Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacle

Applied Mathematics and Computation 348 (2019) 1–11

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Superconvergence analysis of low order nonconforming finite element methods for variational inequality problem with displacement obstacleR Chao Xu a, Dongyang Shi b,∗ a b

Faculty of Mathematics and Physics Education,Luoyang Institute of Science and Technology, Luoyang 471023, PR China School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, PR China

a r t i c l e

i n f o

Keywords: Nonconforming FEMs Variational inequality problem Displacement obstacle Superconvergence analysis

a b s t r a c t Superconvergence analysis of nonconforming finite element methods (FEMs) are discussed for solving the second order variational inequality problem with displacement obstacle. The elements employed have a common typical character, i.e., the consistency error can reach order O(h3/2−ε ), nearly 1/2 order higher than their interpolation error when the exact solution of the considered problem belongs to H 5/2−ε () for any ε > 0. By making full use of special properties of the element’s interpolations and Bramble–Hilbert lemma, the superconvergence error estimates of order O(h3/2−ε ) in the broken H1 -norm are derived. Finally, some numerical results are provided to confirm the theoretical results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction The obstacle problem is important in the field of variational inequality theory and applications, and has been treated by various FEMs. For example, the conforming linear triangular element was applied to solve the second order variational inequality problem with displacement obstacle in [3–5,15]. In particular, the error bound of order O(h3/2−ε ), for any ε > 0, was obtained in [3] for the quadratic finite element under the hypothesis of that the free boundary has finite length. Further, [26] derived the same error bound as [3] for the same quadratic element without the above hypothesis. In [25,27], the nonconforming Crouzeix–Raviart type linear triangular element and rectangular Wilson element were studied and the error estimate was bounded by order O(h). In [18], two Crouzeix–Raviart type nonconforming finite elements were used to this problem on anisotropic meshes and optimal error estimates of order O(h) were also derived by the technique of introducing special auxiliary spaces. The other methods of dealing with the obstacle problem can be found in [2,6,7,23,24,28] and the references therein. But all of the above studies only focused on the convergence analysis. As we know, EQrot 1 element is an important quadrilateral nonconforming finite element and possesses two typical characters: one is that the associated finite element interpolation operator is identical to its traditional Ritz projection operator; the other is that the consistency error can reach order O(h2 ), one order higher than the interpolation error when the exact solution of the problems belongs to H3 (). Thus, it has been employed to cope with different problems [11,14,16,17,19,20,22].

R ∗

The research is supported by NSF of China (No. 11271340, 11671369). Corresponding author. E-mail address: [email protected] (D. Shi).

https://doi.org/10.1016/j.amc.2018.08.015 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

2

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

Particularly, this element was also used to solve the Signorini problem [21], and the superconvergence result of order O(h3/2 ) in the broken H1 -norm was obtained when the exact solution u ∈ H5/2 (). Recently, by imposing a constraint on the Qrot element [13] which keeps the same characters as the EQrot 1 element on 1 square meshes, the so-called CNQrot element was constructed in [8] , which is the simplest quadrilateral element and has 1 only three degrees of freedom as the nonconforming quadrilateral P1 element of [12]. But because this element is defined on the reference element through the bilinear transformation, its implementation is standard compared to [12]. The CNQrot 1 element and nonconforming quadrilateral P1 element are equivalent on a rectangle and are different on a general quadrilateral. Although this CNQrot 1 element has a good performance in solving some problems [8,9,29], it has never been applied to the superconvergence analysis of the second order variational inequality problem with displacement obstacle up to now. rot rot In this paper, as an attempt, we will apply the above three low order EQrot 1 , Q1 and CNQ1 elements to the second order variational inequality problem with displacement obstacle. By virtue of their special properties, we derive the superclose and superconvergence results of order O(h3/2−ε ) in the broken H1 -norm on rectangular meshes under a reasonable regularity assumption u ∈ W 2,∞ () ∩ W s,p (), ∀ 1 < p < ∞, s < 2 + 1p . Finally, some numerical results are provided to confirm the theoretical analysis. We will use the common notations for the Sobolev spaces Wm, p () with norm  · m, p, and semi-norm | · |m,p, , and m W , p (K) with norm  · m,p,K and semi-norm | · |m,p,K , where m and p are nonnegative real numbers. Especially, for p = 2, p will be omitted in the above notations. 2. Variational inequality problem and low order nonconforming finite elements Consider the following variational inequality problem with displacement obstacle: to find u ∈  , such that

a(u, v − u ) ≥ f (v − u ), where

a(u, v ) =

 

