserved.

procedures. We show that all the procedures presented in this paper can be divided into two independent sub-procedures, one of which is for the primary unknown variableuand the other is for the expanded fluxσ. The key point used to explain the split of the procedure isLemma 2.1which was obtained by integration by parts. Similar results have been found and used by [8] to prove the coercivity of least-squares bilinear formats and by [2,3] to establish connections between least-squares and mixed methods. The last two papers also show that not only is the pressure the same as in the Galerkin method, but also the flux is the same as in the mixed method under some conditions on the finite element spaces.

In this paper three procedures were presented for linear parabolic problems. In the first procedure the sub-procedure for the primary unknownuis the same as the standard Galerkin finite element procedure. In the second procedure one of the sub-procedures is for the expanded fluxσonly. The third one is a procedure with second-order approximation in time increment. We give one procedure to deal with the nonlinear problem. For these schemes we give the optimal order error estimates. Finally we give some numerical examples.

The remainder of this paper is organized as follows. In Section2we introduce the split least-squares schemes for linear problems. In Section3, we establish the optimal order error estimates. In Section4, we give a least-squares finite element procedure for nonlinear problems. Finally in Section5we give some numerical examples.

Throughout this paper, the notations of standard Sobolev spaces L2

₍

_Ω

₎

_{, H}k

₍

_Ω

₎

_{and associated normsk · k = k · k}

L2_(Ω), k · k_k= k · k_Hk_(Ω)are adopted as those in [7]. For simplicity we usek · k_L∞(Hm+1₎andk · k_L2(Hm+1₎to representk · k_L∞(J;(Hm+1_(Ω))d₎

andk · k_L2(J;(Hm+1_(Ω))d₎respectively forJ =

(

0

,

T

)

andd ≤ 3. A constantC(with or without subscript) stands for a generic

positive constant independent of the mesh parameterhu,hσand

1

t, it may be different at different occurrence.

2. Least-squares procedure for linear problems

In this section we present three least-squares finite element procedures for linear problems. For simplicity we just consider the homogeneous boundary condition. The same idea can be used to deal with problems with non-homogeneous boundary condition.

Consider the following parabolic problem on a bounded domainΩ⊂Rd

,

_d=2

,

_3:

  

φ

ut−div

(A∇

u

)

=f

,

inΩ×J

,

u=0

,

onΓD×J

,

A∇u·n=0 onΓN×J

,

(2.1) subject to the initial condition

u

(

x

,

0

)

=u₀

(

x

)

onΩ×J

,

(2.2)

where

∂

Ω = ΓD∩ΓN, n is the outward unit normal vector,J =

(

0

,

T]is the time interval and

φ

is a continuous function

satisfying

φ

1 ≤

φ

≤

φ

2with two positive constants

φ

1and

φ

2. We further assume thatA =

(

aij

(

x

))

di,j=1is a bounded, symmetric and positive definite matrix inΩ, i.e., there exist positive constants

α

and

β

such that,

α

kξk2≤

(

Aξ,ξ)≤

β

kξk2

,

∀ξ ∈ Rd

.

(2.3)

In some applications, the problem(2.1)appears as a first-order system for bothuandσ = −A∇u

,

σ =

(σ

1

, . . . ,

σd

)

, as

follows:       

φ

ut+ divσ −f =0

,

inΩ×J

,

σ + A∇u=0

,

inΩ×J

,

u=0

,

onΓD×J

,

σ ·n=0 onΓN×J

.

(2.4)

For example, in the compressible miscible displacement problem [9],urepresents the pressure and_σrepresents the Darcy velocity or flux. In this case the approximations to bothuandσare necessary. We consider the least-squares mixed element approximations for(2.4).

First we consider the first-order approximation in time increment. Let

1

t be a time increment. With tn _{= n}

₁

_t,

un_=u

(

_tn

,

_·

)

_{, put}

δ

tun:= un_−un−1

1

t

,

ρ

n 1:=

φ(δ

tun−unt

).

(2.5) It is clear that

ρ

n 1=O Z tn tn−1 kuttkdt ! =O   4t Z tn tn−1 kuttk2dt !1₂ 

.

(2.6)

Define two function spaces V= {v∈H1

₍

_Ω

₎

_{:v=0 onΓ}

D}

,

(2.7)

From(2.4)we know that forn≥1,

(

un

_,

_σn

₎

_{∈V×W satisfy that} (

φ

−1 2

(φ

un+

1

tdivσn_−Fn 1

)

=0

,

inΩ×J

,

A−1₂

_(σ

n_+A∇un

₎

₌₀

_,

_inΩ×J

_,

(2.9) whereFn 1=

φ

un−1+

1

tfn+

1

t

ρ

n1.

For

(

v

,

τ

)

∈V×W, define the first kind of least-squares functionalJn

1

(

v

,

τ

)

as follows.

Jn₁

(

v

,

τ

)

= k

φ

−12

(φ

_v+

1

tdivτ −F₁n

)

k2+

1

tkA−1₂

_{(τ +}

A∇v

)

k2

.

(2.10)

The least-squares minimization problem corresponding to(2.9)is: find

(

un

,

_σn

)

_∈V×σn_{∈W such that}

Jn₁

(

un

,

σn

)

= inf (v,τ)∈V×WJ

n

1

(

v

,

τ). (2.11)

Define the bilinear forma

(

u

,

σ;v

,

τ

)

corresponding to the least-squares functionalJn

1as

a

(

u

,

σ;v

,

τ)=

(φ

−1

(φ

u+

1

tdivσ

), φ

v+

1

tdivτ

)

+

1

t

(

A−1

(σ + A∇

u

),

τ + A∇v

).

(2.12) The weak statement of the minimization problem(2.11)becomes: find

(

un

,

σn

)

∈V×W such that

a

(

un

,

σn;v

,

τ

)

=

(φ

−1F₁n

, φ

v+

1

tdivτ

),

∀

(

v

,

τ

)

∈V×W

.

(2.13) Noticing the definition ofFn

1,(2.13)becomes

a

(

un

,

σn;v

,

τ

)

=

(

un−1+

1

t

φ

−1

(

fn+

ρ

n₁

), φ

v+

1

tdivτ

),

∀

(

v

,

τ

)

∈V×W

.

