Construction of polynomial extensions in two dimensions and application to the h-p  finite element method

Journal of Computational and Applied Mathematics 261 (2014) 249–270

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Construction of polynomial extensions in two dimensions and application to the h-p finite element method Benqi Guo a,b , Jianming Zhang c,∗ a

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

b

Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada

c

Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming 650500, China

article

abstract

info

Article history: Received 12 April 2013 Received in revised form 27 May 2013

Polynomial extensions play a vital role in the analysis of the p and h-p FEM as well as the spectral element method. In this paper, we construct explicitly polynomial extensions on a triangle T and a square S, which lift a polynomial defined on a side Γ or on whole 1

Dedicated to Professor Ben-yu Guo on the Occasion of his 70th Birthday Keywords: The p and h-p version FEM Polynomial extension Lifting Continuous operator Convolution Sobolev spaces

2 boundary of T or S. The continuity of these extension operators from H00 (Γ ) to H 1 (T ) 1 2

1 2

or H 1 (S ) and from H (∂ T ) to H 1 (T ) or from H (∂ S ) to H 1 (S ) is rigorously proved in a constructive way. Applications of these polynomial extensions to the error analysis for the h-p FEM are presented. © 2013 Elsevier B.V. All rights reserved.

1. Introduction In the analysis of the high-order finite element method (FEM), such as the p and h-p versions of FEM and the spectral element method, we need to construct a globally continuous and piecewise polynomial which has the optimal estimation for its approximation error and satisfies homogeneous or non-homogeneous Dirichlet boundary conditions. The construction of such a polynomial is started with local polynomial projections on each element for the best approximation. A simple union of local polynomial projections is not globally continuous and does not satisfy the Dirichlet boundary conditions. For the continuous Galerkin method in two and three dimensions, we have to adjust these local polynomial projections by a special technique called polynomial extension or lifting. Hence, it is essential for us to build a polynomial extension compatible to FEM subspaces, by which the union of local polynomial projections can be modified to a globally continuous and piecewise polynomial without degrading the best order of approximation error. The polynomial extensions together with local projections have led to the best estimation in the approximation error for the p FEM [1–7] and h-p FEM [8–10] and the best condition number for the preconditioning of the p FEM [11] and the h-p FEM √ [12]. y y An extension F on an equilateral triangle T = {(x, y) | √ − 1 ≤ x ≤ 1 − √ , 0 ≤ y ≤ 3

[f ]

F T ( x, y ) =

∗

√  3

2y

3

√

x+y/ 3

√ x−y/ 3

f (t )dt =



∞

f (t )H (x − t , y)dt = (f ∗ H (·, y))(x) −∞

Corresponding author. E-mail addresses: [email protected], [email protected] (B. Guo), [email protected] (J. Zhang).

0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.09.053

3 2

} was proposed in [4],

250

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

with the characteristic function H (x, y) = a continuous mapping H

1/2

√ 3 2y

y

for − √

3

[f ]

√y 3

, and H (x, y) = 0, otherwise. This extension realizes

(T ) for f ∈ Pp (I ) and FT[f ] |Γ1 = f where Γ1 = {(x, 0) | does not vanish on other sides of T even if f ∈ Pp0 (Γ1 ). Such an extension operator is not

(Γ1 ) → H (T ) such that FT

[f ]

≤ x ≤

1

∈

Pp1

x ∈ I = (−1, 1)}, but FT desired for application to high-order FEM. Hence we need to construct another extension operator which fits the need of the p and h-p FEM. Using the extension operator F it was argued in [4] that there exists a continuous extension operator 1/2 [f ] [f ] RT : H00 (Γ1 ) → H 1 (T ) [11,4] such that RT ∈ Pp1 (T ) for f ∈ Pp0 (Γ1 ) = {φ ∈ Pp (Γ1 ) | φ(−1) = φ(1) = 0} and RT |Γ1 = f , [f ]

RT |∂ T \Γ1 = 0. Unfortunately, the operator RT was not explicitly constructed. Incorporating the operators FT and RT on the triangle T and a bilinear mapping of a standard square S = (−1, 1)2 onto a truncated triangle T˜ (a trapezoid), they implicitly 1/2 generalized the polynomial extension RT to the square S, which realizes a continuous mapping RS : H00 (Γ1 ) → H 1 (S ) such [f ]

[f ]

[f ]

that RS ∈ Pp2 (S ) for f ∈ Pp0 (Γ1 ), and RS |Γ1 = f , RS |∂ S \Γ1 = 0 where Γ1 = {(x, −1) | x ∈ I = (−1, 1)}. Hereafter Pp1 (Ω )

and (Ω ) denote sets of polynomials of total and separate degree ≤ p on domain Ω = T or S, respectively. A polynomial extension from whole boundary of a triangle T was addressed and implicitly proved in [11] with a brief proof, based on the polynomial extension FT of the convolution type, which realized a continuous operator ET : H 1/2 (∂ T ) → [f ] [f ] H 1 (T ) such that ET ∈ Pp1 (T ) for f ∈ Pp (∂ Ω ) and ET |∂ T = f . This polynomial extension was utilized for preconditioning of the p version of finite element method [11]. The proof of the continuity of the operator FT in [11,4] is straightforward and easy to verify because the operator FT is constructed explicitly. On the contrary the proof for the operators RT and ET are not constructive and straightforward since they are not explicitly constructed. Recently we have noticed that the polynomial extension of convolution type has been successfully generalized to tetrahedrons in [6] where the polynomial extension of convolution type on a tetrahedron T was developed by constructing explicitly the extension operator RT . We found that the structure of the extension operator introduced there can be adopted for the polynomial extension operators RT on a triangle in two dimensions, which can make the proof constructive and straightforward. In this paper we would explicitly construct the operators RT and ET on a triangle T and the operators RS and ES on a square S, and give vigorous and constructive proof of the continuity of the operators. For the polynomial extension on a square we adopt a non-convolution approach, namely, use a spectral solution of an eigenvalue–eigenfunction (polynomial) problem and a two-point value problem on the interval (−1, 1), which is totally different from the approach in [11,4]. This approach simplifies the proof of the continuity and unifies the construction of the polynomial extensions in two and three dimensions [10]. The readers, including the beginners in the field of the p and h-p FEM, will be able to follow each step of the proof. The paper is organized as follows. In Section 2, we will construct the polynomial extension RT on a triangle T which lifts a 1/2 polynomial on a side of T , and give a vigorous and constructive proof for the continuity of the operator RT : H00 (I ) → H 1 (T ). We also construct in this section the operator ET on the triangle T in three steps, which lifts a polynomial on the boundary of T , prove the continuity of the operator ET : H 1/2 (∂ T ) → H 1 (T ). The polynomial extension operators RS and ES on a square S = (−1, 1)2 are explicitly defined in Section 3, and the continuity of these two operators is proved vigorously and constructively. In Section 4 we apply the polynomial extensions RT and RS to error analysis for the h-p FEM with quasiuniform meshes, which indicate the importance of the extensions for error estimation of the high-order FEM.

Pp2

2. Polynomial extension on a triangle 2.1. Polynomial extension R from one side of a triangle Let T = {(x, y) | 0 ≤ y ≤ 1 − x, 0 < x < 1} be a right triangle in R2 , and let Γi , 1 ≤ i ≤ 3 denote the side of the triangle T as shown in Fig. 2.1. For any integer p ≥ 0, Pp (Γi ) stands for the space of polynomials of degree ≤ p over Γi . By Pp1 (T ) and Pp2 (T ) we distinguish the polynomial spaces of total degree p and the those of separate degree p over T , respectively. α

by

β

Let L2α,β (Γ1 ) = {(1 − x) 2 x 2 f (x) ∈ L2 (Γ1 )} be the Jacobi-weighted L2 -spaces with α, β > −1. We define an operator FT [f ]

FT (x, y) =

1 y

x+y



f (ξ )dξ .

(2.1)

x

[f ]

[f ]

Obviously, FT (x, 0) = f (x), and FT (x, y) ∈ Pp1 (T ) if f ∈ Pp (Γ1 ). [f ]

Lemma 2.1. Let FT be defined as in (2.1). Then there hold for f ∈ L2 (Γ1 )

∥FT[f ] (x, y)∥L2 (T ) ≤ C ∥f (x)∥L2

,

(2.2)

≤ C ∥f (x)∥L2 (Γ1 ) ,

(2.3)

α,β (Γ1 )

∥FT[f ] (x, y)∥

1

H 2 (T )

∥FT[f ] (x, y)∥H 1 (T ) ≤ C ∥f (x)∥

1

H 2 (Γ1 )

(2.4)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

251

Fig. 2.1. A standard triangular domain T .

with α = 1, β = 0 and α = 0, β = 1. Moreover, if f (0) = f (1) = 0, then

∥FT[f ] (x, y)∥H 1 (T ) ≤ C ∥f (x)∥

1 2 (Γ ) H00 1

.