∀ v ∈ ,

∇ u · ∇vdxdy,

f (v ) =

(2.1)  

f vdxdy,

 = {v ∈ H01 () : v ≥ χ a.e. in ; χ ≤ 0 on ∂ },  ⊂ R2 is bounded convex domain, f and χ are given functions. The variational problem of (2.1) is equivalent to:



−u = f −u ≥ f u≥χ

in in in

+ = { x ∈  : u ( x ) > χ ( x ) } , 0 = { x ∈  : u ( x ) = χ ( x ) } , , u = 0 on ∂ .

(2.2)

Let Th be a regular subdivision of the domain  into rectangular elements K ∈ Th , hK = diam(K ), and h = max hK . K∈Th

 = [−1, 1] × [−1, 1] be the reference element in the (ξ , η) plane with vertices  Let K a1 (−1, −1 ),  a2 (1, −1 ),  a3 ( 1, 1 ),  a4 (−1, 1 ) and edges Fi =  ai  ai+1 (i = 1, 2, 3, 4 (mod 4 )). For a given element K ∈ Th with the center point (xK , yK ), its four  → K is vertices are denoted by ai (xi , yi ) and four edges by Fi = ai ai+1 (i = 1, 2, 3, 4 (mod 4 )). The affine mapping FK : K defined by



x = xK + hx,K ξ , y = yK + hy,K η,

where hx,K and hy,K are the half length of element K along x and y-axis, respectively. Now we introduce the following three low order nonconforming finite elements. Element I: EQrot 1 element a a  a  The EQrot 1 finite element (K , P , ) on K is defined by [11,17]:

 a = {

v1 ,  v2 ,  v3 ,  v4 ,  v5 }, where

 vi =

1 | Fi |

  Fi

a = span{1, ξ , η, ξ 2 , η2 }, P

 vds (i = 1, 2, 3, 4 ),  v5 =

1 |K|

  K

 vdξ dη.

The corresponding finite element space is defined by:

Vha = {vh :  vh = vh |K ◦ FK ∈ Pa ,



F

[vh ]ds = 0,

∀F ⊂ ∂ K, ∀K ∈ Th },

where [vh ] denotes the jump value of vh crossing the edge F if F is an interior edge, and it is equal to vh if F is a boundary edge of ∂ . Element II: Qrot element 1

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

3

The Qrot element space is defined by [13]: 1

Vhb = {vh :  vh = vh |K ◦ FK ∈ Pb ,



F

[vh ]ds = 0,

b = span{1, ξ , η, ξ 2 − η2 }. where P Element III: CNQrot 1 element The CNQrot 1 element space is defined by [8]:



Vhc = {vh ;

vh ∈ Vhb ,

F1

 vh ds +



F3

 vh ds =

 F2

∀F ⊂ ∂ K, ∀K ∈ Th },

 vh ds +

 F4

 vh ds,

∀K ∈ Th }.

The associated interpolation operators over Vh∗ (∗ = a/or b/or c ) are defined as ∗h : v ∈ H 2 () → ∗h v satisfying

 ∗ ∗h |K = ∗K , ∗K v =  v ◦ FK−1 , ∀ v ∈ P∗ . The expressions of ∗h can be found in literature [9,17,30], respectively.  1 It is easy to see that  · h = ( | · |21,K ) 2 is the norm over Vh∗ . K

The Qrot element was proposed by Rannacher and Turek in [13] for solving Stokes problems, which keeps the properties 1 on square meshes that for all vh ∈ Vhb ,

(∇ (u − bh u ), ∇vh ) = 0, and

 K∈Th

∂K

(2.3)