(2.14) Now we consider the second weak formulation different from(2.13). From(2.4)we have that forn≥1,

(

un

,

_σn

)

_∈V×W

satisfy that (

φ

−1₂

_(φ

un+

1

tdivσn_−Fn 1

)

=0

,

inΩ×J

,

A−1₂

_(σ

n_+A∇un_−Gn

₎

₌₀

_,

_in Ω×J

,

(2.15)

whereGn_=σn−1_+A∇un−1_{. For}

₍

_v

_,

_τ

₎

_{∈V×W, define the second kind of least-squares functionalJ}n

2

(

v

,

τ

)

as follows.

Jn₂

(

v

,

τ

)

= k

φ

−12

(φ

_v+

1

tdivτ −Fn

)

k2+

1

tkA−1₂

_{(τ + A∇}

v−Gn

)

k2

.

(2.16)

The least-squares minimization problem corresponding to(2.15)is: find

(

un

,

_σn

)

_{∈V×W such that}

Jn₂

(

un

,

σn

)

= inf (v,τ)∈V×WJ

n

2

(

v

,

τ). (2.17)

Similarly to(2.14), the weak statement of(2.17)is: find

(

un

,

_σn

)

_{∈V×W such that}

a

(

un

,

σn;v

,

τ

)

=

(

un−1+

1

t

φ

−1

(

fn+

ρ

n₁

), φ

v+

1

tdivτ

),

+

1

t

(A

−1σn−1+ ∇un−1

,

τ + A∇v

)

∀

(

v

,

τ

)

∈V×W

.

(2.18)

In order to approximate the formulations(2.14)and(2.18), we need to construct the finite element spaces. LetThuandThσ

be two families of regular finite element partitions of the domainΩ, which are either identical or not. Lethuandhσdenote

the largest of the diameters of the element inThuandThσ respectively. Based onThuandThσ, respectively, we construct the

finite element spaces Vh⊂Vand Wh⊂W with the following approximation properties:

inf vh∈Vh {kv−vhk +huk∇

(

v−vh

)

k} ≤Chmu+1kvkm+1

,

(2.19) inf τh∈Wh kτ − τ_hk ≤Chk+1 σ kτkk+1

,

(2.20) inf τh∈Wh kdiv

(τ − τ

h

)

k ≤Chk1_σkτkk1+1

,

(2.21)

forv∈V∩Hm+1

₍

_Ω

₎

_{andτ ∈W∩}

₍

_Hk1+1

₍

_Ω

₎₎

d_{. It is clear that when assumption(2.20)holds we can deducek}

1=k, and when

Whis selected as any of the Raviart–Thomas mixed element space [17] we can choosek1=k+1. In this paper we always supposek1=k+1 when Whis any of the Raviart–Thomas mixed element space [17] andk1=kotherwise.

We select the initial approximationu0

h∈Vh,σ0h∈Whsuch that ( ku0−u0hkj≤Chm +1−j u ku0km+1

,

j=0

,

1

,

kσ₀−σ0_hk ≤Ch_σk+1kσ₀kk+1

,

(2.22) where_σ0=A∇u₀. The first least-squares finite element procedure based on(2.14)reads as follows.

Scheme (I). Forn≥1 find

(

un

h

,

σnh

)

∈Vh×Whsuch that a

(

un h

,

σ n h;vh

,

τh

)

=

(

unh−1+

1

t

φ

−1_fn

_{, φ}

_v h+

1

tdivτh

),

∀

(

vh

,

τh

)

∈Vh×Wh

.

(2.23)

Based on(2.18)the second least-squares finite element procedure reads as follows.

Scheme (II). Forn≥1 find

(

un h

,

σ n h

)

∈Vh×Whsuch that a

(

un_h

,

σn_h;vh

,

τh

)

=

(

un −1 h +

1

t

φ

−1_fn

_{, φ}

_v h+

1

tdivτh

)

+

1

t

(

A−1σn−1 h + ∇u n−1 h

,

τh+A∇vh

),

∀

(

vh

,

τh

)

∈Vh×Wh

.

(2.24)

Now let us mention about the bilinear forma

(

·

,

·; ·

,

·

)

in the following lemma, which leads to decoupled systems.

Lemma 2.1. For anyu

,

v∈V andσ

,

τ ∈Wwe have that,

a

(

u_,σ;v

,

τ

)

=

(φ

u

,

v

)

+

1

t

(A∇

u

,

∇v

)

+

1

t

(A

−1σ

,

τ

)

+

1

t2

(φ

−1divσ

,

divτ

).

(2.25)

Proof. A direct calculation shows that

a

(

u_,σ;v

,

τ

)

=

(φ

u

,

v

)

+

1

t

(

A∇u

,

∇v

)

+

1

t

(

A−1_σ

_,

_τ

₎

₊

₁

_t2

_(φ

−1_divσ

_,

_divτ

₎

+

1

t

((

u

,

divτ

)

+

(

v

,

divσ

)

+

(

∇u

,

τ

)

+

(

∇v

,

σ

)),

Integrating by parts shows that

(

u

,

div_τ

)

+

(

v

,

div_σ

)

+

(

∇u

,

τ

)

+

(

∇v

,

σ

)

=0

,

(2.26)

which completes the proof. 

UsingLemma 2.1, we have the decoupling equivalent form of each scheme (I) or (II) alternatively by putting_τh=0 and

vh=0 in(2.23)or(2.24).

Equivalent Form of Scheme (I). With the initial guess

(

u0_h

,

σ0_h

)

∈Vh×Wh, forn≥1 find

(

unh

,

σhn

)

∈Vh×Whsuch that for

allvh∈Vhandτh∈Wh

(φ

un h

,

vh

)

+

1

t

(

A∇unh

,

∇vh

)

=

(φ

unh−1+

1

tf n

_,

_v h

),

(2.27)

(A

−1_σn h

,

τh

)

+

1

t

(φ

−1_divσn h

,

divτh

)

=

(

un −1 h +

1

t

φ

−1_fn

_,

_divτ h

).

(2.28)

Equivalent Form of Scheme (II). With the initial guess

(

u0

h

,

σ0h

)

∈Vh×Wh, forn≥1 find

(

unh

,

σ n

h

)

∈Vh×Whsuch that for

allvh∈Vhandτh∈Wh

(φ

un_h

,

vh

)

+

1

t

(

A∇uhn

,

∇vh

)

=

(φ

unh−1+

1

tf n

_,

_v h

)

+

1

t

(σ

nh−1+A∇u n−1 h

,

∇vh

)

(2.29)

(

A−1σn h

,

τh

)

+

1

t

(φ

−1divσnh

,

divτh

)

=

(

A−1σn −1 h

,

τh

)

+

1

t

(φ

−1fn

,

divτh

).