(2.5)

Proof. Letting z = x + y, we have

∥FT[f ] (x, y)∥2L2 (T ) ≤

1



1−x

 dx

0

0 1

 =

1

 dx

0

x

  1  y

2 

x +y

f (ξ )dξ  dy

x

 2  z  1    dz . f (ξ ) d ξ z − x  x

Applying Hardy’s inequality 327 of [13], b

 a

 2  x   1    dx ≤ 4 g (ξ ) d ξ x − a  a

b

|g (x)|2 dx a

with a = x, b = 1, and g (ξ ) = f (ξ ), we have 1 −x

 0

  1  y

x+y x

2    f (ξ )dξ  dy =

1 x

 z 2  1     f (ξ )dξ  dz ≤ 4 |f (z )|2 dz , (z − x)2  x x 1

which implies

∥FT[f ] (x, y)∥2L2 (T ) ≤ C

1



1



|f (z )|2 dz = C

dx 0

1



x

z |f (z )|2 dz .

(2.6)

0

For the estimation (2.2) with α − 1 = β = 0, we introduce a mapping M,

 M :

xˆ = 1 − x − y, yˆ = y,

which maps T onto itself. Then [f ]

F T ( x, y ) =

=

1 y 1 yˆ

x +y



f (ξ )dξ = x



1 yˆ



1−ˆx

f (ξ )dξ 1−ˆx−ˆy

xˆ

1

fˆ (1 − ξˆ )(−dξˆ ) =

yˆ

xˆ +ˆy

xˆ +ˆy



ˆ fˆ (ξˆ )dξˆ = F [f ] (ˆx, yˆ ),

xˆ

ˆ

where fˆ (ξˆ ) = f (1 − ξˆ ). Applying (2.2) to F [f ] (ˆx, yˆ ) we have ˆ

∥FT[f ] (x, y)∥2L2 (T ) = ∥FT[f ] (ˆx, yˆ )∥2L2 (T ) ≤ C

1



z |fˆ (z )|2 dz = C 0

1



(1 − x)|f (x)|2 dx. 0

A combination of (2.6) and (2.7) yields (2.2) with α − 1 = β = 0 and α = β − 1 = 0.

(2.7)

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B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

To prove (2.3)–(2.5) we extend f on entire x-axis R = (−∞, ∞) such that

∥f ∥

1

H 2 (R)

≤ C ∥f ∥

1

H 2 (Γ1 )

.

[f ]

Then FT (x, y) is written as a convolution [f ]

FT (x, y) =

+∞



f (t )H (x − t , y)dt = (f ∗ H (·, y))(x) −∞

with H (x, y) =

 1 y

,

−y < x < 0, otherwise.

0

[f ]

˜ (ξ , y) we denote the Fourier transform of f (x), FT (x, y) and H (x, y) with respect to the variable x. By f˜ (ξ ), F˜ (ξ , y) and H Then ˜ (ξ , y) F˜ (ξ , y) = f˜ (ξ )H with

˜ (ξ , y) = √1 H y 2π



1 eiξ y − 1 e−iξ x dx = √ . 2π iξ y −y 0

Let Ω = R × I = {(x, y)| − ∞ < x < ∞, 0 < y < 1} (resp. {(ξ , y)| − ∞ < x < ∞, 0 < y < 1}). There holds

∥FT[f ] ∥2H 1 (T ) ≤ ∥FT[f ] ∥2H 1 (Ω ) = ∥F˜ ∥2H 1 (Ω ) =  + Ω

 Ω

|f˜ (ξ )|2 |H˜ (ξ , y)|dξ dy

|f˜ (ξ )|2 |ξ |2 |H˜ (ξ , y)|2 dξ dy +

 Ω

2    2 ∂ ˜ ˜ |f (ξ )|  H (ξ , y) dξ dy. ∂y

(2.8)

˜ (ξ , y)| ≤ C , we have Noting that |H  Ω

|f˜ (ξ )|2 |H˜ (ξ , y)|2 dξ dy ≤ C



∞ −∞

|f˜ (ξ )|2 dξ = C ∥f˜ ∥2L2 (R) ≤ C ∥f ∥2L2 (Γ ) . 1

(2.9)

Letting z = yξ , we obtain

∥H˜ (ξ , ·)∥2L2 (I ) = =

1

 0

1

π |ξ |

ξ

 0

ξ

 iz   e − 1 2   dz 2π |ξ | 0  iz   for |ξ | ≤ 1, C 1 − cos z C dz = Φ (ξ ) = 1 for |ξ | > 1  z2 |ξ |

|H˜ (ξ , y)|2 dy =

1



(2.10)

which implies

 Ω

|f˜ |2 |ξ H˜ (ξ , y)|2 dξ dy ≤



∞

−∞

|f˜ (ξ )|2 |ξ |2 ∥H˜ (·, y)∥2L2 (I ) dξ ≤ C

≤ C ∥f ∥2 1

H 2 (R)

≤ C ∥f ∥2 1

H 2 (Γ1 )

Note that

∂  1 iξ yeiξ y − eiξ y + 1 H (ξ , y) = √ ∂y y2 iξ 2π and

 2 2 2  ∂    = |ξ | 2(1 − cos z ) + z − 2z sin z H (ξ , y )  ∂y  2π z2

.



∞

|ξ ||f˜ (ξ )|2 dξ −∞

(2.11)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

253

which imply

 2  |ξ | ∂  2(1 − cos z ) + z 2 − 2z sin z    dy = |ξ | H (ξ , y ) dz  ∂y  2π 0 z4 0  C |ξ |2 for |ξ | ≤ 1, ≤ Φ2 (ξ ) = C |ξ | for |ξ | > 1

|H˜ (ξ , ·)|2H 1 (I ) =

1



(2.12)

and



2   ∞  ∂ H (ξ , y) dξ dy ≤ C |f˜ (ξ )|2 |H˜ (·, y)|2H 1 (I ) dξ |f˜ (ξ )|2   ∂y −∞ Ω  ∞ |ξ ||f˜ (ξ )|2 dξ ≤ C ∥f ∥2 1 ≤C

H 2 (R)

−∞

≤ C ∥f ∥2 1

H 2 (Γ1 )

.

(2.13)

Combining (2.8)–(2.13) we have (2.4). If f (0) = f (1) = 0, we extend f (x) by zero outside of [0, 1]. Then

∥f ∥

1

H 2 (R)

≤ C ∥f ∥

1 2 (Γ ) H00 1

,

which together with (2.11) and (2.13) yields (2.5). Due to (2.10) and (2.12) it holds that

∥H˜ (ξ , ·)∥2H 1 (I ) = ∥H˜ (ξ , ·)∥2L2 (I ) + |H˜ (ξ , ·)|2H 1 (I ) ≤ Φ1 (ξ ) + Φ2 (ξ ) =



C C |ξ |

for |ξ | ≤ 1, for |ξ | > 1.

By interpolation space theory we have

∥H˜ (ξ , ·)∥

1

1 H 2 (I )

1

≤ ∥H˜ (ξ , ·)∥L22 (I ) ∥H˜ (ξ , ·)∥H2 1 (I ) ≤ C ,

which implies



∥F˜ (ξ , ·)∥2 1 dξ = H 2 (I )

R

 R

|f˜ (ξ )|2 ∥H˜ (ξ , ·)∥2 1 dξ ≤ C ∥f˜ ∥2L2 (R) .

(2.14)

H 2 (I )

Also we have by (2.10)



∥F˜ (·, y)∥2 1

 

H 2 (R)

I

 |ξ | |f˜ (ξ )|2 |H˜ (ξ , y)|2 dξ dy

dy = I

R

 ≤ R

|ξ | |f˜ (ξ )|2 ∥H˜ (ξ , ·)∥2L2 (I ) dξ ≤ C

 R

|f˜ (ξ )|2 dξ ≤ C ∥f˜ ∥2L2 (R) .

(2.15)

A combination of (2.14)–(2.15) leads to



∥F˜ ∥2 1

H 2 (Ω )

This yields (2.3).

≤C

∥F˜ (ξ , ·)∥2 1 dξ + H 2 (I )

R



∥F˜ (·, y)∥2 1

H 2 (R)

I

 dy

≤ C ∥f˜ ∥L2 (R) ≤ C ∥f ∥2L(Γ1 ) .



