ON THE SUPERCONVERGENCE OF BILINEAR FINITE ELEMENT

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INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 6, Number 1, Pages 1–9

ON THE SUPERCONVERGENCE OF BILINEAR FINITE ELEMENT FOR SECOND ORDER ELLIPTIC PROBLEMS WITH GENERAL BOUNDARY CONDITIONS MINGXIA LI AND SHIPENG MAO Abstract. In this paper, the superconvergence of bilinear finite element for general second order elliptic problems with general boundary conditions is studied to avoid a sub-optimal superconvergence, we construct an auxiliary equation to eliminate the effect of boundary terms. As a result, the O(h2 ) superconvergence rate is obtained under the regularity assumption that u ∈ H 3 (Ω) when almost uniform rectangular meshes is employed, which improves the previous results. Key Words. finite element methods; general boundary conditions; superconvergence

1. Introduction Since Oganesyan and Rukhovetz [15] first proved the superconvergence for linear element approximations on a uniform triangle mesh in 1969, superconvergence for the finite element solutions has been an active research area in numerical analysis for more than thirty years. The main target in the superconvergence study is to improve the existing approximation accuracy by applying certain postprocessing techniques which are easy to implement. Roughly speaking, the superconvergence analysis has developed into several different methods in recent years, for example: superconvergence on locally point-symmetric meshes [1, 17, 18]; element analysis method [7]; error expansions with special integral identities [11, 12]; the Zienkiewicz-Zhu gradient patch recovery method (SPR) [10, 21, 22, 23, 24]; L2 projection method [19, 20]; computer based methods and etc., interested readers are referred to [2, 8, 9] for the surveys on the development and various results of superconvergence. Furthermore, superconvergence is also recognized as a useful tool in a posterior error estimations, mesh refinement and adaptivity. In this paper, we aim to give a minor complementarity of the superconvergence of the lowest order rectangular element, i.e, the conforming bilinear element. It is known that bilinear finite element approximation is often used for it’s simplicity and nice structure accompanied with high accuracy. Its superconvergence for the second order elliptic problems have been obtained by many authors (for example, in [7, 11, 12, 18] and etc.). When the exact solution u ∈ H 3 (Ω), O(h2 ) superconvergence rate on uniform rectangular meshes can be proved for pure Dirichlet conditions; however, as to other boundary conditions, for example, the Neumann or the Robin 3 boundary conditions, only O(h 2 ) superconvergence rate has been proved if under the same assumptions. The object of this paper is to improve the latter result when general boundary conditions are considered. Generally, after the process of Received by the editors February 5, 2007 and, in revised form, July 10, 2009. 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. 1

2

M. LI AND S. MAO

asymptotic expansions, there will appear some residual boundary terms which can not be eliminated. The classical technique is to transform these boundary terms to 3 the whole domain by the trace like theorems, which results in a non-optimal O(h 2 ) 1 convergence rate under H 3 regularity (quasi-optimal convergence rate O(h2 |lnh| 2 ) can be proved if the solution is more smoother, i.e., u ∈ H 4 (Ω), cf. [11, 12]). We overcome this drawback by introducing an auxiliary equation to convert the residual boundary terms into the integral on whole domain, and optimal convergence rate of O(h2 ) can be obtained only with H 3 regularity on almost uniform rectangular meshes. The rest of this paper is organized as follows. In Section 2, after introducing the general second order elliptic problems and some notations, we are dedicated the superconvergence of bilinear finite element approximation based on the work in [11, 12]. In Section 3, we carry out a numerical test to verify our result. 2. Superconvergence for the general boundary conditions Let Ω be a convex polygonal domain in the xy-plane. Throughout this paper, we will use the standard Sobolev space W m,p (Ω) (m ∈ N) with the norm and the seminorm (see, for example, [3]) given by   p1 X Z kψkm,p,Ω =  |Dα ψ|p dxdy  , 0≤|α|≤m

Ω

and

 p1

 |ψ|m,p,Ω = 

X Z

|α|=m

|Dα ψ|p dxdy  ,

Ω

respectively. As usual, we will drop the index p when it is 2, and write H m (Ω) instead of W m,2 (Ω). The two-dimensional general second order elliptic problem with general boundary conditions can be written as: find u, such that  −div(A∇u) + β · ∇u + γu = f, in Ω,     u = g1 , on Γ1 , (2.1)  A∇u · n = g2 , on Γ2 ,    A∇u · n + σu = g3 , on Γ3 , where