∂u v ds ≤ Ch2 |u|3, vh h , u ∈ H 3 (), ∂n h

(2.4)

which can lead to the superclose result

bh u − uh h ≤ ch2 |u|3 ,

(2.5)

where uh be the finite element solution of the second order problem. rot The EQrot 1 element can be regarded as the Q1 element adding a nonconforming bubble function [14], which extends the orthogonal property (2.3) to rectangular meshes for a new degree of freedom of average value on the each element K was introduced in [11]. In additional, this element possesses the anisotropic properties and can be applied to anisotropic meshes [17]. But the Qrot element does not have this property and a counter-example was constructed in [1]. 1 For the CQrot 1 element, the orthogonal property (2.3) is no longer valid and can be replaced by

(∇ (u − ch u ), ∇vh ) ≤ Ch2 |u|3 vh h , ∀vh ∈ Vhc ,

(2.6)

which ensures that the superclose result (2.5) holds true. Particularly, superconvergence results of this element can be derived on quadrilateral meshes satisfying the usual Bi-section condition [8]. 3. Superconvergence analysis of the EQrot 1 finite element In this section, we will study the superconvergence analysis of the EQrot 1 finite element. To do this, we introduce the following important lemmas. Lemma 3.1. [11, 17] For vh ∈ Vha , u ∈ H2 (), there holds

(∇ (u − ah u ), ∇vh ) = 0.

(3.1)

Lemma 3.2. For vh ∈ Vha , u ∈ H 5/2−ε (), we have



K∈Th

∂K

∂u v ds ≤ Ch3/2−ε |u|5/2−ε, vh h . ∂n h

Proof. Let P0F (v ) = |F1|



K∈Th

∂K

 F

vds, then

  ∂u v ds = ∂n h K∈T F ⊂∂ K

  = −

h

 F

∂u v ds, ∂n h 

∂u [v − P0F2 (vh )]dy ∂ x h K∈Th  

∂u ∂u + [vh − P0F3 (vh )]dx − [vh − P0F4 (vh )]dy F3 ∂ y F4 ∂ x F1

∂u [v − P F1 (v )]dx + ∂y h 0 h

F2

(3.2)

4

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11 4 .  = Ai .

(3.3)

K∈Th i=1

∂ v

Since  vh ∈ span{1, ξ , η, ξ 2 , η2 }, ∂ηh ∈ span{1, η}. We can see that

  1 1 1 1 F2     vh (1, η ) − P v = v ( 1 , η ) dt − v (1, t )dt h h 0 2 −1 2 −1 h  1 η  1 η 1 ∂ vh 1 ∂ vh = (1, z )dzdt = (−1, z )dzdt 2 −1 t ∂ z 2 −1 t ∂ z F4  = vh (−1, η ) − P v. 0 h

(3.4)

Hence, for a given element K, there holds





[vh − P0F2 (vh )] = [vh − P0F4 (vh )] . F2

(3.5)

F4

A direct computation implies



K

 vh (xK + hx,K , y ) − P0F2 (vh (xK + hx,K , y )) dxdy = 0.

For simplicity, let φ2 = [vh − P02 (vh )] F



and φ4 = [vh − P04 (vh )] , then φ2 = φ4 . Note that φ 2 and φ 4 only rely on variable F

F2

(3.6)

y, we can derive by (3.5) and (3.6) that



F4



∂u ∂u [vh − P0F2 (vh )]dy − [vh − P0F4 (vh )]dy ∂ x F2 F4 ∂ x  yK +hy,K ∂u = (xK + hx,K , y )φ2 (xK + hx,K , y )dy yK −hy,K ∂ x

A2 + A4 =



∂u (x − hx,K , y )φ4 (xK − hx,K , y )dy ∂x K  yK +hy,K  xK +hx,K  ∂  ∂u = (x, y )φ2 (xK + hx,K , y ) dxdy yK −hy,K xK −hx,K ∂ x ∂ x  2 ∂ u = φ (x + hx,K , y )dxdy 2 2 K K ∂x   2  2 ∂ u K ∂ u = − P ( ) φ2 (xK + hx,K , y )dxdy 0 2 ∂ x2 K ∂x  2   ∂ 2u   1 ∂ u ≤ (2hx,K ) 2  2 − P0K ( 2 ) φ2 (xK + hx,K , y ) 0,F2 ∂x ∂ x 0,K 3 / 2 −ε ≤ ChK |u|5/2−ε,K |vh |1,K . −

yK +hy,K

yK −hy,K

(3.7)

Similarly,

A1 + A3 ≤ Ch3K/2−ε |u|5/2−ε,K |vh |1,K .