Remark 2.1. The estimations (2.2) with α = β = 0 and (2.4)–(2.5) were proved in [11,4]. We have improved the proof for ˜ (ξ , ·)∥L2 (I ) and |H˜ (ξ , ·)|H 1 (I ) . Moreover, we have proved the new estimations these estimations by a precise estimate of ∥H (2.4) and (2.2) with α − 1 = β = 0 and α = β − 1 = 0 in terms of ∥f ∥L2 (I ) and ∥f ∥L2 (I ) , respectively. The latter plays an α,β

essential role in the proof of Theorem 2.2 and polynomial extensions in three dimensions [10]. 1

[f ]

2 (Γ1 ) → H 1 (T ) which lifts a polynomial f (x) on Γ1 into the triangle The operator FT realizes a continuous mapping: H00

[f ]

T , but FT does not vanish on other sides Γ2 and Γ3 of T . In the application to the continuous Galerkin finite element method, one needs an extension which vanishes on Γ2 ∪ Γ3 . To this end we further introduce an operator RT by [f ]

RT (x, y) = x(1 − x − y)F 1



f

ξ (1−ξ )

[f ]



( x, y ) =

x(1 − x − y) y

x +y

 x

[f ]

f (ξ ) dξ ξ (1 − ξ )

2 for all f ∈ H00 (Γ1 ). Obviously, RT (x, y)|Γ1 = f (x) and RT (x, y)|Γ2 ∪Γ3 = 0.

(2.16)

254

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

[f ]

[f ]

1,023

Theorem 2.2. Let f (x) ∈ Pp0 (Γ1 ), and let RT (x, y) be constructed as in (2.16). Then RT (x, y) ∈ Pp

Ψ |Γ2 ∪Γ3 = 0} ⊂

Pp1

(T ) = {Ψ ∈ Pp1 (T ) |

[f ]

(T ) and RT (x, y)|Γ1 = f (x), and

∥R[Tf ] (x, y)∥L2 (T ) ≤ C ∥f (x)∥L2

(2.17)

α,β (Γ1 )

with α − 1 = β = 0 and α = β − 1 = 0, and

∥R[Tf ] (x, y)∥H 1 (T ) ≤ C ∥f ∥

1 2 (Γ ) H00 1

.

(2.18)

[f ]

[f ]

[f ]

Proof. It is easily seen that RT (x, y) ∈ Pp1 (T ), RT (x, y)|Γ1 = f (x) and RT (x, y) |Γ2 ∪Γ3 = 0. Since

   1  y

∥R[Tf ] (x, y)∥2L2 (T ) ≤

T

x +y

x

2  |f (ξ )|dξ  dxdy = ∥F [|f |] (x, y)∥2L2 (T ) ,

(2.17) follows that from Lemma 2.1. [f ] [f ] [f ] To prove (2.18) we decompose RT (x, y) into (1 − x − y)R1 (x, y) and xR2 (x, y), i.e., x(1 − x − y)

[f ]

RT (x, y) =

x +y



y

x

f (ξ ) [f ] [f ] dξ = (1 − x − y)R1 (x, y) + xR2 (x, y) ξ (1 − ξ )

with x +y



x

[f ]

R1 (x, y) =

y

f (ξ )

ξ

x

1−x−y

[f ]

dξ ,

R 2 ( x, y ) =

y

x +y

 x

f (ξ ) 1−ξ

dξ .

(2.19)

Therefore,

∂ R[Tf ] ∂ R[f ] ∂ R[f ] = −R[1f ] (x, y) + (1 − x − y) 1 + R2[f ] (x, y) + x 2 , ∂x ∂x ∂x where

∂ R[1f ] 1 = ∂x y

x +y



f (ξ )

ξ

x

dξ +

x y(x + y)

f (x + y) −

f ( x) y

and

∂ R[2f ] 1 =− ∂x y

x +y

 x

f (ξ ) 1−ξ

f (x + y)

dξ +

y

−

(1 − x − y)f (x) . y(1 − x)

By Lemma 2.1, we obtain for i = 1, 2

  x +y  1  [|f |]  ∥R[i f ] (x, y)∥L2 (T ) ≤  | f (ξ )| d ξ y  2 ≤ ∥F (x, y)∥L2 (T ) ≤ C ∥f (x)∥L2 (Γ1 ) . x L (T )

(2.20)

Noting that

∂ FT[f ] 1 = (f (x + y) − f (x)), ∂x y we have

∂ F [f ] 1 ∂ R[1f ] = T + ∂x ∂x y

x+y

 x

f (ξ )

ξ

dξ −

f (x + y) x+y

.

(2.21)

Due to (2.2) with α = β − 1 = 0, we have

  1  y

x +y x

f (ξ )

  1 dξ  ≤ C ∥x− 2 f (x)∥L2 (Γ1 )  ξ L2 (T )

(2.22)

and

   1  1 −y  1  1  f (x + y) 2 |f (x + y)|2 |f (z )|2   = dy dx = dy dz  x+y 2 2 (x + y) z2 0 0 0 y L (T )  1  z  1 |f (z )|2 |f (z )|2 1 = dz dy = dz = C ∥x− 2 f (x)∥L2 (Γ1 ) . 2 0

0

z

0

z

(2.23)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

255

By (2.22)–(2.23) and (2.5) we get

   ∂ R[f ]   1     ∂x  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

.

(2.24)

Similarly, there holds

∂ F [f ] 1 ∂ R[2f ] = T − ∂x ∂x y

f (ξ )

x +y



dξ +

1−ξ

x

f (x) 1−x

.

Due to (2.2) with α − 1 = β = 0, we have

  1  y

f (ξ )

x +y

  dξ  1−ξ 2

1

≤ C ∥(1 − x)− 2 f (x)∥L2 (Γ1 ) .

L (T )

x

(2.25)

Note that there holds

   f (x) 2   1 − x 2

1

 =

1−x

 dx

L (T )

0

0

|f (x)|2 dy = (1 − x)2

1

 0

|f (x)|2 dx 1−x

which together with (2.25) and (2.5) leads to

   ∂ R[f ]   2     ∂x  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

.

(2.26)

Therefore, a combination of (2.20), (2.24) and (2.26) leads to

   ∂ R[f ]   T     ∂x  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

.

(2.27)

We next to show that

   ∂ R[f ]   T     ∂y  2

≤ C ∥f ∥

L (T )

1 2 (Γ ) H00 1

.

(2.28)

Since 1 ∂ FT[f ] =− 2 ∂y y

x +y



1

f (ξ )dξ +

y

x

f (x + y)

and

∂ R[f ] ∂ R[Tf ] ∂ R[f ] = −R[1f ] (x, y) + (1 − x − y) 1 + x 2 ∂y ∂y ∂y with

∂ R[1f ] x =− 2 ∂y y

x +y



f (ξ )

ξ

x

dξ +

xf (x + y) y(x + y)

∂ R[2f ] 1−x =− 2 ∂y y

,

x +y

 x

f (ξ ) 1−ξ

dξ +

f (x + y) y

,

we have

∂ R[1f ] ∂ F [f ] 1 − T = 2 ∂y ∂y y

x+y





f (ξ ) 1 − x

x



ξ

dξ −

f (x + y) x+y

.

(2.29)

Since 0 ≤ 1 − ξx ≤ ξ for x ≤ ξ ≤ x + y, there holds y

  1   y2

x+y



f (ξ ) 1 − x

x



ξ

   1  dξ  ≤ y

x +y x

|f (ξ )| dξ = F ξ



|f (ξ )| ξ



(x, y),

which together with (2.22) leads to

  1   y2

x +y



f (ξ ) 1 − x

x

ξ



  dξ  2

L (T )



≤ ∥F

|f (ξ )| ξ



1

(x, y)∥L2 (T ) ≤ C ∥x− 2 f (x)∥L2 (Γ1 ) .

(2.30)

256

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

Fig. 2.2. Mapping between equilateral triangle T and right triangle  T.

By (2.23), (2.30) and (2.5), we get

   ∂ R[f ]   1     ∂y  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

.