A=

α11 α21

α12 α22

,

β = (β1 , β2 ), αij (i, j = 1, 2), βi (i = 1, 2), γ, σ are given functions, α12 = α21 , and A is uniformly elliptic, i.e., 2 X

αi,j ξi ξj ≥ θ|ξi |2 , ∀ξ ∈ R2 a.e. in Ω,

(2.2)

i,j=1

where θ is a positive constant. In this paper, we assume αi,j (i, j = 1, 2), βi (i = 1, 2) ∈ W 1,∞ (Ω) and γ, σ ∈ L∞ (Ω). For the purpose of this paper, we assume Ω to be a rectangular domain. The right, top, left and bottom boundaries are denoted by ∂Ωi (i = 1, · · · , 4), respectively (as shown in Figure 2 .1).

SUPERCONVERGENCE OF THE GENERAL SECOND ELLIPTIC PROBLEMS IN 2D

3

Figure 2.1

Let Th be a rectangular partition of Ω and e ∈ Th an arbitrary element with the edges le,1 = {(x, y) : x = xe + he , ye − ke ≤ y ≤ ye + ke }; le,2 = {(x, y) : xe − he ≤ x ≤ xe + he , y = ye + ke }; le,3 = {(x, y) : x = xe − he , ye − ke ≤ y ≤ ye + ke }; le,4 = {(x, y) : xe − he ≤ x ≤ xe + he , y = ye − ke }. Assume V = H 1 (Ω),

V0 = HΓ11 (Ω) = {v ∈ H 1 (Ω), v = 0 on Γ1 },

¯ vh |e ∈ Q1 (e), ∀ e ∈ Th }, Vh = {vh ∈ C(Ω),

and Vh0 = {vh ∈ Vh , vh |Γ1 = 0}.

Then the variational form of (2.1) is: find u ∈ V , such that   Z Z 2 2 X X   αij ∂i u∂j v + βi ∂i uv + γuv dxdy + Ω

i,j=1

Z =

Z f v dxdy +

Ω

g2 v ds + Γ2

i,j=1

Z =

g3 v ds,

∀ v ∈ V0 ,

Γ3

Γ3

i=1

Z f vh dxdy +

Ω

Z g3 vh ds, ∀ vh ∈ Vh0 .

g2 vh ds + Γ2

(2.3)

Z

∂ ∂ where ∂1 = ∂x , and ∂2 = ∂y . The associated discrete problem is: find uh ∈ Vh , such that   Z Z 2 2 X X  αij ∂i uh ∂j vh + βi ∂i uh vh + γuh vh  dxdy + Ω

σuv ds

Γ3

i=1

Γ3

σuh vh ds (2.4)

4

M. LI AND S. MAO

We define the average function of αij as R αij |e =

αij , |e|

e

where |e| is the area of the element e, and |αij − αij | ≤ Chkαij k1,∞ . Let h = maxe {ke , he }. Further suppose the mesh satisfy the almost uniform rectangular mesh in the following sense: 3

he − h = O(h 2 ),

3

ke − h = O(h 2 ).

(2.5)

Let uI denote the bilinear interpolation of u, and set w = u − uI . Based on the result of [12], there holds Z ∂i w∂i vh dxdy = O(h2 )kuk3,e kvh k1,e , i = 1, 2. (2.6) e

So we have Z αii ∂i w∂i vh dxdy Z XZ αii ∂i w∂i vh dxdy + (αii − αii )∂i w∂i vh dxdy = Ω

e∈Th

=

X

Ω

e

(2.7)

2

O(h )kuk3,e kvh k1,e + O(h)kwk1,Ω kvh k1,Ω

e∈Th

= O(h2 )kuk3,Ω kvh k1,Ω ,

i = 1, 2.

Theorem 1. Assume Vh is the bilinear finite element space, and αi,j (i, j = 1, 2) ∈ W 1,∞ (Ω), we have Z (α12 wx vhy + α21 wy vhx ) dxdy = O(h2 )kuk3,Ω kvh k1,Ω , ∀ vh ∈ Vh . (2.8) Ω

Proof. It has been proved in [11, 12] that Z Z 1 X 2 he uxx vhxy dxdy + O(h2 )|u|3,Ω |vh |1,Ω , wx vhy dxdy = 3 e Ω

∀ vh ∈ Vh .