(3.8)

Together, we obtain 4 

Ai ≤ Ch3/2−ε |u|5/2−ε, vh h ,

(3.9)

K∈Th i=1

which is the desired result. Define the closed convex nonempty set ha as:

    ha = vh ∈ Vha : vh ds ≥ χ ds, vh dxdy ≥ χ dxdy, F ⊂ ∂ K, K ∈ Th }. F

F

K

K

Then we consider the approximation problem of (2.1): to find uh ∈ ha , such that

ah ( uh , vh − uh ) ≥ f ( vh − uh ), where ah (u, v ) =



K∈Th

K

∀vh ∈ ha ,

(3.10)

∇ u · ∇vdxdy.

Now we are ready to prove the main result of this section for the EQrot 1 element.



C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

5

Theorem 3.1. Let u and uh be the solutions of (2.1) and (3.10) respectively, u ∈ W s,p () ∩ W 2,∞ (), ∀ 1 < p < ∞, s < 2 + 1p , and the data f ∈ H1 () ∩ L∞ (), χ ∈ Ws, p () ∩ W2, ∞ (). Then under the hypothesis of finite length of the free boundary, the following superclose error estimate holds

ah u − uh h ≤ Ch3/2−ε.

(3.11)

Proof. Applying Lemma 3.1,

ah u − uh 2h = ah (ah u − uh , ah u − uh ) = ah (ah u − u, ah u − uh ) + ah (u, ah u − uh ) − ah (uh , ah u − uh ) . ≤ ah (u, ah u − uh ) − ( f, ah u − uh ) = Eh (u, ah u − uh ). Let wh =

ah u

− uh , and use Green’s formula,



Eh (u, wh ) = =

K∈Th

 K∈Th

(3.12)

K

(∇ u · ∇ wh − f wh )dxdy

 ∂u w ds + ∂n h K∈T

∂K

h

 K

(−u − f )wh dxdy =. I1 + I2 ,

(3.13)

where n is the outward unit normal vector over ∂ K.



For the term I1 , by use of Lemma 3.2, we have

I1 ≤ Ch3/2−ε |u|5/2−ε, wh h ≤ Ch3−ε |u|25/2−ε, +

1 wh 2h . 4

(3.14)

In order to estimate the term I2 , we let w = −u − f and

 + h = { K ∈ Th : K ⊂ + }, 0h = { K ∈ Th : K ⊂ 0 }, −h = { K ∈ Th : K ∩ + = φ , K ∩ 0 = φ}.

Then, I2 can be rewritten as

I2 = (w, wh ) = (w, ah (u − χ ) − (u − χ )) + (w, u − χ ) + (w, ah χ − uh ) . = I21 + I22 + I23 . Firstly, under the hypothesis of finite length of the free boundary which indicates that the total number of K ∈ more than O(h−1 ), we have

I21 = ≤ ≤



K∈

− h

K∈

− h

 

(3.15)

− h

is no

(w, ah (u − χ ) − (u − χ )) Ch2 w0,∞, ah (u − χ ) − (u − χ )0,∞, Ch4 w0,∞, u − χ 2,∞,

K∈− h

≤ Ch3 (|u|22,∞, +  f 20,∞, + |χ |22,∞, ).

(3.16)

Secondly, from (2.2), we know that (w, u − χ )(x ) = 0, ∀x ∈ , then

I22 = 0.

(3.17)

Thirdly, notice that w(x ) = 0, ∀x ∈ + , we have

I23 =



 K∈0h ∪− h

K

w(ah χ − uh )dxdy.

Since w ≥ 0, we have P0K w ≥ 0 and



K

P0K w(ah χ − uh )dxdy = P0K w

 K

(3.18)

(ah χ − uh )dxdy ≤ 0.