(2.30)

Note that each Scheme (I) or (II) is split into two independent symmetric positive definite systems. Sub-procedure(2.27) is the same as the standard Galerkin finite element procedure for parabolic problems. Sub-procedure(2.30)is a procedure for the unknown fluxσn

hwith first-order approximation in time increment.

It clear that both problems(2.23)and(2.24)have a unique solution. Now we consider the second-order approximation in time increment. Let

ρ

n 2:=

φ



δ

tun−u n−1₂ t  +1 2div

(σ

n_+σn−1

₎

_{− divσ}n−1₂

_,

(2.31) which can be estimated as

ρ

n 2=O  

1

t 3 2 Z tn tn−1

(

|uttt|2+ |divσtt|2

)

dt !1₂ 

.

(2.32)

From(2.4)we know that forn≥1,

(

un

_,

_σn

₎

_{∈V×W satisfy that}

    

φ

−1₂ 

φ

un+

1

t 2 divσ n_−Fn 2  =0

,

inΩ×J

,

A−1₂

_(σ

n_+A∇un_−Gn

₎

₌₀

_,

_inΩ×J

_,

(2.33)

whereGn_{is the same as in(2.15),}

F₂n=

φ

un−1+

1

tfn−12 −

1

t 2 divσ

n−1₊

₁

_t

_ρ

n

2

.

(2.34)

For

(

v

,

τ

)

∈V×W, define the least-squares functionalJn

3

(

v

,

τ

)

as follows. Jn₃

(

v

,

τ

)

=

φ

−1₂ 

φ

v+

1

t 2 divτ −F n 2  2 +

1

t 2 kA −1₂

_{(τ + A∇}

v−Gn

)

k2

.

(2.35)

The least-squares minimization problem corresponding to(2.33)is: find

(

un

_,

_σn

₎

_{∈V×W such that}

Jn₃

(

un

,

σn

)

= inf

v∈V,τ∈WJ

n

3

(

v

,

τ

).

(2.36)

Define the bilinear formb

(

·

,

·; ·

,

·

)

as

b

(

u

,

σ;v

,

τ

)

=  u+

1

t 2

φ

−1_divσ

_{, φ}

_v+

1

t 2 divτ  +

1

t 2

(

A −1_{σ + ∇u}

_,

_{τ + A∇v}

_).

_(2.37)

Noticing the definition ofFn

2in(2.34), the weak statement of the minimization problem(2.36)is: find

(

un

,

σn

)

∈V×W such that b

(

un

,

σn;v

,

τ

)

=  un−1+

1

t

φ

−1  fn−12 −1 2divσ n−1₊

_ρ

n 2 

, φ

v+

1

t 2 divτ 

,

+

1

t 2

(

A −1_σn−1_{+ ∇u}n−1

_,

_{τ + A∇v}

₎

_∀

₍

_v

_,

_τ

₎

_∈V×W

_.

_(2.38) Then the corresponding least-squares finite element procedure reads as follows.

Scheme (III). With the initial guess

(

u0_h

,

σ0_h

)

∈Vh×Wh, forn≥1 find

(

unh

,

σhn

)

∈Vh×Whsuch that

b

(

un h

,

σ n h;vh

,

τh

)

=  un−1 h +

1

t

φ

−1_fn−1₂ − 1 2divσ n−1 h 

, φ

vh+

1

t 2 divτh  +

1

t 2

(

A −1_σn−1 h + ∇u n−1 h

,

τh+A∇vh

),

∀

(

vh

,

τh

)

∈Vh×Wh

.

(2.39)

Similarly toLemma 2.1we know that the following lemma holds.

Lemma 2.2. For anyu

,

v∈V andσ,τ ∈Wwe have that,

b

(

u_,σ;v

,

τ

)

=

(φ

u

,

v

)

+

1

t 2

(

A∇u

,

∇v

)

+

1

t 2

(

A −1_σ

_,

_τ

₎

₊

1

t 2 2

(φ

−1_divσ

_,

_divτ

_).

_(2.40)

UsingLemma 2.2we have a decoupling equivalent form of Scheme (III).

Equivalent Form of Scheme (III). With the initial guess

(

u0_h

,

σ0_h

)

∈Vh×Wh, forn≥1 find

(

unh

,

σhn

)

∈Vh×Whsuch that

(φ

un_h

,

vh

)

+

1

t 2

(

A∇u n h

,

∇vh

)

= 

φ

un_h−1+

1

tfn−12 −

1

t 2 divσ n−1 h

,

vh  +

1

t 2

(σ

n−1 h +A∇u n−1 h

,

∇vh

),

∀vh∈Vh

.

(2.41)

(

A−1σn h

,

τh

)

+

1

t 2

(φ

−1_divσn h

,

divτh

)

=

(

A−1σnh−1

,

τh

)

+

1

t 

φ

−1_fn−1₂ −1 2divσ n−1 h 

,

divτh 

,

∀τ_h∈Wh

.

(2.42)

Then this scheme also can be split into two independent sub-procedures. Sub-procedure(2.42)is a procedure for the unknown fluxσn

hwith second-order approximation in time increment.

Remark 2.3. Results similar toLemma 2.1orLemma 2.2have been found and used by [8] to prove the coercivity of least-squares bilinear formats and by [2,3] to establish connections between least-squares and mixed methods.

3. Error estimates

In this section we give the error estimates for the schemes described in Section2. We first discuss the error estimate for Scheme (I) in the followingTheorem 3.1.

Theorem 3.1. Suppose

(

un

h

,

σnh

)

∈ Vh × Wh is the solution of Scheme

(

I

)

. Under the assumption ku0h − u0k =

O

(

h_um+1−jku0k_Hm+1−j

),

j=0

,

1, there exists a positive constantCindependent ofhu,hσand

1

tsuch that

kun_h−unks≤Chum+1−s

(

kukL∞(Hm+1₎+ kutkL2(Hm+1₎

)

+C

1

tkuttkL2(L2)

,

s=0

,

1

,

(3.1) kσn_h−σk +

1

t12kdiv

(σ

n h−σ n

₎

_{k ≤ C}

₍

_hk+1 σ kσnkk+1+

1

t 1 2_hk1 σkσnkk1+1+

1

tkuttkL2(L2)

)

+Cmin{hm_u

, 1

t−12_hm+1 u }

(

kukL∞(Hm+1₎+ kutkL2(Hm+1₎

).