(2.31)

Similarly, there holds

  x +y ∂ R[2f ] ∂ F [f ] 1 − x x+y f (ξ ) 1 − =− 2 dξ + 2 f (ξ )dξ ∂y ∂y y 1−ξ y x x  x +y  x +y 1 f (ξ ) 1 f (ξ ) =− 2 dξ = − 2 dξ . (1 − x − (1 − ξ )) (ξ − x) y x 1−ξ y x 1−ξ ξ −x

≤ 1 for x ≤ ξ ≤ x + y, we obtain y    x +y    |f (ξ )|  1 x+y |f (ξ )|  1 f (ξ ) 1−ξ ≤ − − x d ξ d ξ = F (x, y), (ξ )  y2 1−ξ  y x 1−ξ x

Since

which together with (2.2) implies

   1 −  y2

f (ξ )

x+y

(ξ − x) x

1−ξ

 



dξ  

≤ ∥F

|f (ξ )| 1−ξ



L2 (T )

1

∥L2 (T ) ≤ C ∥(1 − x)− 2 f (x)∥L2 (Γ1 ) .

(2.32)

By (2.5) and (2.32) we get

   ∂ R[f ]   2     ∂y  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

.

(2.33)

Therefore, combining (2.20), (2.31) and (2.33) we have

   ∂ R[f ]   T     ∂y  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ) H00 1

. 

We now consider the lifting theorem on an equilateral triangle T shown as in Fig. 2.2. The operators FT and RT are defined as [f ]

FT (x, y) =

√  3

2y

y 3

x+ √ y 3

f (t )dt

(2.34)

x− √

and [f ]

RT (x, y) =



y



1−x− √ 3

1+x− √

√ 

y 3 1−x− √

=

y

2y

y 1+x− √

3



f

(1−t 2 )

F

3



3



 



y 3

x+ √ y x− √ 3

( x, y ) f (t ) 1 − t2

dt .

(2.35)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

[f ]

257

[f ]

It is easy to verify that FT (x, y) and RT (x, y) on the equilateral triangle T are the images of those on the right triangle  T defined as in (2.1) and (2.16) under a linear mapping M of T onto  T,

M :

   1 y  ξ = , 1+x− √ 2

3

y  η = √ . 3

i (i = 1, 2, 3). Therefore, we have the lifting theorem on the equilateral triangle T . M maps T onto  T and Γi onto Γ Corollary 2.3. Let T is an equilateral triangle shown as in Fig. 2.2, and let FT and RT be the operators defined as in (2.34) and [f ] [f ] (2.35). If f (x) ∈ Pp (Γ1 ), then FT (x, y) ∈ Pp1 (T ), FT (x, y)|Γ1 = f (x), and

∥FT[f ] (x, y)∥H 1 (T ) ≤ C ∥f ∥

1

H 2 (Γ1 )

.

(2.36) 1,023

[f ]

Furthermore, If f (x) ∈ Pp0 (Γ1 ), then RT (x, y) ∈ Pp

∥R[Tf ] (x, y)∥H 1 (T ) ≤ C ∥f ∥

1 2 (Γ ) H00 1

(T ) ⊂ Pp1 (T ) and R[Tf ] (x, y)|Γ1 = f (x), and

.

(2.37)

Remark 2.2. It has been an open question that

∥R[Tf ] (x, y)∥

1

H 2 (T )

≤ C ∥f ∥L2 (I ) .

Recently it was claimed for f ∈ Pp (Γ1 ) in [14] that

∥R[Tf ] (x, y)∥

1

H 2 (T )

≤ C log p∥f ∥L2 (I ) .

2.2. Polynomial extension ET from whole boundary of a triangle We shall construct an extension ET of a polynomial defined on whole boundary ∂ T of a triangle T , and prove that ET is a continuous operator: H 1/2 (∂ T ) → H 1 (T ), which lifts a polynomial f ∈ Pp (∂ T ) = {f ∈ C 0 (∂ T ), f |Γi ∈ Pp (Γi ), 1 ≤ i ≤ 3} to the interior of T . 1

2 We introduce the Sobolev space H00 (Γ1 , 0) by 1

1

1

2 H00 (Γ1 , 0) = {v ∈ H 2 (Γ1 )|x− 2 v ∈ L2 (Γ1 )}

(2.38)

with the norm

2 (Γ ,0) H00 1

|v(x)|2



= ∥v∥2 1

∥v∥2 1

+

H 2 (Γ1 )

x

Γ1

1

dx.

[f ]

2 Lemma 2.4. Let f (x) ∈ H00 (Γ1 , 0), and let R1 (x, y) be given by (2.19). Then

∥R[1f ] (x, y)∥H 1 (T ) ≤ C ∥f (x)∥

1 2 (Γ ,0) H00 1

.

(2.39)

Proof. By (2.21)–(2.23), (2.29)–(2.30) and Theorem 2.2 we get

   ∂ R[f ]   1     ∂x  2

≤ C ∥f (x)∥

L (T )

1 2 (Γ ,0) H00 1

and

(2.20) and (2.40) lead immediately to (2.39).

   ∂ R[f ]   1     ∂y  2

L (T )

≤ C ∥f (x)∥

1 2 (Γ ,0) H00 1

.

(2.40)



Theorem 2.5. There exists a linear operator ET : H 1/2 (∂ T ) → H 1 (T ) such that ET |∂ T = f and ET and [f ]

∥ET[f ] ∥H 1 (T ) ≤ C ∥f ∥

1

H 2 (∂ T )

,

where C is a constant independent of f and p.

[f ]

∈ Pp1 (T ) for f ∈ Pp (∂ T ), (2.41)

258

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

Proof. Let f ∈ Pp (∂ T ) and fi = f |Γi , 1 ≤ i ≤ 3. F [fi ] stand for a lifting operator associated to the side Γi , defined as in (2.1). [f ]

We shall construct ET by three concrete steps. [f ]

[f ]

[f ]

First of all, we lift f1 by FT 1 . Due to Lemma 2.1, FT 1 ∈ Pp (T ), FT 1 |Γ1 = f1 and [f ]

∥FT 1 ∥H 1 (T ) ≤ C ∥f1 ∥

1

H 2 (Γ1 )

.

(2.42) [f ]

Secondly, we shall simultaneously lift f1 and f3 . Let ψ3 (y) = f3 (y) − FT 1 |Γ3 = f3 (y) − F [f1 ] (0, y), and let [ψ3 ]

[f ]

φ13 (x, y) = FT 1 (x, y) + R1

(y, x) = F [f1 ] (x, y) +



y x

y+x

y

ψ3 (ξ ) dξ . ξ

It is easy to see that ψ3 (0) = f3 (0) − f1 (0) = 0 and φ13 |Γi = fi , i = 1, 3, and we further claim that

∥ψ3 ∥

≤ C ∥f ∥

1 2 (Γ ,0) H00 3

1

H 2 (Γ1 ∪Γ3 )

,

(2.43)

1

2 (Γ3 , 0) is defined as in (2.38). Note that due to (2.42) where H00

∥ψ3 ∥

= ∥f3 − F [f1 ] ∥

1

H 2 (Γ3 )

≤ ∥f3 ∥

1

H 2 (Γ3 )

1

H 2 (Γ3 )

≤ ∥f3 ∥

1

H 2 (Γ3 )

+ ∥F [f1 ] ∥

+ C ∥F [f1 ] ∥H 1 (T ) ≤ ∥f3 ∥

1

H 2 (Γ3 )

1

H 2 (Γ3 )

+ C ∥f 1 ∥

1

H 2 (Γ1 )

C ∥f ∥

1

H 2 (Γ1 ∪Γ3 )

.

1

2 Hence, due to the definition of H00 (Γ3 , 0), it remains to prove that

1



|ψ3 (y)|2 y

0

|(f3 − F [f1 ] |Γ3 )(y)|2

1

 dy =

y

0

dy ≤ C ∥f ∥2 1

H 2 (Γ1 ∪Γ3 )

.

(2.44)

Since F [f1 ] |Γ1 = f1 , F [f1 ] (y, 0) = f1 (y), and (f3 − F [f1 ] |Γ3 )(y) = f3 (y) − F [f1 ] (0, y), we have

(f3 − F [f1 ] |Γ3 )(y) = f3 (y) − F [f1 ] (0, y) = (f3 (y) − f1 (y)) + (F [f1 ] (y, 0) − F [f1 ] (0, y)). 1

It is known that (e.g. see [11,15]) the following norm is an equivalent norm for the space H 2 (Γ1 ∪ Γ3 )

∥f ∥

1 H2

(Γ1 ∪Γ3 )

 ≈ ∥f 1 ∥2 1

H 2 (Γ1 )

+ ∥ f 3 ∥2 1

H 2 (Γ3 )

+ D(f1 , f3 )

 21

,

where D(f1 , f3 ) =



1

(f1 (t ) − f3 (t ))2 t

0

dt .