Further, we can derive

I23 ≤



 K∈

0∪ h



− h

K

(w − P0K w )(ah χ − uh )dxdy

(3.19)

6

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

=

  K∈

+

K

− h

(w − P0K w )[ah (χ − u ) − (χ − u )]dxdy

  K

K∈− h

(w − P0K w )[(χ − u ) − P0K (χ − u )]dxdy 



+

K∈

0∪ h



− h

K

(w − P0K w )[(ah u − uh ) − P0K (ah u − uh )]dxdy

1 ≤ Ch3−ε (|w|21/2−ε, + |u|22,∞, +  f 20,∞, + |χ |22,∞, ) + ah u − uh 2h 4   K K + (w − P0 w )[(χ − u ) − P0 (χ − u )]dxdy. K∈− h

(3.20)

K

By use of the same techniques of error analysis as in [26], we know that

 

K∈

− h

K

(w − P0K w )[(χ − u ) − P0K (χ − u )]dxdy

≤ Ch3−ε ||w||1/p−ε1 ,p, χ − u2+1/t −ε2 ,t , ,

 2+ε1 

(3.21)

where 1t = ε3 , p = 1 + ε1 , ε = ε1 1+ε + ε2 + ε3 > 0. 1 Substituting (3.21) into (3.20) yields

I23 ≤ Ch3−ε +

1 ah u − uh 2h . 4

(3.22)

Finally, collecting (3.13)–(3.17) and (3.22) leads to

1 2

|Eh (u, wh )| ≤ Ch3−ε + ah u − uh 2h ,

(3.23)

which together with (3.12) shows

uh − ah uh ≤ Ch3/2−ε .

(3.24)

The proof is complete.  Now we will use the proper interpolation postprocessing operator developed in [10,11] to get the global superconvergence estimate. Suppose T2h to be a rectangular meshes, and Th can be obtained by dividing each element of T2h into four equal  elements Ki (i = 1, 2, 3, 4 ). Let T ∈ T2h , T = 4i=1 Ki . L1 , L2 , L3 and L4 denote the four edges of T . Then as [10,11], we define a the interpolation operator 2h on the partition T2h :

 a   2h u|aT ∈ P2 (T ), ∀T ∈ T2h , (2h u − u )ds = 0, i = 1, 2,3, 4, L i a a K1 ∪K3 (2h u − u )d xd y = 0, K2 ∪K4 (2h u − u )d xd y = 0,

where P2 (T ) denotes the set of polynomials of degree 2. It has been shown in [10] that a2h defined above satisfies the following properties:

a2h ah u = a2h u,

u − a2h uh ≤ Chr |u|r+1, , 0 ≤ r ≤ 2,

a2h vh ≤ C vh , ∀v ∈ Vha .

(3.25)

(3.26)

Theorem 3.2. Under the assumptions of Theorem 3.1, we can get the following global superconvergence result

u − a2h uh h ≤ Ch3/2−ε .

(3.27)

Proof. Noticing that

u − a2h uh = u − a2h ah u + a2h ah u − a2h uh ,

(3.28)

by (3.26), we have

u − a2h ah uh = u − a2h uh ≤ Ch3/2−ε .

(3.29)

Consequently, it follows from (3.11) and (3.25) that

a2h uh − a2h ah uh = a2h (uh − ah u )h ≤ C uh − ah uh ≤ Ch3/2−ε . Then the desired result can be derived via (3.28)–(3.30).

(3.30) 

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

7

4. Superconvergence analysis of the Qrot and CNQrot 1 elements 1 In this section, we will study the superconvergence phenomenon of the Qrot and CNQrot 1 elements. As a representative, 1 rot we will pay more attention to the CNQ1 element, and do some modifications in the proof of Theorem 3.1. It can be checked that Lemma 3.2 still holds for CNQrot 1 element. Thus we only need to give the following important estimate through a new approach instead of the conventional idea. Lemma 4.1. If u ∈ H 5/2−ε (), for all vh ∈ Vhc , there holds

(∇ (u − ch u ), ∇vh ) ≤ Ch3/2−ε |u|5/2−ε vh h .