(3.2)

Proof. Since Scheme (I) is equivalent to(2.27)and(2.28), from the error estimates of the finite element method for parabolic problems (see [18] and [19] for example), we know that(3.1)holds.

We next considerσi h−σ

i_{, 1≤i≤n≤} T

1t. Subtracting(2.14)from(2.23)and settingvh=0, usingLemma 2.1, we have

(

A−1

(

σi h−σ i

_),

_τ h

)

+

1

t

(φ

−1div

(

σih−σ i

_),

_divτ h

)

=

(

ui −1 h −u i−1

_,

_divτ h

)

−

1

t

(φ

−1

ρ

i1

,

divτh

)

∀τh∈Wh = −

(

∇

(

ui_h−1−ui−1

),

τh

)

−

1

t

(φ

−1

_ρ

i 1

,

divτh

).

(3.3) Letσi

I∈Whbe an interpolant ofσisuch that

( kσi_I−σik ≤Ch_σk+1kσik_k+1

,

kdiv

(σ

i I−σ i

₎

_{k ≤Ch}k1 σkσikk1+1

.

(3.4) Denote by

ξ

i σ=σih−σ i I

.

(3.5)

Settingτh=

ξ

i_σ =σih−σiIin(3.3), and using the



-inequality,(2.6), we have

kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2 =

(

_A−1

(σ

i_−σi I

)

;

ξ

i σ

)

+

1

t

(φ

−1div

(σ

i−σiI

)

;div

ξ

i σ

)

−

(

∇

(

uih−1−u i−1

_{), ξ}

i σ

)

+

1

t

(φ

−1

ρ

i1

,

div

ξ

iσ

)

≤ 1 2  kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2  +C h kσi−σi_Ik2+

1

tkdiv

(σ

i_−σi I

)

k 2_{+ k∇}

₍

_ui−1 h −u i−1

₎

_k2₊

₁

_tk

_ρ

i 1k2 i ≤ 1 2  kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2  +C

1

t2kuttk2L2(L2) +Ch2(k+1) σ kσik2k+1+

1

thσ2k1kσik2k1+1+ k∇

(

u i−1 h −u i−1

₎

_k2

.

(3.6) By using −

(

∇

(

u_hi−1−ui−1

), ξ

i_σ

)

=

(

ui_h−1−ui−1

,

div

ξ

i σ

)

≤ C

1

t−1_2kui−1 h −u i−1_k

₁

_t1_2k

_φ

1_2div

_ξ

i σk

,

we have the following estimate instead of(3.6),

kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2≤ 1 2  kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2  +C

1

t2kuttk2L2(L2) +C  h2_σ(k+1)kσik_k2₊₁+

1

th2_σk1kσik2 k1+1+

1

t−1kuih−1−u i−1_k2

.

(3.7)

Then, using(3.1)and(3.7), and the positive definiteness ofA, we have that

k

ξ

i σk +

1

t 1 2kdiv

ξ

i σk ≤C

(

hkσ+1kσikk+1+

1

t 1 2hk1 σkσikk1+1+

1

tkuttkL2(L2)

)

+Cmin{hm_u

, 1

t−12_hm+1 u }

(

kukL∞(Hm+1₎+ kutkL2(Hm+1₎

).

(3.8)

Combining(3.8)with(3.4)completes the proof. 

For the error estimates for Scheme (II), for anyi≤n≤₁T

twe define the auxiliary projectionu˜ i

h∈Vhsatisfying

(

A∇

(

u˜i_h−ui

),

∇vh

)

=0

,

∀vh∈Vh

.

(3.9)

From this definition we have that

(

A∇

δ

t

(

u˜ih−u

i

_),

_∇v

h

)

=0

,

∀vh∈Vh

.

(3.10)

From [7] it is easy to see that that

       k ˜ui_h−uik_j≤Ch_um+1−jkuik_m+1

,

j=0

,

1

,

k

δ

t

(

u˜ih−u i

₎

_k j≤Chmu+1−j 1

1

t Z ti ti−1 kutk2m+1dt !1₂

,

j=0

,

1

.

(3.11) Theorem 3.2. Suppose

(

un

h

,

σnh

)

∈Vh×Whis the solution of Scheme

(

II

)

. The initial guess satisfieskσ0h−σ0k =O

(

huk1kσ0kHk1

)

.

Whenhu,hσand

1

tare sufficiently small, there exists a positive constantCindependent ofhu,hσand

1

tsuch that kσn_h−σnk + n X i=1

1

tkdiv

(σ

i h−σ i

₎

_k2 !1₂ ≤C

(

hk1_σkσk L∞(Hk1 +1)+h k+1 σ kσtkL2(Hk+1₎+

1

tkuttkL2(L2)

).

(3.12)

Further, ifu0_h = ˜u0_hholds there, we have that

kun_h−unk ≤Ch_um+1

(

kuk_L∞(Hm+1₎+ kutkL2(Hm+1₎

)

+Chk1_σkσ k_L∞(Hk1 +1)+Ch

k+1

σ kσtkL2(Hk+1₎+C

1

tkuttkL2(L2)

.

(3.13)

Proof. Subtracting(2.18)from(2.24)we have that

a

(

ui_h−ui

,

σi_h−σi;vh

,

τh

)

=

(

uih−1−u i−1

_{, φ}

_v h+

1

tdivτh

)

+

1

t

(

A−1

(σ

ih−1−σ i−1

₎

_{+ ∇}

₍

_ui−1 h −u i−1

_),

_τ h +A∇vh

),

−

1

t

(φ

−1

_ρ

i 1

, φ

vh+

1

tdivτh

),

∀

(

vh

,

τh

)

∈Vh×Wh

.