(2.45)

Therefore 1



|f3 (y) − f1 (y)|2 y

0

dy ≤ ∥f ∥2 1

H 2 (Γ1 ∪Γ3 )

and 1



|F [f1 ] (y, 0) − F [f1 ] (0, y)|2 y

0

dy = D(F [f1 ] |Γ1 , F [f1 ] |Γ3 )

≤ C ∥F [f1 ] ∥2 1

H 2 (Γ1 ∪Γ3 )

≤ C ∥F [f1 ] ∥2H 1 (T ) ≤ C ∥f1 ∥2 1

H 2 (Γ1 )

,

which imply (2.44). By Lemma 2.4 and (2.19) there hold [ψ3 ]

∥R1

(y, x)∥H 1 (T ) ≤ C ∥ψ3 ∥

1 2 (Γ ,0) H00 3

≤ C ∥f ∥

1

H 2 (Γ1 ∪Γ3 )

and [ψ3 ]

∥φ13 ∥H 1 (T ) = ∥F [f1 ] + R1 ≤ C (∥f1 ∥

1

H 2 (Γ1 )

[ψ3 ]

(y, x)∥H 1 (T ) ≤ ∥F [f1 ] ∥H 1 (T ) + ∥R1 + ∥ψ3 ∥

1 2 (Γ ,0) H00 3

Therefore, the claim (2.43) follows immediately.

) ≤ C ∥f ∥

1

H 2 (Γ1 ∪Γ3 )

(y, x)∥H 1 (T )

.

(2.46)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

259

Fig. 2.3. Mapping M of right triangle  T onto right triangle T .

[ψ2 ]

Let ψ2 = f2 − φ13 |Γ2 , By Lemma 2.1 there exists an extension Ψ2 (x, y) = FT

Ψ2 (x, y) |Γ2 = Ψ2 (x, 1 − x) = ψ2 (x),

∈ Pp1 (T ), such that Ψ2 |Γ2 = ψ2 . Note that

0 ≤ x ≤ 1,

and let 

F2

ψ2 (ξ ) ξ (1−ξ )

[ψ2 ]

Then xyF2



xyF2



x+y−1

ψ2 (ξ ) dξ . ξ (1 − ξ )

x



1

(x, y) =

1−y

(x, y) ∈ Pp1 (T ) vanishing on Γi , i = 1, 3, and

ψ2 (ξ ) ξ (1−ξ )





|Γ2 = x(1 − x)F2

ψ2 (ξ ) ξ (1−ξ )



|Γ2 = ψ2 (x).

We now introduce the lifting operator ET by  [f ]

ET = φ13 + xyF2

ψ2 (ξ ) ξ (1−ξ )



.

(2.47) [f ]

It is not difficult to verify that ET |Γi = fi , i = 1, 3, and  [f ]

ET |Γ2 = φ13 |Γ2 +xyF2

ψ2 ξ (1−ξ )



|Γ2 = φ13 |Γ2 +ψ2 = φ13 |Γ2 +f2 − φ13 |Γ2 = f2 .

Let M is a bilinear mapping:



x = 1 − x˜ − y˜ , y = x˜ ,

which maps T˜ onto T , Γ˜ 1 onto Γ2 , Γ˜ 2 onto Γ3 and Γ˜ 3 onto Γ1 as shown in Fig. 2.3. Noting that 

xyF2

ψ2 (ξ ) ξ (1−ξ )



◦M =

x˜ (1 − x˜ − y˜ )



−˜y

1−˜x−˜y

1−˜x



= x˜ (1 − x˜ − y˜ )FT

ψ˜ 2 (η) η(1−η)

ψ2 (ξ ) x˜ (1 − x˜ − y˜ ) dξ = ξ (1 − ξ ) y˜



x˜ +˜y

x˜

ψ2 (1 − η) dη η(1 − η)

 [ψ˜ 2 ]

= RT

,

˜ 2 (˜x) = ψ2 ◦ M = ψ2 (1 − x˜ ), we have by Theorem 2.2, where FT and RT are defined by (2.1) and (2.16) and ψ [ψ˜ 2 ]

∥R T

∥H 1 (T˜ ) ≤ C ∥ψ˜ 2 ∥

1 2 (Γ˜ ) H00 1

.

Since  [f ]

∥ET ∥H 1 (T ) ≤ ∥φ13 ∥H 1 (T ) + ∥xyF2

ψ2 (ξ ) ξ (1−ξ )



 ∥H 1 (T ) ≤ C ∥f ∥

 1

H 2 (Γ1 ∪Γ3 )

+ ∥ψ˜ 2 ∥

1 2 (Γ˜ ) H00 1

,

we need to show

∥ψ˜ 2 ∥

1 2 (Γ˜ ) H00 1

≤ C ∥f ∥

1

H 2 (∂ T )

.

(2.48)

260

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

Note that

∥ψ˜ 2 ∥

 1

H 2 (Γ˜ 1 )



≤ C ∥ψ2 ∥ 1 ≤ C ∥f2 ∥ 1 + ∥φ13 ∥ 1 H 2 (Γ2 ) H 2 (Γ2 ) H 2 (Γ2 )   ≤ C ∥f2 ∥ 1 + ∥f ∥ 1 ≤ C ∥f ∥ 1 H 2 (Γ2 )

H 2 (Γ1 ∪Γ3 )

 ≤ C ∥f2 ∥

H 2 (Γ1 ∪Γ2 ∪Γ3 )

 + ∥φ13 ∥H 1 (T )

1

H 2 (Γ2 )

,

and it remains to show 1

 0

|ψ˜ 2 (˜x)|2 dx˜ ≤ C ∥f ∥2 1 1 − x˜ H 2 (∂ T )

(2.49)

|ψ˜ 2 (˜x)|2 dx˜ ≤ C ∥f ∥2 1 . x˜ H 2 (∂ T )

(2.50)

and 1

 0

[f ]

[ψ3 ]

[f ]

Let f˜ = f ◦ M , F˜T 1 = FT 1 ◦ M and R˜ 1 f˜1 (˜x) = f2 (1 − x˜ ),

[ψ3 ]

= R1

f˜3 (˜y) = f1 (1 − y˜ ),

◦ M, and let f˜i be restriction of f˜ on Γ˜ i , 1 ≤ i ≤ 3. Then f˜2 (˜x) = f3 (˜x)

and [f ]

[ψ3 ]

[f ]

F˜T 1 (˜x, y˜ ) = FT 1 (1 − x˜ − y˜ , x˜ ),

R˜ 1

[ψ3 ]

(˜x, y˜ ) = R1

[f ]

(˜x, 1 − x˜ − y˜ ).

[f ]

[ψ3 ]

[f ]

Since f˜3 (˜x) = f1 (1 − x˜ ) = FT 1 (1 − x˜ , 0) = F˜T 1 (0, x˜ ) = F˜T 1 |Γ˜3 , R˜ 1 there holds

[ψ ] [ψ ] [ψ ] (0, x˜ ) = R˜ 1 3 |Γ˜3 = 0 and R˜ 1 3 (˜x, 0) = R˜ 1 3 |Γ˜1 ,

[ψ ] [f ] ψ˜ 2 (˜x) = ((f2 − φ13 |Γ2 ) ◦ M )(˜x) = f˜1 (˜x) − φ˜ 13 |Γ˜1 = f˜1 (˜x) − F˜T 1 (˜x, 0) − R˜ 1 3 (˜x, 0) [ψ ] [ψ ] [f ] [f ] = f˜1 (˜x) − f˜3 (˜x) + F˜T 1 (0, x˜ ) − F˜T 1 (˜x, 0) + R˜ 1 3 (0, x˜ ) − R˜ 1 3 (˜x, 0). 1

Noting that the following norm is an equivalent norm for the space H 2 (Γ1 ∪ Γ2 ∪ Γ3 )

 ∥f ∥

1

H 2 (Γ1 ∪Γ2 ∪Γ3 )

≈

3  i=1

∥fi ∥

2 1

+

H 2 (Γi )

 12

3 

D(fi , fj )

,

i,j=1,i
where D(fi , fj ) is defined as in (2.45), we have by (2.42) and (2.43) 1

 0

1

 0

|f˜1 (˜x) − f˜3 (˜x)|2 dx˜ = D(f˜1 , f˜3 ) ≤ C ∥f˜ ∥2 1 ≤ C ∥f ∥2 1 , x˜ H 2 (Γ˜ 1 ∪Γ˜ 3 ) H 2 (Γ2 ∪Γ1 ) |F˜ [f1 ] (0, x˜ ) − F˜ [f1 ] (˜x, 0)|2 dx˜ = D(F˜ [f1 ] |Γ˜3 , F˜ [f1 ] |Γ˜1 ) ≤ C ∥F˜ [f1 ] ∥2 1 x˜ H 2 (Γ˜ 3 ∪Γ˜ 1 ) ≤ C ∥F˜ [f1 ] ∥2H 1 (T˜ ) ≤ C ∥F [f1 ] ∥2H 1 (T ) ≤ C ∥f1 ∥2 1

H 2 (Γ1 )

,

and 1

 0

[ψ3 ]

|R˜ 1

[ψ3 ]

(0, x˜ ) − R˜ 1 x˜

(˜x, 0)|2

[ψ3 ]

dx˜ = D(R˜ 1

[ψ ] [ψ ] |Γ˜3 , R˜ 1 3 |Γ˜1 ) ≤ C ∥R˜ 1 3 ∥2 1

H 2 (Γ˜ 3 ∪Γ˜ 1 )

[ψ3 ] 2

˜ [ψ3 ] 2

≤ C ∥R1

∥H 1 (T ) ≤ C ∥R1

∥H 1 (T ) ≤ C ∥f ∥2 1

H 2 (Γ1 ∪Γ3 )

.