(4.1)

Proof. It is easy to see that

(∇ (u − ch u ), ∇vh )     = (u − ch u )x vhx dxdy + (u − ch u )y vhy dxdy K

K∈Th

K

.  = ( B1 + B2 ).

(4.2)

K∈Th

For the term B1 , we consider the functional

B ( u,  vh ) =



 K

 c ( u− u )ξ  v hξ d ξ d η .

(4.3)

According to the Sobolev embedding theorem,

|B (  u,  vh )| ≤ C  u5/2−ε,K| vh |1,K.

(4.4)

A direct computation shows that

B ( u,  vh ) = 0,

), ∀  ∀ u ∈ P2 (K vh ∈ Vhc ,

(4.5)

which gives by Bramble–Hilbert lemma that

|B (  u,  vh )| ≤ C | u|5/2−ε,K| vh |1,K. Hence

B1 =

(4.6)

    c    c  2 ( u −  u ) v ( h h /h ) d ξ d η ≤ C ( u −  u ) v d ξ d η   ξ hξ x,K y,K x,K ξ hξ K

K

≤ C | u|5/2−ε,K| vh |1,K ≤ C h3/2−ε |u|5/2−ε,K |vh |1,K .

(4.7)

Similarly,

B2 ≤ Ch3/2−ε |u|5/2−ε,K |vh |1,K .

(4.8) 

Combining with (4.2) and (4.7)–(4.8), we can derive the desired result. Now we define the closed convex nonempty set hc as£º

hc = {vh ∈ Vhc : vh (ai ) ≥ χ (ai ), i = 1, 2, 3, 4, K ∈ Th }. Then we consider the approximation problem of (2.1) as: to find uh ∈ hc , such that

ah ( uh , vh − uh ) ≥ f ( vh − uh ),

∀vh ∈ hc .

(4.9)

Theorem 4.1. Let u and uh be the solutions of (2.1) and (4.9) respectively, under the condition of Theorem 3.1, we have the following superconvergence error estimate

ch u − uh h ≤ Ch3/2−ε .

(4.10)

Proof. Applying Lemma 4.1, we can derive

ch u − uh 2h = ah (ch u − uh , ch u − uh ) = ah (ch u − u, ch u − uh ) + ah (u, ch u − uh ) − ah (uh , ch u − uh ) ≤ Ch3/2−ε |u|5/2−ε ch u − uh h + Eh (u, ch u − uh ).

(4.11) 

Now we only need to show the proof of different part in Theorem 4.1 for the

CNQrot 1

element, i.e., the term I23 .

8

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

1

u(x,y)

0.8 0.6 0.4 0.2 0 1 1

0

0 −1

−1

y−axis

x−axis Fig. 1. The exact solution u.

Table 1 Numerical results of the EQrot 1 nonconforming element. h

u − uh h

uh − ah uh

u − a2h uh h

1/8 1/12 1/16 1/24 1/32 1/48 1/64 Order

0.200113727 0.123388946 0.092651538 0.056999216 0.042199340 0.027671942 0.019977367 1.1020

0.115181262 0.065602440 0.048359674 0.021868256 0.014090156 0.008299681 0.0 0246460 0 1.7361

0.183568070 0.096575504 0.077933046 0.034568966 0.023776421 0.013140271 0.008884752 1.4720

Table 2 Numerical results of the CNQrot 1 nonconforming element. h

u − uh h

uh − ch uh

|u − c2h uh |1

1/8 1/12 1/16 1/24 1/32 1/48 1/64 Order

0.422301336 0.283442272 0.214032266 0.142257166 0.106869335 0.071298791 0.053562233 0.9944

0.093046483 0.047742967 0.029858111 0.017553208 0.013064470 0.0 053450 06 0.003957028 1.5150

0.201403643 0.079692173 0.084802338 0.042499495 0.028511609 0.011704393 0.011541580 1.3999

In fact, as to the term I23 , notice that w(x ) = 0, ∀x ∈ + , we have





I23 =

K

K∈0h ∪− h

w(ch χ − uh )dxdy.