e∈Th

An application of integrating by parts gives Z Z Z Z 1 X 2 wx vhy dxdy = he − uxxy vhx dxdy + − uxx vhx dx 3 Ω e l2 l4 (2.9) e∈Th + O(h2 )|u|3,Ω |vh |1,Ω . Noticing that for any two adjacent elements ek and el , αij |ek − αij |el = O(h), and invoking the following inequality (cf. [12]), 1

kvh k1,∂Ω ≤ Ch− 2 kvh k1,Ω ,


5

by (2.5) and (2.9) , it can be derived that Z α12 wx vhy dxdy Ω Z XZ = α12 wx vhy dxdy + (α12 − α12 )wx vhy dxdy e∈Th

=

X

e

Ω

O(h2e )kuk3,e kvh k1,e + h

e∈Th

+ O(h2 )

X

O(h2e )kuk2,∂e kvh k1,∂e

e∈Th

Z

Z

α12 uxx vhx dx + O(h)kwk1,Ω kvh k1,Ω Z Z =O(h2 )kuk3,Ω kvh k1,Ω + O(h2 ) − (α12 − α12 )uxx vhx dx ∂Ω2 ∂Ω4 Z Z 2 + O(h ) − α12 uxx vhx dx ∂Ω2 ∂Ω4 Z Z 2 2 =O(h )kuk3,Ω kvh k1,Ω + O(h ) − α12 uxx vhx dx. −

∂Ω2

(2.10)

∂Ω4

∂Ω2

∂Ω4

By the same argument, we have Z Z 2 2 α21 wy vhx dxdy = O(h )kuk3,Ω kvh k1,Ω + O(h ) Ω

∂Ω1

Z −

α21 uyy vhy dy,

∂Ω3

(2.11) which, together with (2.10), we can obtain Z (α12 wx vhy + α21 wy vhx ) dxdy Ω Z Z =O(h2 )kuk3,Ω kvh k1,Ω + O(h2 ) − α12 uxx vhx dx ∂Ω2 ∂Ω4 Z Z 2 + O(h ) − α21 uyy vhy dy ∂Ω1 ∂Ω3 Z =O(h2 )kuk3,Ω kvh k1,Ω + O(h2 ) α12 uss vhs ds,

(2.12)

∂Ω

where s is the unit tangentialR derivative along ∂Ω. Now, the key ingredient is to estimate the boundary term ∂Ω α12 uss vhs ds of (2.12). To this end, let’s consider the auxiliary equation: −∆φ = 0, in Ω, (2.13) φ = α12 uss , on ∂Ω. After some applications of Green’s formula, we have Z Z φvhs ds = φ ∇vh · s ds ∂Ω Z∂Ω = φ curl vh · n ds ∂Ω Z Z = ∇φ curl vh dxdy + φ div (curl vh ) dxdy Ω ZΩ = ∇φ curl vh dxdy, Ω

(2.14)

6

M. LI AND S. MAO

where

∂v ∂v ,− , ∂y ∂x and n is the unit normal derivative of ∂Ω. It follows from the regularity result of problem (2.13) and the trace theorem that curl v =

|φ|1,Ω ≤ C|α12 uss | 12 ,∂Ω ≤ Ckuk3,Ω .

(2.15)

Then the desired result can be obtained by a combination of (2.12) – (2.15), which completes the proof of Theorem 1. Now, we are in the position to bound other terms. Noticing that βi (i = 1, 2) ∈ W 1,∞ (Ω) and γ, σ ∈ L∞ (Ω), and by the interpolation error estimates, it is easy to verify (c.f. [12]) that ∀ vh ∈ Vh , the following estimates hold: Z X Z X Z 2 2 βi ∂i wvh dxdy = − w∂i (βi vh ) dxdy+ wvh β·n ds = O(h2 )kuk3,Ω kvh k1,Ω , Ω i=1

Ω i=1

∂Ω

(2.16) Z

2

γwvh dxdy = O(h )kuk2,Ω kvh k1,Ω ,

(2.17)

Ω

and Z

σwvh ds = O(h2 )kuk2,Γ3 kvh k0,Γ3 = O(h2 )kuk3,Ω kvh k1,Ω .

(2.18)

Γ3

Together with (2.7) and (2.8), we obtain   Z 2 2 X X  αij ∂i (uh − uI )∂j vh + βi ∂i (uh − uI )vh + γ(uh − uI )vh  dxdy Ω

i,j=1

i=1

Z σ(uh − uI )vh ds

+ Γ3

 Z =

2 X

 Ω

αij ∂i w∂j vh +

i,j=1

2 X

 Z βi ∂i wvh + γwvh  dxdy +

=O(h2 )kuk3,Ω kvh k1,Ω ,

σwvh ds Γ3

i=1

∀ vh ∈ Vh0 .