Then the above inequalities lead to (2.50). Noting that [f ] [f ] [f ] [f ] [f ] FT 1 (0, x˜ ) = FT 1 (0, y) = FT 1 |Γ3 = F˜T 1 |Γ˜2 = F˜T 1 (˜x, 1 − x˜ ), [ψ3 ]

ψ3 (˜x) = ψ3 (y) = R1

[ψ3 ]

(y, x)|Γ3 = R˜ 1

[ψ3 ]

(˜x, y˜ )|Γ˜2 = R˜ 1

(˜x, 1 − x˜ ),

and [f ]

[f ]

ψ3 (˜x) = ψ3 (y) = f3 (y) − F [f1 ] (x, y)|Γ3 = f˜2 (˜x) − F˜T 1 (˜x, y˜ )|Γ˜2 = f˜2 (˜x) − F˜T 1 (˜x, 1 − x˜ )|,

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

261

Fig. 3.1. Square S = (−1, 1)2 and sides Γi , 1 ≤ i ≤ 4.

we rewrite [ψ ] [f ] ψ˜ 2 (˜x) = ((f2 − φ13 |Γ2 ) ◦ M )(˜x) = f˜1 (˜x) − φ˜ 13 |Γ˜1 = f˜1 (˜x) − F˜T 1 (˜x, 0) − R˜ 1 3 (˜x, 0) [ψ ] [f ] [f ] [f ] = f˜1 (˜x) − f˜2 (˜x) + f˜2 (˜x) − FT 1 (0, x˜ ) + FT 1 (0, x˜ ) − F˜T 1 (˜x, 0) − R˜ 1 3 (˜x, 0) [ψ ] [f ] [f ] [f ] = f˜1 (˜x) − f˜2 (˜x) + f˜2 (˜x) − F˜T 1 (˜x, 1 − x˜ ) + F˜T 1 (˜x, 1 − x˜ ) − F˜T 1 (˜x, 0) − R˜ 1 3 (˜x, 0) [ψ ] [ψ ] [f ] [f ] = f˜1 (˜x) − f˜2 (˜x) + R˜ 1 3 (˜x, 1 − x˜ ) − R˜ 1 3 (˜x, 0) + F˜T 1 (˜x, 1 − x˜ ) − F˜T 1 (˜x, 0).

Then we have 1

 0

1

 0

|f˜1 (˜x) − f˜2 (˜x)|2 ≤ C ∥f ∥2 1 , dx˜ = D(f˜1 , f˜2 ) ≤ C ∥f˜ ∥2 1 1 − x˜ H 2 (Γ2 ∪Γ3 ) H 2 (Γ˜ 1 ∪Γ˜ 2 ) |F˜ [f1 ] (˜x, 1 − x˜ ) − F˜ [f1 ] (˜x, 0)|2 dx˜ = D(F˜ [f1 ] |Γ˜2 , F˜ [f1 ] |Γ˜1 ) 1 − x˜ [f ] [f ] [f ] ≤ C ∥F˜T 1 ∥2H 1 (T˜ ) ≤ C ∥FT 1 ∥2H 1 (T ) ≤ C ∥f1 ∥2 1 ≤ C ∥F˜T 1 ∥2 1 H 2 (Γ˜ 2 ∪Γ˜ 1 )

H 2 (Γ1 )

,

and 1

 0

[ψ3 ]

|R˜ 1

[ψ3 ]

(˜x, 1 − x˜ ) − R˜ 1 1 − x˜

(˜x, 0)|2

[ψ3 ]

dx˜ = D(R˜ 1

[ψ ] |Γ˜2 , R˜ 1 3 |Γ˜1 )

[ψ3 ] 2

≤ C ∥R˜ 1 which imply (2.49) and (2.48) follow immediately.

∥

[ψ3 ] 2

1

H 2 (Γ˜ 2 ∪Γ˜ 1 )

≤ C ∥R˜ 1

[ψ3 ] 2

∥H 1 (T˜ ) ≤ C ∥R1

∥H 1 (T ) ≤ C ∥f ∥2 1

H 2 (Γ1 ∪Γ3 )

,



Remark 2.3. The extension ET can be constructed on an equilateral triangle, and (2.41) holds. 1

Remark 2.4. The extension ET is constructed in three steps. We introduce three continuous extensions: H 2 (Γ1 ) → H 1 (T ), 1 2

1 2

H00 (Γ3 , 0) → H 1 (T ) and H00 (Γ2 ) → H 1 (T ). Using these extensions we first lift f1 from Γ1 to T , then f1 and f3 form Γ1 ∪ Γ3 to T , and finally f from whole boundary ∂ T to T . Remark 2.5. The techniques and arguments for the extension operator ET on a triangle can be generalized to the construction of extension on a tetrahedron [6], in which there are 4 steps for a tetrahedron, instead of three steps, i.e., lifting a polynomial from a face to a tetrahedron T ; lifting polynomial on two faces to T ; lifting polynomial from three faces to T , and lifting polynomial from all faces of T to T . It is worth indicating that the proof of continuity of ET in [6] was incomplete because the proof of the continuity of the lifting operator constructed at last two steps, which is the most difficult and essential part, was not provided. Fortunately, the proof of the continuity of the extension ET on a triangle from all sides to T fills this gap and can be generalized to the extension on a tetrahedron. 3. Polynomial extension on a square 3.1. Polynomial extension RS from one side of a square Let S = (−1, 1)2 be a standard square, and let Γi , 1 ≤ i ≤ 4 be its sides shown in Fig. 3.1. For an integer p ≥ 0, Pp2 (S )

is the set of polynomial of separate degree ≤ p, and Pp2,0 (S ) denotes its subset of polynomial ϕ(x, y) ∈ Pp2 (S ) vanishing on its boundary ∂ S =

4

i =1

Γi .

262

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

α,β

Let Jn

(x) be the Jacobi polynomial of degree n;

(−1)n (1 − x)−α (1 + x)−β dn (1 − x)n+α (1 + x)n+β , 2n n! dxn with weights α, β > −1, and let Jnα,β (x) =

1 − x2 2,2 ϕi (x) =  Ji−1 (x), γi2−,12 2,2

1

n≥0

(3.1)

i = 1, 2, 3, . . . ,

2 ,2

(3.2)

25 i(i+1)

where γi−1 = −1 |Ji−1 (x)|2 (1 − x2 )2 dx = (2i+3)(i+2)(i+3) . Due to the properties of the Jacobi polynomials, there holds

(ϕi (x), ϕj (x))L2 (I ) = δij ,

1 ≤ i, j ≤ p − 1,

(3.3)

and ϕi (x), i = 1, 2 · · · p − 1 form an orthonormal basis of Pp0 (I ) in the space L2 (I ). We consider an eigenvalue problem

 ′′ −u (x) = λu(x) u(−1) = u(1) = 0,

in I = (−1, 1),

(3.4)

and its spectral solution (λp , ψp ) with ψp ∈ Pp0 (I ) which satisfies



ψp′ q′ dx = λp I



ψp qdx,

∀q ∈ Pp0 (I ).