Since w ≥ 0, we have P0K w ≥ 0 and



K

P0K w(ch χ − uh )dxdy = P0K w

= P0K w

4  |K | 

4

(4.12)

 K

(ch χ − uh )dxdy

 χ ( ai ) − uh ( ai ) ≤ 0.

i=1

Then with the same arguments as (3.20)–(3.22), we have

I23 ≤

 K∈0h ∪− h



K

(w − P0K w )(ch χ − uh )dxdy

(4.13)

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

9

1

uh(x,y)

0.8 0.6 0.4 0.2 0 1 1

0

0 −1

−1

y−axis

x−axis

Fig. 2. The numerical results of

EQrot 1

element with h = 1/32.

1

uh(x,y)

0.8 0.6 0.4 0.2 0 1 1

0

0 −1

−1

y−axis Fig. 3. The numerical results of

≤ Ch3−ε +

x−axis CNQrot 1

element with h = 1/32.

1 ch u − uh 2h . 4

Thus we can prove Theorem 4.1 as Theorem 3.1 similarly. The proof is complete.

(4.14) 

Remark 1. For the Qrot element, Theorem 4.1 also holds true on square subdivision. On the other hand, like [9] and [10], 1 g rot we can introduce the corresponding interpolation postprocessing operators 2h (g = c/or b) of the CNQrot 1 element and Q1 element defined on the partition T2h respectively, and then obtain the similar result as Theorem 3.2 directly. Remark 2. By further analysis, we know by the definition of EQrot 1 element that

(P0K w, ah (u − χ ) − (u − χ )) = 0, then

I21 =

 K∈− h

(w, ah (u − χ ) − (u − χ ))

(4.15)

10

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11 0

10

−1

Errors

10

−2

10

||u−uh||h ||u − h

||u−

a h

u||

h

a u || 2h h h

−3

10

−2

−1

10

0

10 Mesh size h

10

Fig. 4. Numerical results of EQrot 1 nonconforming element. 0

10

−1

Errors

10

−2

10

||u−uh||h ||uh− |u−

c h

u||h

c u | 2h h 1

−3

10

−2

−1

10

10 Mesh size h

0

10

Fig. 5. Numerical results of CNQrot 1 nonconforming element.

=



(w − P0K w, ah (u − χ ) − (u − χ )).

(4.16)

K∈− h

Using the similar skills as (3.21), we can derive the same results as Theorem 3.1. Here we omit the hypothesis of finite rot length of the free boundary in estimation. The same results can also be obtained for CNQrot 1 and Q1 elements by use of their definitions and the technique (4.13). 5. Numerical results In order to investigate the numerical behaviors of these nonconforming finite elements, we consider the following example [2]: Let  = (−1.5, 1.5 )2 , f = −2, χ = 0 and u = r 2 /2 − ln(r ) − 1/2 on ∂ , where r 2 = x2 + y2 for (x, y) ∈ R2 . Then the

C. Xu and D. Shi / Applied Mathematics and Computation 348 (2019) 1–11

11

exact solution u is given by



u=

r 2 /2 − ln(r ) − 1/2, 0,

if in

r ≥ 1, r < 1,

(5.1)

