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

aSchool of Mathematics, Shandong University, Jinan, Shandong, 250100, China

bDepartment of Mathematics, Kyungpook National University, Daegu 702-701, South Korea

cDepartment 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 variableσ. 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 thea posteriorerror 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

IThe 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:hxrui@sdu.edu.cn(H. Rui),skim@knu.ac.kr(S.D. Kim),sckim@changwon.ac.kr(S. Kim).

0377-0427/$ – see front matter©2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.cam.2008.03.030

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

)

^{, Hk}

(

^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,h_σand

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⊂R^d

,

^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∈R^d

.

^(2.3)

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

,

σ =

(

σ1

, . . . ,

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 andσrepresents 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:= ^u

n−uⁿ⁻¹

1

^t

, ρ

ⁿ1:=

φ(δ

tuⁿ−uⁿ_t

).

^(2.5)

It is clear that

ρ

ⁿ1=O

Z tⁿ tⁿ−1

ku_ttkdt

!

=O



 4t Z tⁿ

tⁿ−1

ku_ttk²dt

!¹₂



.

^(2.6)

Define two function spaces

V= {v∈H¹

(

^Ω

)

^{:v=0 onΓ}D}

,

^(2.7)

W= {τ ∈

(

^L2

(

^Ω

))

^{d: div}τ∈_L²

(

^Ω

),

τ·_n=_{0 onΓN}}

.

^(2.8)

From(2.4)we know that forn≥1,

(

^un

,

^σn

)

^∈V×Wsatisfy that (

φ

⁻¹²

(φ

^un+

1

^tdivσⁿ−F₁ⁿ

)

⁼⁰

,

^inΩ×J

,

A⁻¹²

(

σⁿ+A∇uⁿ

)

⁼⁰

,

^inΩ×J

,

^(2.9)

whereF₁ⁿ=

φ

^un−1+

1

^tfn+

1

^t

ρ

ⁿ1.

For

(

^v

,

τ

)

^∈V×W, define the first kind of least-squares functionalJ₁ⁿ

(

^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∈Wsuch that

Jⁿ₁

(

^un

,

σⁿ

)

⁼₍ ^inf

v,τ)∈_V×_WJⁿ₁

(

^v

,

τ

).

^(2.11)

Define the bilinear forma

(

^u

,

^σ;v

,

^τ

)

corresponding to the least-squares functionalJⁿ₁as

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×Wsuch that}

a

(

^un

,

σⁿ;v

,

τ

)

⁼

(φ

^−1F1ⁿ

, φ

^v+

1

^tdivτ

),

^∀

(

^v

,

τ

)

^∈V×W

.

^(2.13) Noticing the definition ofFⁿ₁,(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−F₁ⁿ

)

⁼⁰

,

^inΩ×J

,

A⁻¹²

(

σⁿ+A∇uⁿ−Gⁿ

)

⁼⁰

,

^inΩ×J

,

^(2.15)

whereGⁿ=σⁿ⁻¹+A∇uⁿ⁻¹. For

(

^v

,

τ

)

^∈V×W, define the second kind of least-squares functionalJⁿ₂

(

^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×Wsuch that}

Jⁿ₂

(

^un

,

σⁿ

)

⁼₍ ^inf

v,τ)∈_V×_WJⁿ₂

(

^v

,

τ

).

^(2.17)

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

(

^un

,

σⁿ

)

^{∈V×Wsuch 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. LetTh_uandThσ

be two families of regular finite element partitions of the domainΩ, which are either identical or not. Leth_uandh_σdenote the largest of the diameters of the element inThuandThσ respectively. Based onThuandThσ, respectively, we construct the finite element spaces Vh⊂VandWh⊂_Wwith the following approximation properties:

inf

v_h∈V_h{kv−v_hk +h_uk∇

(

^v−vh

)

^{k} ≤Chm}u⁺¹kvk_m+1

,

^(2.19)

τhinf∈Wh

kτ−τhk ≤Ch^k_σ⁺¹kτk_k+1

,

^(2.20)

τhinf∈W_hkdiv

(

τ−τh

)

^{k ≤Chk1}_σ^kτk_k1+1

,

^(2.21)

forv∈_V∩H^m+1

(

^Ω

)

^andτ ∈_W∩

(

^Hk1+1

(

^Ω

))

^d. It is clear that when assumption(2.20)holds we can deducek₁=k, and when W_his selected as any of the Raviart–Thomas mixed element space [17] we can choosek₁=k+1. In this paper we always supposek₁=k+1 whenW_his any of the Raviart–Thomas mixed element space [17] andk₁=kotherwise.