(3.14)

Settingvh=0, usingLemma 2.1and the divergence theorem, we have forτh∈Whthat

(

A−1

(σ

i h−σ i

_),

_τ h

)

+

1

t

(φ

−1div

(σ

ih−σ i

_),

_divτ h

)

=

(

A−1

(σ

ih−1−σ i−1

_),

_τ h

)

−

1

t

(φ

−1

ρ

i1

,

divτh

),

(3.15)

which can be written as

(

A−1

ξ

i σ

,

τh

)

+

1

t

(φ

−1div

ξ

i_σ

,

divτh

)

=

(

_A−1

ξ

i_σ−1

,

τ_h

)

+

1

tA−1

δ

t

(

σi−σiI

),

τh  +

1

t

φ

−1div

(

σi_−σi I

),

divτh  −

1

t

(φ

−1

ρ

i₁

,

divτh

)

≤1 2kA −1₂

_ξ

i−1 σ k2+ 1 2kA −1_2τ hk2+

1

t 2 kA −1₂

_δ

t

(σ

i−σiI

)

k 2₊

1

t 2 kτhk 2₊

₁

_tk

_φ

−1_2div

_(σ

i_−σi I

)

k 2 +

1

t 4 k

φ

−1₂ divτhk2+

1

tk

φ

− 1 2

ρ

i 1k2+

1

t 4 k

φ

−1₂ divτhk2

,

(3.16)

where the notation

δ

tis defined in(2.5).

Note that

φ

is bounded below and above, 0

< φ

1≤

φ

≤

φ

2. Then puttingτh=

ξ

i_σin(3.16)and using the



-inequality we

have kA−1₂

_ξ

i σk2+

1

tk

φ

− 1 2_div

ξ

i σk2 ≤ 1+

1

t 2 kA −1₂

_ξ

i σk2+

1

t 2 k

φ

−1₂ div

ξ

i σk2+ 1 2kA −1₂

_ξ

i−1 σ k2 +C

1

thk

δ

_t

(

σ − σ_I

)

i_k2_{+ kdiv}

₍

_{σ − σ} I

)

ik2+ k

ρ

i1k2 i

.

(3.17) Since k

δ

_t

(σ − σ

_I

)

ik = 1

1

t Z ti ti−1

(σ − σ

I

)

tdt ≤Chk_σ+1 1

1

t Z ti ti−1 kσ_tk2 k+1dt !1₂

,

applying(3.4)and(2.6)to(3.17)we have

k_A−1₂

_ξ

i σk2+

1

tk

φ

− 1 2div

ξ

i σk2 ≤ kA− 1 2

ξ

i−1 σ k2+

1

tkA− 1 2

ξ

i σk2+C

1

t2 Z ti ti−1 kuttk2dt +C " h2(_σk+1) Zti ti−1 kσ_tk2 k+1dt+

1

t hσ2k1kσk2L∞(Hk1 +1) #

.

(3.18)

Carrying out summation fori=1

,

2

, . . . ,

nwe have that kA−1₂

_ξ

n σk2+ n X i=1

1

tk

φ

−12_div

ξ

n σk2 ≤ n X i=1

1

tkA−1₂

_ξ

i σk2+ kA− 1 2

(σ

0 h−σ 0 I

)

k 2_+C

₁

_t2Z T 0 kuttk2dt +C  h2_σ(k+1) Z T 0 kσtk2k+1dt+hσ2k1kσ k2L∞(Hk1 +1₎ 

.

(3.19) Noticingσ0 h−σ0I =

(σ

0

h−σ0

)

+

(σ

0−σ0I

)

, using Gronwall’s inequality shows that

k

ξ

n_σk2+ n X i=1

1

tkdiv

ξ

i σk2≤C h h2_σk1kσk2 L∞(Hk1 +1)+h 2(k+1) σ kσtk2L2(Hk+1₎+

1

t2kuttk2L2(L2) i

.

(3.20)

Combining with(3.4)completes the proof of(3.12). Now we consider the estimate ofui_h−ui_{, fori≤n≤} T

1t. Lettingτh =0 in(3.14), usingLemma 2.1and the divergence

theorem lead to

(φ(

ui_h−ui

),

vh

)

+

1

t

(

A∇

(

uih−u i

_),

_∇v h

)

=

(φ(

uii−1−u i−1

_),

_v h

)

−

1

t

(ρ

i1

,

vh

)

+

1

t

(σ

ih−1−σ i−1

_,

_∇v h

)

+

1

t

(

A∇

(

ui−1 h −u i−1

_),

_∇v h

),

∀vh∈Vh

.

(3.21)

With the use of the definition ofeu i h, we have

(φ(

ui_h− ˜ui_h

),

vh

)

+

1

t

(

A∇

(

uih− ˜u i h

),

∇vh

)

=

(φ(

ui_h−1− ˜u_hi−1

),

vh

)

+

1

t

(φδ

t

(

ui− ˜uih

),

vh

)

−

1

t

(

div

(σ

i −1 h −σ i−1

_),

_v h

)

−

1

t

(ρ

i1

,

vh

)

+

1

t

(

A∇

(

ui_h−1− ˜u_hi−1

),

∇vh

),

∀vh∈Vh

.

(3.22) Let

ξ

i u=u i h− ˜u i h

,

η

i u=u i_{− ˜u}i h

.

(3.23)

With the choicevh=

ξ

ui =uih− ˜uihin(3.22), it follows that

k

φ

12

ξ

i uk 2₊

₁

_tkA1 2∇

ξ

i_uk2 ≤ 1+

1

t 2 k

φ

1 2

ξ

i uk 2₊

1

t 2 kA 1 2∇

ξ

i_uk2+1 2k

φ

1 2

ξ

i−1 u k 2₊

1

t 2 kA 1 2∇

ξ

i_u−1k2 +C

1

thk

δ

_t

(

ui_{− ˜u}i h

)

k 2_{+ kdiv}

_(σ

i−1 h −σ i−1

₎

_k2_{+ k}

_ρ

i 1k2 i

,

(3.24)

which can be reduced to k

φ

12

ξ

i uk 2₊

₁

_tkA1 2∇

ξ

i_uk2 ≤ k

φ

12

ξ

i−1 u k 2₊

₁

_tk

_φ

1 2

ξ

i uk 2₊

₁

_tkA1 2∇

ξ

i_u−1k2 +C

1

thk

δ

t

(

ui− ˜uih

)

k 2_{+ kdiv}

_(σ

i−1 h −σ i−1

₎

_k2_{+ k}

_ρ

i 1k2 i

.