(2.19) By virtue of Garding’s Inequality (cf. [3]), we know that there exists a constant K < ∞ such that θ (2.20) a(v, v) + Kkvk20,Ω ≥ kvk21,Ω . 2 where   Z Z 2 2 X X   a(u, v) := αij ∂i u∂j v + βi ∂i uv + γuv dxdy + σuv ds, Ω

i,j=1

i=1

and θ is the coefficient in (2.2). By (2.19) and (2.20), we have θ kuh − uI k21,Ω 2 ≤a(uh − uI , uh − uI ) + Kkuh − uI k20,Ω =a(u − uI , uh − uI ) + Kkuh − u + u − uI k20,Ω ≤Ch2 kuk3,Ω kuh − uI k0,Ω .

Γ3


7

Therefore the following superapproximation estimate holds: kuh − uI k1,Ω ≤ Ch2 kuk3,Ω .

(2.21)

Our next goal is to construct S4 a postprocessing operator in order to get global superconvergence. Let e˜ = i=1 ei (see Figure 2.2), and Π22h : C(˜ e) → Q2 (˜ e) be defined as Π22h w(Zi ) = w(Zi ),

i = 1, · · · , 9,

(2.22)

where Zi (i = 1, · · · , 9) are the nodes of the small elements.

7

4 e1

3 e2

8

6

9 e4

e3

1

5

Figure 2.2

2

the big element e˜

It can be checked that the interpolation defined as (2.22) is well-posed. Furthermore, it has the following properties (cf. [12]):  2 Π v = Π22h v,   2h I kΠ22h vh k1,Ω ≤ Ckvh k1,Ω , ∀ vh ∈ Vh ,   2 kΠ2h v − vk1,Ω ≤ Ch2 kvk3,Ω ,

(2.23)

where vI ∈ Vh is the bilinear interpolation of v. Then we can get the following global superconvergence theorem. Theorem 2. Assume u ∈ H 3 (Ω) ∩ V , uh ∈ Vh be the solution of (2.2) and (2.3), respectively. Th is an partition of Ω which satisfies (2.4), then the superconvergence estimate holds kΠ22h uh − uk1,Ω ≤ Ch2 kuk3,Ω .

(2.24)

Proof. A combination of superapproximation result (2.21) and the properties of Π22h gives kΠ22h uh − uk1,Ω =kΠ22h uh − Π22h uI + Π22h uI − uk1,Ω ≤kΠ22h uh − Π22h uI k1,Ω + kΠ22h uI − uk1,Ω ≤kΠ22h (uh − uI )k1,Ω + kΠ22h u − uk1,Ω

(2.25)

≤Ckuh − uI k1,Ω + Ch2 kuk3,Ω ≤Ch2 kuk3,Ω .

8

M. LI AND S. MAO

3. Numerical test In this numerical test, we take Ω = [0, 1] × [0, 1], Γ1 = [0, 1] × {0} is the Dirichlet boundary, Γ2 = {0} × [0, 1] ∪ {1} × [0, 1] is the Neumann boundary, and Γ3 = [0, 1] × {1} is the Robin boundary. Let u = sinπxcosπy, 1 + ex 0.5 A= , 0.5 1 + ey which is symmetric and positive definite, β = (0, 0), γ = ex + ey and σ = ex+y . Let Th be a uniform rectangular partition of Ω which satisfies our assumptions, then the errors of convergence, superapproximation and superconvergence (in H 1 semi-norm) are laid out below, which are used to validate our result. Table 3.1 m×n |u − uh |1 |uh − uI |1 |u − Π22h uh |1

2×4 4 × 8 8 × 16 16 × 32 32 × 64 0.7940 0.3986 0.1992 0.0996 0.0498 0.2374 0.0643 0.0165 0.0042 0.0010 0.7200 0.1727 0.0414 0.0102 0.0026

0

10

Energy norm error

−1

10

−2

10

convergence superapproximation superconvergence

−3

10

0

10

1

10

2

10 Numerber of elements

3

10

4

10

Figure 3.1

Acknowledgment Mingxia Li is indebted much to her advisor Prof. Qun Lin and his group for their kindly advice and encouragement. Without their help, this paper would have been impossible. References [1]

J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math. Comp. 31 (1977), 94-111.


[2]

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[16] [17]

[18] [19] [20] [21] [22] [23]

[24]

9

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Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, East Road, Zhongguancun, Beijing, 100190, People’s Republic of China E-mail: [email protected] Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, East Road, Zhongguancun, Beijing, 100190, People’s Republic of China E-mail: [email protected]