We select the initial approximationu⁰_h∈_Vh,σ⁰h∈_{Whsuch that} (ku₀−u⁰_hk_j≤Ch^m_u^+1−jku₀k_m+1

,

^j=0

,

¹

,

kσ0−σ⁰hk ≤Ch^k_σ⁺¹kσ0k_k+₁

,

^(2.22)

whereσ0=A∇u₀. The first least-squares finite element procedure based on(2.14)reads as follows.

Scheme (I).Forn≥1 find

(

^unh

,

σⁿh

)

^∈Vh×W_hsuch that

a

(

^unh

,

σⁿh;v_h

,

τh

)

⁼

(

^unh⁻¹+

1

^t

φ

^−1fn

, φ

^vh+

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

(

^unh

,

^σnh

)

^{∈Vh×Whsuch that}

a

(

^unh

,

σⁿh;v_h

,

τh

)

⁼

(

^unh⁻¹+

1

^t

φ

^−1fn

, φ

^vh+

1

^tdivτh

)

+

1

^t

(

^A−1σⁿh⁻¹+ ∇uⁿ_h⁻¹

,

τh+A∇v_h

),

^∀

(

^vh

,

τh

)

^∈Vh×W_h

.

^(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∈Vandσ

,

^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−1σ

,

τ

)

⁺

1

^t2

(φ

^−1divσ

,

^divτ

)

+

1

^t

((

^u

,

^divτ

)

⁺

(

^v

,

^divσ

)

⁺

(

^∇u

,

τ

)

⁺

(

^∇v

,

σ

)),

Integrating by parts shows that

(

^u

,

^divτ

)

⁺

(

^v

,

^divσ

)

⁺

(

^∇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 puttingτh=0 and v_h=0 in(2.23)or(2.24).

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

(

^u0h

,

σ⁰h

)

^∈Vh×W_h, forn≥1 find

(

^unh

,

σⁿh

)

^∈Vh×W_hsuch that for allv_h∈V_handτh∈W_h

(φ

^unh

,

^vh

)

⁺

1

^t

(

^A∇unh

,

^∇vh

)

⁼

(φ

^unh⁻¹+

1

^tfn

,

^vh

),

^(2.27)

(

A⁻¹σⁿ_h

,

τh

)

⁺

1

^t

(φ

^−1divσⁿ_h

,

^divτh

)

⁼

(

^unh⁻¹+

1

^t

φ

^−1fn

,

^divτh

).

^(2.28) Equivalent Form of Scheme (II).With the initial guess

(

^u0h

,

σ⁰h

)

^{∈Vh×Wh, forn≥1 find}

(

^unh

,

σⁿh

)

^∈Vh×Whsuch that for allv_h∈V_handτh∈W_h

(φ

^unh

,

^vh

)

⁺

1

^t

(

^A∇unh

,

^∇vh

)

⁼

(φ

^unh⁻¹+

1

^tfn

,

^vh

)

⁺

1

^t

(

σⁿh⁻¹+A∇uⁿ_h⁻¹

,

^∇vh

)

^(2.29)

(

^A−1σⁿh

,

τh

)

⁺

1

^t

(φ

^−1divσⁿh

,

^divτh

)

⁼

(

^A−1σⁿh⁻¹

,

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 fluxσⁿhwith 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

ⁿ2:=

φ

δ

tuⁿ−uⁿ⁻

1 2 t

+¹

2div

(

σⁿ+σⁿ⁻¹

)

^{− div}σⁿ⁻¹²

,

^(2.31)

which can be estimated as

ρ

ⁿ2=O





1

^{t32 Z t}

n

tⁿ−1

(

^{|uttt|2+ |div}σtt|²

)

^dt

!¹₂



.

^(2.32)

From(2.4)we know that forn≥1,

(

^un

,

^σn

)

^∈V×Wsatisfy that







φ

⁻¹²

φ

^un+

1

^t

2 divσⁿ−F₂ⁿ

=0

,

^inΩ×J

,

A⁻¹²

(

σⁿ+A∇uⁿ−Gⁿ

)

⁼⁰

,

^inΩ×J

,

(2.33)

whereGⁿis the same as in(2.15), F₂ⁿ=

φ

^un−1+

1

^{tfn−12 −}

1

^t

2 divσⁿ⁻¹+

1

^t

ρ

ⁿ2

.