and pictured in Fig. 1. We use the projection SOR algorithm employed in [18] to solve this problem. rot The square mesh subdivision of  is implemented and the numerical results of EQrot 1 and CNQ1 elements are listed in Tables 1 and 2, and pictured in Figs. 2–5. From Tables 1 and 2, we can see that the average convergence rate in the broken H1 -norm is of order O(h), and the superclose and superconvergence errors are near to O(h3/2 ), which confirm the theoretical analysis. References [1] T. Apel, S. Nicaise, J. Schöberl, Crouzeix–Raviart type finite elements on anisotropic meshes, Numer. Math. 89 (2001) 193–223. [2] D. Braess, C. Carstensen, R.H.W. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem, Numer. Math. 107 (2007) 455–471. [3] F. Brezzi, W.W. Hager, P.A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977) 431–443. [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [5] R.S. Falk, Error estimates for the approximation of a class of variational inequalites, Math. Comp. 28 (1974) 963–971. [6] C. Gräser, R. Kornhuber, Multigrid methods for obstacle problems, J. Comput. Math. 27 (2009) 1–44. [7] T. Gudi, K. Porwal, A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems, Math. Comput. 83 (2014) 579–602. [8] J. Hu, Z.C. Shi, Constrained quadrilateral nonconforming rotated Q1 element, J. Comput. Math. 23 (2005) 561–586. [9] H.P. Liu, N.N. Yan, Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for stokes equations, Adv. Comput. Math. 29 (2008) 375–392. [10] Q. Lin, J.F. Lin, Finite element Methods: Accuracy and Improvement, Science Press, Beiing, 2006. [11] Q. Lin, L. Tobiska, A.H. Zhou, Superconvergence and extropolation of nonconforming low order elements applied to the poisson equation, IMA J. Numer. Anal. 25 (2005) 160–181. [12] C. Park, D. W. Sheen, P1 nonconforming quadrilateral finite element methods for second-order elliptic problems, SIA M. J. Numer. Anal. 41 (2003) 624–640. [13] R. Rannacher, S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods PDEs 8 (1992) 97–111. [14] U. Risch, Superconvergence of a nonconforming low order finite element, Appl. Numer. Math. 54 (2005) 324–338. [15] J.F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987. [16] D.Y. Shi, L.F. Pei, Low order Crouzeix-Raviart type nonconforming finite element methods for approximating Maxwell’s equations, Int. J. Numer. Anal. Mod. 5 (2008) 373–385. [17] D.Y. Shi, S.P. Mao, S.C. Chen, Anisotropic nonconforming finite element with some superconvergence results, J. Comput. Math. 23 (2005) 261–274. [18] D.Y. Shi, C.X. Wang, Q.L. Tang, Anisotropic Crouzeix-Raviart type nonconforming finite element methods to variational inequality problem with displacement obstacle, J. Comput. Math. 33 (2015) 86–99. [19] D.Y. Shi, J.J. Wang, Unconditional superconvergence analysis for nonlinear hyperbolic equation with nonconforming finite element, Appl. Math. Comput. 305 (2017) 1–16. [20] D.Y. Shi, J.J. Wang, F.N. Yan, Unconditional superconvergence analysis for nonlinear parabolic equation with EQrot 1 nonconforming finite element, J. Sci. Comput. 70 (2017) 85–111. [21] D.Y. Shi, C. Xu, EQrot 1 nonconforming finite element appproximation to Signorini problem, Sci. China Math. 56 (2013) 1301–1311. [22] D.Y. Shi, C. Xu, J.H. Chen, Anisotropic nonconforming EQrot 1 quadrilateral finite element approximation to second order elliptic problems, J. Sci. Comput. 56 (2013) 637–653. [23] F. Wang, W.M. Han, X.L. Cheng, Discontinuous Galerkin methods for solving elliptic variational inequalities, SIA M. J. Numer. Anal. 48 (2010) 708–733. [24] F. Wang, W.M. Han, J. Eichholz, X.L. Cheng, A posteriori error estimates for discontinuous Galerkin methods of obstacle problems, Nonlinear Anal. Real. World Appl. 22 (2015) 664–679. [25] L.H. Wang, Error estimates of two nonconforming finite elements for the obstacle problem, J. Comput. Math. 4 (1986) 11–20. [26] L.H. Wang, On the quadratic finite element approximation to the obstacle problem, Numer. Math. 92 (2002) 771–778. [27] L.H. Wang, On the error estimate of nonconforming finite element approximation to the obstacle problem, J. Comput. Math. 21 (2003) 481–490. [28] A. Weiss, B.I. Wohlmuth, A posteriori error estimator for obstacle problems, SIAM J. Sci. Comput. 32 (2010) 2627–2658. [29] C. Xu, D.Y. Shi, X. Liao, Low order nonconforming mixed finite element method for nonstationary incompressible Navier–Stokes equations, Appl. Math. Mech. Engl.Ed. 37 (2016) 1095–1112. [30] X.J. Xu, On the accuracy of nonconforming quadrilateral q1 element approximation for the Navier-Stokes problem, SIAM. J. Numer. Anal. 38 (20 0 0) 17–39.