(3.25)

Summing(3.25)fromi=1 tonand noticing(3.12)we have

k

φ

12

ξ

n uk 2₊

₁

_tkA12∇

ξ

n uk 2 _≤Xn i=1

1

tk

φ

12

ξ

i uk 2_{+ k}

_φ

1₂

_ξ

0 uk 2₊

₁

_tkA1_2∇

_ξ

0 uk 2_+C

₍

_h2(m+1) u kutk2_L2(Hm+1₎ +

1

t2kuttk2_L2(L2)

)

+C n X i=1

1

tkdiv

(σ

i−1 h −σ i−1

₎

_k2 ≤ n X i=1

1

tk

φ

12

ξ

n uk2+C h h2(_um+1)kutk2L2(Hm+1₎+

1

t2kuttk2L2(L) i +Chh2_σk1kσk2 L∞(Hk1 +1)+h 2(k+1) σ kσtk2L2(Hk+1₎ i

.

(3.26)

Therefore we can apply Gronwall’s inequality to(3.26). Hence it follows that k

ξ

n uk +

1

t 1 2k∇

ξ

n uk ≤C h hm+1 u kutkL2(Hm+1₎+

1

tkuttkL2(L2) i +Chhk1 σkσ kL∞(Hk1 +1₎+h_σk+1kσtkL2(Hk+1₎ i

.

Finally, combining(3.27)with(3.11)completes the proof. 

Remark 3.3. Instead ofu0_h = ˜u0_hif we supposeku0_h−u0_k

j=O

(

hmu+1−j

)

, from the proof we know that replacing(3.13)we have

a estimate

kun_h−unk ≤C

(

hm_u+1+

1

t12_hm

u +hk1σ +

1

t

).

Now we give the error estimate for Scheme (III). For this purpose, defineσ˜i_h∈Whsuch that

(

σ˜i_h−σi

,

τ_h

)

+

(φ

−1div

(

σ˜_hi −σi

),

divτh

)

=0

,

∀τh∈Wh

.

(3.27)

It is clear thatσ˜i_hexist uniquely. By splittingσ˜i_h−σi_asσ˜i

h−σi=

(

σ˜ i

h−σiI

)

+

(σ

iI−σi

)

and using(3.27), we have

(

σ˜i_h−σi_I

,

τh

)

+

(φ

−1div

(

σ˜ih−σ i

I

),

divτh

)

=

(σ

i−σiI

,

τh

)

+

(φ

−1div

(σ

i−σiI

),

divτh

)

≤ 1 2kσ i_−σi Ik 2₊1 2kτhk 2₊ 1 2k

φ

−1₂ div

(σ

i_−σi I

)

k + 1 2k

φ

−1₂ divτhk2

,

∀τh∈Wh

.

(3.28) Let_τh=σih−σ i

I. We have the following error estimate,

k ˜σi

h−σ

i_{k + kdiv}

₍

_σ˜i_−σi

₎

_{k ≤Ch}k1

σkσikk1+1

.

(3.29)

From(3.28)we also get that

(δ

t

(

σ˜ih−σ

i

I

),

τh

)

+

(φ

−1div

δ

t

(

σ˜ih−σ i

I

),

divτh

)

=

(δ

t

(σ

i−σiI

),

τh

)

+

(φ

−1div

δ

t

(σ

i−σiI

),

divτh

),

∀τh∈Wh

.

(3.30)

Letτh=

δ

t

(σ

ih−σiI

)

. We have the following error estimate similarly,

k

δ

_t

(

σ˜i−σi

₎

_{k + kdiv}

_δ

t

(

σ˜i−σi

)

k ≤Chk1_σ 1

1

t Z ti ti−1 kσ_tk_k1+1dt !1₂

.

(3.31)

Theorem 3.4. Suppose

(

un

h

,

σnh

)

∈Vh×Whis the solution of Scheme (III). Under the assumptionkσ0h−σ0k =O

(

hk

+1 σ kσ0kHk+1

)

, then kσn h−σk + " _n X i=1

1

tkdiv

(

σi h−σ i_+σi−1 h −σ i−1

₎

_k2 #1₂ ≤Chhk1_σkσk L∞(Hk1 +1₎+h k+1 σ kσtkL∞(Hk+1₎ i +

1

t2hku(3)t kL2(L2)+ kσttkL2(H1) i

.

(3.32) Moreover, ifkdiv

(σ

0

h−σ0

)

k =O

(

hk1σkdivσ0kHk1

)

andu0h= ˜u0hwe have

kun_h−unk ≤Chm_u+1hkuk_L∞(Hm+1₎+ kutkL2(Hm+1₎ i +

1

t2hkut(4)kL2(L2)+ kσ( 3) t kL2(H1)+ ku( 3) t kL∞(L2)+ kσttkL∞(H1) i +hk1_σkσ k L∞(Hk1 +1)+h k+1 σ kσtkL2(Hk1 +1)+ kσ kL∞(Hk+1₎

.

(3.33)

HereCdenotes a positive constantCindependent ofhu,hσand

1

t. Proof. First note that subtracting(2.38)from(2.39)leads to

b

(

ui_h−ui

,

σ_hi −σi;vh

,

τh

)

=  ui_h−1−ui−1−

1

t 2

φ

−1

₍

_div

_(σ

i−1 h −σ i−1

_{)), φ}

_v h+

1

t 2 divτh  +

1

t 2  A−1

(σ

i−1 h −σ i−1

₎

_{+ ∇}

₍

_ui−1 h −u i−1

_),

_τ h+A∇vh 

,

−

1

t 

φ

−1

_ρ

i 2

, φ

vh+

1

t 2 divτh 

,

∀

(

vh

,

τh

)

∈Vh×Wh

.

(3.34)

UsingLemma 2.2and(3.34)with a chosenvh=0, it follows that

1

t 2

(

A −1

_(σ

i h−σ i

_),

_τ h

)

+ 

1

_t 2 2

(φ

−1_div

_(σ

i h−σ i

_),

_divτ h

)

=

1

t 2

(

u i−1 h −u i−1

_,

_divτ h

)

− 

1

_t 2 2

(φ

−1_div

₍

_σi−1 h −σ i−1

_),

_divτ h

)

+

1

t 2

(

A −1

_(σ

i−1 h −σ i−1

_),

_τ h

)

+

1

t 2

(

∇

(

u i−1 h −u i−1

_),

_τ h

),

−

(1

t

)

2 2

(φ

−1

_ρ

i 2

,

divτh

),

∀

(

vh

,

τh

)

∈Vh×Wh

.