^(2.34)

For

(

^v

,

τ

)

^∈V×W, define the least-squares functionalJⁿ₃

(

^v

,

τ

)

as follows.

Jⁿ₃

(

^v

,

τ

)

⁼

φ

⁻¹²

φ

^v+

1

^t

2 divτ−Fⁿ₂

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×Wsuch that}

Jⁿ₃

(

^un

,

σⁿ

)

^{= inf}

v∈_V,τ∈_WJⁿ₃

(

^v

,

τ

).

^(2.36)

Define the bilinear formb

(

^·

,

^{·; ·}

,

^·

)

^as

b

(

^u

,

σ;v

,

τ

)

^=u+

1

^t

2

φ

^−1divσ

, φ

^v+

1

^t

2 divτ+

1

^t

2

(

^A−1σ+ ∇u

,

τ+A∇v

).

^(2.37)

Noticing the definition ofF₂ⁿin(2.34), the weak statement of the minimization problem(2.36)is: find

(

^un

,

^σn

)

^∈V×Wsuch that

b

(

^un

,

σⁿ;v

,

τ

)

^=un−1+

1

^t

φ

^{−1fn−12 −1}

2divσⁿ⁻¹+

ρ

ⁿ2

, φ

^v+

1

^t 2 divτ

,

+

1

^t

2

(

^A−1σⁿ⁻¹+ ∇uⁿ⁻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

(

^u0h

,

σ⁰h

)

^∈Vh×W_h, forn≥1 find

(

^unh

,

σⁿh

)

^∈Vh×W_hsuch that b

(

^unh

,

σⁿh;v_h

,

τh

)

^=unh⁻¹+

1

^t

φ

^{−1fn−12− 1}₂^divσⁿh⁻¹

, φ

^vh+

1

^t 2 divτh

+

1

^t

2

(

^A−1σnh⁻¹+ ∇uⁿ_h⁻¹

,

^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∈Vandσ

,

τ ∈Wwe have that, b

(

^u,σ;v

,

τ

)

⁼

(φ

^u

,

^v

)

⁺

1

^t

2

(

^A∇u

,

^∇v

)

⁺

1

^t

2

(

^A−1σ

,

τ

)

⁺

1

^t 2

2

(φ

^−1divσ

,

^divτ

).

^(2.40)

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

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

(

^u0h

,

^σ0h

)

^{∈Vh×Wh, forn≥1 find}

(

^unh

,

^σnh

)

^{∈Vh×Whsuch that}

(φ

^unh

,

^vh

)

⁺

1

^t

2

(

^A∇unh

,

^∇vh

)

⁼

φ

^unh⁻¹+

1

^{tfn−12 −}

1

^t

2 divσⁿh⁻¹

,

^vh

+

1

^t

2

(

σⁿh⁻¹+A∇uⁿ_h⁻¹

,

^∇vh

),

∀v_h∈_Vh

.

^(2.41)

(

^A−1σⁿh

,

τh

)

⁺

1

^t

2

(φ

^−1divσⁿh

,

^divτh

)

⁼

(

^A−1σⁿh⁻¹

,

τh

)

⁺

1

^t

φ

^{−1fn−12 −1}₂^divσⁿh⁻¹

,

^divτh

,

∀τh∈W_h

.

^(2.42)

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

(

^unh

,

σⁿh

)

^{∈ V}h × W_h is the solution of Scheme

(

^I

)

. Under the assumption ku⁰_h − u⁰k = O

(

^hmu^+1−jku⁰k_Hm+1−j

),

^j=0

,

1, there exists a positive constantCindependent ofhu,h_σand

1

^{tsuch that}

kuⁿ_h−uⁿk_s≤Ch^m_u^+1−s

(

^kukL∞(^Hm+1)+ kutk_L2(^Hm+1)

)

^+C

1

^tkuttkL2(^L2)

,

^s=0

,

¹

,

^(3.1) kσⁿh−σk +

1

^t12kdiv

(

σⁿh−σⁿ

)

^{k ≤ C}

(

^hk_σ^+1kσⁿk_k+1+

1

^t12hk_σ^1kσⁿk_k1+1+

1

^tkuttk_L2(L2)

)

+Cmin{h^m_u

, 1

^t−12hmu⁺¹}

(

^kukL∞(^Hm+1)+ kutk_L2(^Hm+1)

).

^(3.2)