(3.5)

I

Selecting the basis {ϕi (x), i = 1, 2, . . . , p − 1} as in (3.2) and letting ψp (x) = system of linear algebraic equations

− →

− →

p−1 i =1

ci ϕi (x), we have the corresponding

− →

K C = λM C = λ C ,

− →

p−1

where C = (c1 , c2 , . . . , cp−1 )T , K = (kij )i,j=1 with kij = I ∇ϕi ∇ϕj dx. Here we used the orthonormality of ϕn (x) in L2 (S ) which implies the matrix M = I. Therefore the spectral solution of eigenvalue problem (3.5) is equivalent to the eigenvalue problem of matrix K . Since K is symmetric and positive definite, the eigenvalues λp,k > 0, k = 1, 2, . . . , p − 1 and the

− →(k)

corresponding eigenvectors C



= (c1(k) , c2(k) , . . . , cp(k−)1 )T are orthogonal. We normalize these eigenvectors such that

p−1

 (k) (l) − →(k) − →(l) (C , C )= ci ci = δk,l ,

1 ≤ k, l ≤ p − 1.

i =1

Np

(k)

Consequently, λp,k > 0 and ψp,k = n=1 cn ϕn (x1 , x2 ), k = 1, 2, . . . , p − 1 are the spectral solutions of the problem (3.5). Due to the properties of eigenvalues and vectors of K , we have the following lemma. Lemma 3.1. The problem (3.5) has p − 1 real eigenvalues, and the corresponding eigen-polynomials {ψp,k (x), 1 ≤ k ≤ p − 1} are orthogonal in L2 (I ) and H 1 (I ), which form a L2 -orthonormal basis of Pp0 (I ). Proof. The problem (3.5) has p − 1 real eigenvalues because the corresponding stiffness matrix K is positive definite, and there hold for 1 ≤ k, k′ ≤ Np

(ψp,k , ψp,k′ )L2 (I ) =

p−1  p−1 

(k) (k′ )

ci cj

− →(k) − →(k′ ) (ϕi , ϕj )L2 (I ) = ( C , C ) = δk,k′

j=1 i=1

and

 S

ψp′ ,k ψp′ ,k′ dx = λk



ψp,k ψp,k′ dx = λk δk,k′ . S

Therefore, {ψp,k , k = 1, 2, . . . , p − 1} is an orthogonal base in L2 (I ) and H 1 (I ), and is orthonormal in L2 (I ).



We next consider a two-point boundary value problem



−vp′′,k (y) + λp,k vp,k (y) = 0, vp,k (−1) = 1, vp,k (1) = 0,

y ∈ I = (−1, 1),

(3.6)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

263

and its spectral solution φp,k (y) ∈ 0 Pp (I ) = {ϕ ∈ Pp (I ) | ϕ(1) = 0} such that φp,k (−1) = 1, φp,k (1) = 0, and

 I

(φp′ ,k q′ + λp,k φp,k q)dy = 0 ∀q ∈ Pp0 (I ),

which is equivalent to find φp,k = φ˜ p,k +

 I

1−y 2

(3.7)

with φ˜ p,k ∈ Pp0 (I ) satisfying

(φ˜ p′ ,k (y)q′ (y) + λp,k φ˜ p,k (y)q(y))dy = −

λp,k



2

(1 − y)q(y)dy.

(3.8)

I

Since the corresponding bilinear form is coercive and continuous on H01 (I ) × H01 (I ), the variational equation has unique solution φ˜ p,k (x3 ) ∈ Pp0 (I ) for each λp,k .

Lemma 3.2. Let λp,k be an eigenvalue of the problem (3.5), and let φp,k (x3 ) be the corresponding solution of two-point value problem (3.6). Then



1

−1

 (|φp′ ,k (y)|2 + λp,k |φp,k (y)|2 )dy ≤ C λp,k ,

Proof. We refer readers to [16,10].

k = 1, 2, . . . , Np .

(3.9)



Let f (x) ∈ Pp0 (Γ1 ) with Γ1 = {(x, −1)| − 1 ≤ x ≤ 1}. Since {ψp,k (x), 1 ≤ k ≤ p − 1} is an orthogonal basis of Pp0 (I ), f (x) =

p−1 

βk ψp,k (x)

k=1

with βk = [f ]

1

 I

2 (Γ1 ) → H 1 (S ) f (x)ψp,k (x)dx. We introduce the following extension operator RS : H00

RS (x, y) =

p−1 

βk ψp,k (x)φp,k (y).

(3.10)

k=1

Obviously, [f ]

p−1 

[f ]

RS |Γ1 = RS (x, −1) =

βk ψp,k (x) = f (x),

k=1

where Γ1 = {(x, −1)| − 1 < x < 1} is a side of the square S = (−1, 1)2 . [f ]

[f ]

[f ]

[f ]

Theorem 3.3. Let f ∈ Pp0 (Γ1 ), and let RS be given as in (3.10). Then RS ∈ Pp2 (S ), RS |Γ1 = f , RS |∂ S \Γ1 = 0, and

∥R[Sf ] ∥H 1 (S ) ≤ C ∥f ∥

1 2 (Γ ) H00 1

,

(3.11)

where C is a constant independent of p and f . [f ]

Proof. Since ψp,k ∈ Pp0 (I ) and φp,k ∈ 0 Pp (I ), RS |Γ1 = f and vanish on ∂ S \ Γ1 . Due to the orthogonality of the ψp,k in L2 (S ) and H 1 (S ), and by using (3.5) and Lemma 3.2 we have

∥R[Sf ] ∥2L2 (S ) =

p−1 

βk2 

k=1

1

λp,k

and

|R[Sf ] |2H 1 (S ) = =

 D

    ∂ RS f  ∂x

p−1 

βk2



Np

=

 k=1

|ψp,k |2 dx I

k=1

βk2

 I

 2  [f ] 2  ∂ R  +  S   dxdy   ∂y   I

|φp′ ,k |2 dy +

 I

|ψp′ ,k |2 dx

(|φp′ ,k |2 + λp,k |φp,k |2 )dy ≤ C

p−1  k=1



 |φp,k |2 dy

I

 βk2 λp,k .

264

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

Therefore

∥R[Sf ] ∥2H 1 (S ) ≤ C

p−1 

βk2 (1 +

 λp,k ).

(3.12)

k=1

Note that

∥f ∥2L2 (Γ ) =

Np 

1

βk2 ,

∥f ∥2H 1 (Γ ) = 1

0

k=1

Np 

βk2 (1 + λp,k ).

k=1

By interpolation space theory [17,18,16,19]

∥f ∥2 1

≈

2 (Γ ) H00 1

Np 

1

βk2 (1 + λp,k ) 2 ≈

k=1

Np 

βk2 (1 +

 λp,k ),

k =1

which together with (3.12) implies (3.11).



Analogously, we consider spectral solutions in either Pp2 (Γ1 ) or 0 Pp2 (Γ1 ) = {ϕ ∈ Pp (Γ1 ) | ϕ(1) = 0} for the correspond√

ing eigen-value problem (3.5). Obviously, {

(2i+1) 2

(1−x) 2,0 Ji−1 (x), 1 γi2−,01

Li (x), 0 ≤ i ≤ p} and { 

2,0

≤ i ≤ p} are the orthonormal bases

of Pp2 (Γ1 ) and 0 Pp2 (Γ1 ), respectively, where Ji−1 (x1 ) denote the Legendre and the Jacobi polynomials with α − 2 = β = 0. The arguments for Theorem 3.3 can be carried out except replacing Pp0 (Γ1 ) by Pp2 (Γ1 ) or 0 Pp2 (Γ1 ). Therefore, we have the following two theorems which are in parallel to Theorem 3.3. Theorem 3.4. Suppose that f ∈ Pp (Γ1 ) and f = Pp (I ), I = (−1, 1). Let [f ]

R˜ S (x, y) =

p 

p

k=0

βk ψp,k (x) where ψp,k (x) are the eigenfunctions of the problem (3.5) on

βk ψp,k (x)φp,k (y)

k=0

[f ]

where φp,k (y) are the spectral solutions of two-point boundary value problem (3.6) on Pp (I ), then R˜ S

˜ [f ]

∈ Pp2 (S ) such that

[f ]

RS |Γ1 = f , RS |Γ3 = 0, and

∥R˜ [Sf ] ∥H 1 (D) ≤ C ∥f ∥

1

H 2 (Γ1 )

,

(3.13)

where C is a constant independent of p and f . Theorem 3.5. Suppose that f ∈ 0 Pp2 (Γ1 ) and f = 0

Pp (I ), I = (−1, 1). Let [f ]

Rˆ S (x, y) =

p 

p

k=1

βk ψp,k (x) where ψp,k (x) are the eigenfunctions of the problem (3.5) on

βk ψp,k (x)φp,k (y)

k=1

[f ]

where φp,k (y) are the spectral solutions of two-point boundary value problem (3.6) on 0 Pp (I ), then Rˆ S [f ]

∈ Pp2 (S ) such that

[f ]

Rˆ S |Γ1 = f , RS |Γ2  Γ3 = 0, and

∥Rˆ [Sf ] ∥H 1 (S ) ≤ C ∥f ∥

1 2 (Γ ,1) H00 1

,

(3.14)

where C is a constant independent of p and f , and

∥f ∥2 1 2 (Γ ,1) H00 1

= ∥f ∥2 1

H 2 (Γ1 )

 + Γ1

|f |2 dx. 1−x

(3.15)

Remark 3.1. The lifting operator using solutions of the eigenvalue problem (3.4) was first proposed by Canuto and Funaro 1

2 for the extension of a function in H00 (Γ1 ) on a side Γ1 of a square to its interior [16] to develop the Schwarz algorithm for spectral methods. This technique has been adopted to establish polynomial extension operators on a square S = (−1, 1)2 in dimensions and on a cube D = (−1, 1)3 in three dimensions [10] and has become a powerful tool for the p and h-p finite element methods.