(3.35)

Let us split intoσi

h−σi=

ξ

σi −

η

iσwhere

ξ

i

σ =σih− ˜σ i

h∈Wh

,

η

i_σ =σi− ˜σih

.

(3.36)

By the definition ofσ˜i_hin(3.27), we have

(φ

−1_div

_(η

i σ+

η

iσ−1

),

divτh

)

= −

(η

i_σ+

η

i_σ−1

,

τh

).

It is clear that

(

ui_h−1−ui−1

,

divτh

)

+

(

∇

(

uih−1−u i−1

_),

_τ h

)

=0

.

Hence(3.35)reduces to: for all_τh∈Wh

(A

−1

_(ξ

i σ−

ξ

σi−1

),

τh

)

+

1

t 2

(φ

−1_div

_(ξ

i σ+

ξ

iσ−1

),

divτh

)

=

1

t

(

A−1

δ

t

η

i_σ

,

τh

)

−

1

t 2

(η

i σ+

η

iσ−1

,

τh

)

−

1

t

(φ

−1

ρ

i2

,

divτh

).

(3.37)

Lettingτh=

ξ

i_σ+

ξ

_σi−1∈Whin(3.37)and using the Cauchy inequality, we have

k_A−1₂

_ξ

i σk2− kA− 1 2

ξ

i−1 σ k2+

1

t 2 k

φ

−1₂ div

(ξ

i σ+

ξ

iσ−1

)

k2≤

1

tk

ξ

iσ+

ξ

iσ−1k2+

1

t 4 k

φ

−1₂ div

(ξ

i σ+

ξ

iσ−1

)

k2 +

1

t ₁ 2kA −1

_δ

t

η

i_σk2+ 1 8k

η

i σ+

η

iσ−1k2+ k

φ

− 1 2

ρ

i 2k2 

.

(3.38)

Summing(3.38)fori=1

,

2

, . . . ,

nwe can deduce that kA−1₂

_ξ

n σk2+ 1 2 n X i=1

1

tk

φ

−12_div

(ξ

i σ+

ξ

σi−1

)

k2 ≤C n X i=1

1

tk

ξ

i_σk2+Ck

ξ

0_σk2+C n X i=1

1

t[k

δ

t

η

i_σk2+ k

η

i_σk2+ k

ρ

i2k2] +C

1

tk

η

0σk2

.

(3.39) Using Gronwall’s Lemma we can get that,

k

ξ

n_σk2+ n X i=1

1

tkdiv

(ξ

i σ+

ξ

iσ−1

)

k2≤Chσ2(k+1)

(

kσk2L∞(Hk+1₎+ kσtk2L2(Hk+1₎

)

+C

1

t4

(

ku( 3) t k2L2(L2)+ kdivσttk2L2(L2)

).

(3.40)

Combining with(3.4)completes the proof of(3.32).

Choosingτh= ₁1_t

(ξ

_σi −

ξ

_σi−1

)

=

δ

t

ξ

i_σin(3.37)we have that

1

tkA−1₂

_δ

t

ξ

i_σk2+ 1 2k

φ

−1₂ div

ξ

i σk2− 1 2k

φ

−1₂ div

ξ

i−1 σ k2 =

1

t

(A

−1

δ

t

η

i_σ

, δ

t

ξ

i_σ

)

−

1

t 2

(η

i σ+

η

iσ−1

, δ

t

ξ

i_σ

)

−

1

t

(φ

−1

_ρ

i 2

,

div

δ

t

ξ

i_σ

),

≤C

1

tk_A−12

δ

t

ξ

i_σk

(

k

δ

t

η

i_σk + k

η

i_σ+

η

i −1 σ k

)

−

(φ

−1

ρ

i2

,

div

ξ

i σ

)

+

(φ

−1

ρ

i2−1

,

div

ξ

i−1 σ

)

−

1

t

(φ

−1

δ

t

ρ

i2

,

div

ξ

i−1 σ

),

(3.41) where we have used the equivalence

1

t

(φ

−1

ρ

i₂

,

div

δ

t

ξ

i_σ

)

=

(φ

−1

ρ

i2

,

div

ξ

iσ

)

−

(φ

−1

ρ

i2−1

,

div

ξ

iσ−1

)

−

1

t

(φ

−1

δ

t

ρ

i2

,

div

ξ

iσ−1

).

For convenience we introduce a notation

ρ

0

2and

δ

t

ρ

12 = ρ1

2−ρ02

1t . Making summation overi = 1

,

2

, . . . ,

nand using the

Cauchy inequality result in

n X i=1

1

tk_A−12

δ

_t

ξ

i σk2+ 1 2k

φ

−1 2div

ξ

n σk2≤C n X i=1

1

tk_A−12

δ

_t

ξ

i σk2

(

k

δ

t

η

i_σk + k

η

i_σ+

η

i −1 σ k

)

+ 1 2k

φ

−1 2div

ξ

0 σk2 −

(φ

−1

ρ

n₂

,

div

ξ

n σ

)

+

(φ

−1

ρ

02

,

div

ξ

0σ

)

+ n X i=2

1

t

(φ

−1

δ

t

ρ

i2

,

div

ξ

i−1 σ

)

+

1

t

(φ

−1

δ

t

ρ

12

,

div

ξ

0σ

).

(3.42) Since

(φ

−1

_ρ

0

2

,

div

ξ

0σ

)

+

1

t

(φ

−1

δ

t

ρ

12

,

div

ξ

0σ

)

=

(φ

−1

ρ

12

,

div

ξ

0σ

),

using the



-inequality we have that

n X i=1

1

tkA−1₂

_δ

t

ξ

i_σk2+ 1 2k

φ

−1₂ div

ξ

n σk2≤ 1 2 n X i=1

1

tkA−1₂

_δ

t

ξ

i_σk2+ 1 4k

φ

−1₂ div

ξ

n σk2+Ckdiv

ξ

0σk2 +C n X i=1

1

t

(

k

δ

_t

η

i σk2+ k

η

iσk2

)

+C

(

k

η

0σk2+ k

ρ

n2k2+ k

ρ

12k2

)

+C n X i=2

1

tk

δ

_t

ρ

i 2k2+C n X i=1

1

tk

φ

−1₂ div

ξ

i σk2

.

(3.43) Moving the first two terms of the right-hand side to the left side, then Gronwall’s inequality results in

kdiv

ξ

n σk2+ n X i=1

1

tk

δ

_t

ξ

i_σk2 ≤ Ch2_σ(k+1)

(

kσ k2 L∞(Hk+1₎+ kσtk2L2(Hk+1₎

)

+Ch2_σk1kdivσ0k2_Hk1 +C

1

t4

(

kut(4)k2L2(L2)+ kσ (3) t k2L2(L2)+ ku (3) t k2L∞(L2)+ kσttk2L∞(L2)

).

(3.44)

Now we consider the estimate ofun h−u n_{. Choosingτ} h=0 in(3.34)we have that,

(φ(

ui_h−ui

),

vh

)

+

1

t 2

(

A∇

(

u i h−u i

_),

_∇v h

)

=

(φ(

uih−1−u i−1

_),

_v

₎

₋

₁

_t

_(ρ

i 2

,

vh

)

−

1

t

(

div

(σ

ih−1−σ i−1

_),

_v h

)

+

1

t 2

(

A∇

(

u i−1 h −u i−1

_),

_∇v h

),

∀vh∈Vh

.

(3.45) Denote by

ξ

n u=u n h− ˜u n h

,

η

n u=u n_{− ˜u}n h

.

(3.46)

From(3.45)we have that

(φξ

i u

,

vh

)

+

1

t 2

(

A∇

ξ

i u

,

∇vh

)

=

(φξ

i −1 u

,

vh

)

+

1

t

(φδ

t

η

iu

,

vh

)

−

1

t

(

div

(σ

ih−1−σ i−1

_),

_v h

)

−

1

t

(ρ

i₂

,

vh

)

+

1

t 2

(A∇ξ

i−1 u

,

∇vh

),

∀vh∈Vh

.

(3.47)

Settingvh=

ξ

iuand using the Cauchy inequality we have that

k

φ

12

ξ

i uk 2₊

1

t 2 kA 1 2∇

ξ

i uk 2 _≤ 1 2k

φ

1 2

ξ

i uk 2₊1 2k

φ

1 2

ξ

i−1 u k 2₊

1

t 6 k

φ

1 2

ξ

i uk 2_+C

₁

_tk

_δ

t

η

iuk 2 +

1

t 6 k

φ

1 2

ξ

i uk 2_+C

₁

_tkdiv

_(σ

i−1 h −σ i−1

₎

_k2 +

1

t 6 k

φ

1 2

ξ

i uk 2_+C

₁

_tk

_ρ

i 2k2+

1

t 4 kA 1 2∇

ξ

i uk 2₊

1

t 4 kA 1 2∇

ξ

i−1 u k 2 = 1+

1

t 2 k

φ

1 2

ξ

i uk 2₊

1

t 4 kA 1 2∇

ξ

i_uk2+1 2k

φ

1 2

ξ

i−1 u k 2₊

1

t 4 kA 1 2∇

ξ

i_u−1k2 +C

1

t[k

δ

_t

η

_uik2+ kdiv

(σ

i−1 h −σ i−1

₎

_k2_{+ k}

_ρ

i 2k2]

,

(3.48) then k

φ

12

ξ

i uk 2₊

1

t 2 kA 1 2∇

ξ

i_uk2 ≤ k

φ

12

ξ

i−1 u k 2₊

₁

_tk

_φ

1 2

ξ

i uk 2₊

1

t 2 kA 1 2∇

ξ

i_u−1k2 +C

1

t[k

δ

_t

η

i uk 2_{+ kdiv}

₍

_σi−1 h −σ i−1

₎

_k2_{+ k}

_ρ

i 2k2]

.

(3.49)

Summing(3.49)overi=1

,

2

, . . . ,

n, we have that k

φ

12

ξ

n uk 2₊

1

t 2 kA 1 2∇

ξ

n_uk2≤ n X i=1

1

tk

φ

12

ξ

n uk 2_{+ k}

_φ

1 2

ξ

0 uk 2₊

1

t 2 kA 1 2∇

ξ

_u0k2 +Ch2(m+1) u kutk2_L2(Hm+1₎+

1

t2

(

ku(3)t k2L2(L2)+ kdivσttk 2 L2(L2)

)

+C n X i=1

1

tkdiv

(σ

i−1 h −σ i−1

₎

_k2

_.

_(3.50) Since div

(σ

i−1 h −σ i−1

₎

_{= div}

_(ξ

i−1 σ

)

+ div

(

σ˜i −1 h −σ i−1

_),

noticing(3.44)and(3.31), by Gronwall’s inequality shows that

kun_h− ˜un_hk +

1

t12k∇

ξ

n_uk ≤Chm_u+1kutkL2(Hm+1₎+C

1

t2

(

ku(_t4)k_L2(L2)+ kσ(_t3)k_L2(H1)

)

+C

1

t2

(

kut(3)kL∞(L2)+ kσttkL∞(H1)

)

+Chk1σkσkL∞(Hk1 +1)+Ch

k+1

σ

(

kσtkL2(Hk+1₎+ kσk_L∞(Hk+1₎

).

Combining with(3.11)completes the proof.  4. Least-squares procedure for nonlinear problems

In this section we give a least-squares finite element procedure for nonlinear parabolic problems. We consider the following problem on a bounded domainΩ⊂Rd_:

  

φ(

u

)

ut−div

(

A

(

u

)

∇u

)

=f

(

u

),

inΩ×J

,

u=0

,

onΓD×J

,

A

(

u

)

∇u·n=0 onΓN×J

,

(4.1)

subject to the initial condition

u

(

x

,

0

)

=u₀

(

x

)

onΩ×J

.

(4.2)

The coefficient

φ(

u

)

is a strictly positive function and the coefficient matrixA

(

u

)

=

(

aij

(

u

))

di,j=1is a bounded, symmetric and positive definite matrix, i.e., there exist two positive constants

φ

1and

φ

2and two positive constants

α

and

β

such that, for

u∈_R1

φ

1≤

φ(

u

)

≤

φ

2

,

α

kξk2≤

(

A

(

u

)ξ,

ξ

)

≤

β

kξk2

,

∀ξ ∈ Rd

.

(4.3) In general the coefficients

φ(

u

)

,A

(

u

)

andf

(

u

)

are also dependent on time variabletand space variablex. Since our main purpose is to consider the nonlinearity, for convenience we just consider the dependence of the coefficients onu.