Remark 3.2. A different polynomial extension on a square via implicitly defined convolution-type extension RT on a triangle and a bilinear mapping M of a square onto a truncated triangle TH shown in Fig. 3.2 was proposed in [4].

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

265

Fig. 3.2. Mapping of square S onto trapezoid TH .

3.2. Polynomial extension ES from whole boundary of a square 1

We should next construct a continuous extension ES : H 2 (∂ S ) → H 1 (S ) which lifts f ∈ Pp (∂ S ) = {ϕ ∈ C 0 (∂ S ) | ϕ |Γi = fi ∈ Pp (Γi ), 1 ≤ i ≤ 4} to the interior of S. Theorem 3.6. Let S = [−1, 1]2 be the square with sides denoted by γi , 1 ≤ i ≤ 4, and let f ∈ Pp (∂ S ). Then there exists an [f ]

[f ]

extension ES such that ES ∈ Pp2 (S ) and ES |∂ S = f , and

∥ES[f ] ∥H 1 (S ) ≤ C ∥f ∥

1

H 2 (∂ S )

,

(3.16)

where C is a constant independent of p and f . [f ]

Proof. By Theorem 3.4 the polynomial extension U1 = R˜ S 1 lifts f1 on the side Γ1 such that U1 ∈ Pp2 (S ), U1 |Γ1 = f1 , U1 |Γ3 = 0, and

∥U1 ∥H 1 (S ) ≤ C ∥f1 ∥

1

H 2 (Γ1 )

.

Similarly, there is an extension which lifts f3 on the side Γ3 such that U1 ∈ Pp2 (S ), U3 |Γ1 = f3 , U3 |Γ1 = 0, and

∥U1 ∥H 1 (S ) ≤ C ∥f1 ∥

1

H 2 (Γ3 )

.

Let g2 = f2 − U1 |Γ2 − U3 |Γ2 and g4 = f4 − U1 |Γ4 − U3 |Γ4 , then g2 vanishes at the end points of Γ2 and g4 vanishes at the [f ]

[f ]

end points of Γ4 . By Theorem 3.3, there exist extensions U2 = RS 2 and U4 = RS 4 , which lift g2 and g4 on the sides Γ2 and Γ4 into the interior of S such that U2 , U4 ∈ Pp2 (S ), U2 |Γ2 = g2 , U2 |Γi = 0, i = 1, 3, 4; U4 |Γ4 = g4 , U4 |Γj = 0, j = 1, 2, 3, and

∥U2 ∥H 1 (S ) ≤ C ∥g2 ∥

1 2 (Γ ) H00 2

,

∥U 4 ∥H 1 ( S ) ≤ C ∥g 4 ∥

1 2 (Γ ) H00 4

.

Let [f ]

ES = U1 + U2 + U3 + U4 .

(3.17)

[f ]

Then ES ∈ Pp2 (S ) and ES = f on ∂ S, and

∥ES[f ] ∥H 1 (S ) ≤ ∥U1 ∥H 1 (S ) + ∥U2 ∥H 1 (S ) + ∥U3 ∥H 1 (S ) + ∥U4 ∥H 1 (S )  ≤ C ∥f1 ∥

1

H 2 (Γ1 )

+ ∥g2 ∥

1 2 (Γ ) H00 2

+ ∥ f3 ∥

1

H 2 (Γ3 )

+ ∥ g4 ∥

 1 2 (Γ ) H00 4

.

(3.18)

We note that

∥g 2 ∥2 1 2 (Γ ) H00 2

= ∥g2 ∥2 1

H 2 (Γ2 )



1

+ −1

|g2 (y)|2 dy + 1−y



1

−1

|g2 (y)|2 dy 1+y

(3.19)

266

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

and

∥g2 ∥

1

H 2 (Γ2 )

= ∥f2 − U1 |Γ2 − U3 |Γ2 ∥ ≤ ∥f2 ∥

1 H2

(Γ2 )

1

H 2 (Γ2 )

≤ ∥f2 ∥

1

H 2 (Γ2 )

+ ∥U1 ∥H 1 (S ) + ∥U3 ∥H 1 (S ) ≤ C

+ ∥U1 ∥

3 

∥fi ∥

i=1

1

H 2 (Γ2 )

1

H 2 (Γi )

+ ∥ U3 ∥

1

H 2 (Γ2 )

.

(3.20)

For the second term of (3.20) we write g2 (y) = f2 (y) − U1 (x, y)|Γ2 − U3 (x, y)|Γ2

= f2 (y) − f3 (y) + f3 (y) − U1 (1, y) − U3 (1, y) = (f2 (y) − f3 (y)) + (U3 (y, 1) − U3 (1, y)) − U1 (1, y).

(3.21) 1

It is known that (e.g. see [11,15]) the following norm is an equivalent norm for the space H 2 (Γ2 ∪ Γ3 )

∥f ∥

1 H2

(Γ2 ∪Γ3 )

 ≈ ∥f 2 ∥2 1

H 2 (Γ2 )

+ ∥ f 3 ∥2 1

H 2 (Γ3 )

+ D(f2 , f3 )

 21

,

where D(f2 , f3 ) =



(f2 (t ) − f3 (t ))2 dt . 1−t

1

−1

Therefore



1

−1

|f2 (y) − f3 (y)|2 dy ≤ C ∥f ∥2 1 1−y H 2 (Γ2 ∪Γ3 )

(3.22)

|U3 (y, 1) − U3 (1, y)|2 dy = D(U3 |Γ3 , U3 |Γ2 ) ≤ C ∥U3 ∥2 1 ≤ C ∥U3 ∥2H 1 (S ) ≤ C ∥f3 ∥2 1 . 1−y H 2 (Γ3 ) H 2 (Γ2 ∪Γ3 )

(3.23)

and



1

−1

1

By the definition of Sobolev–Slobodeckij norm for the space H 2 (Γ2 ∪ Γ3 ), there holds

∥U1 ∥

2 1 H2

|U1 (X ) − U1 (Y )|2 dXdY |X − Y |2 Γ2 ∪Γ3 Γ2 ∪Γ3          |U1 (X ) − U1 (Y )|2 2 dXdY = ∥U1 ∥L2 (Γ ) + + + + 2 |X − Y |2 Γ2 Γ2 Γ3 Γ3 Γ2 Γ3 Γ3 Γ2  1 1 |U1 (1, y)|2 = ∥U1 ∥2 1 dxdy +2 2 2 H 2 (Γ2 ) −1 −1 (x − 1) + (y − 1)  1 2 |U1 (1, y)|2 +2 = ∥U1 ∥2 1 tan−1 dy. 1−y 1−y H 2 (Γ2 ) −1 U1 2L2 (Γ ∪Γ ) 2 3

=∥ ∥

(Γ2 ∪Γ3 )





+

Therefore,

∥U1 ∥2 1

H 2 (Γ2 )



1

+ −1

|U1 (1, y)|2 dy ≈ ∥U1 ∥2 1 ≤ C ∥U1 ∥2H 1 (S ) ≤ C ∥f1 ∥2 1 . 1−y H 2 (Γ2 ∪Γ3 ) H 2 (Γ1 )

(3.24)

A combination of (3.21)–(3.24) leads to immediately



1

−1

|g2 (y)|2 dy ≤ C ∥f ∥2 1 . 1−y H 2 (Γ1 ∪Γ2 ∪Γ3 )

For the third term of (3.20), we rewrite g2 (y) = f2 (y) − U1 (x, y)|Γ2 − U3 (x, y)|Γ2

= f2 (y) − f1 (y) + f1 (y) − U1 (1, y) − U3 (1, y) = (f2 (y) − f1 (y)) + (U1 (y, −1) − U1 (1, y)) − U3 (1, y).

(3.25)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

267

As an analogue to (3.22)–(3.24), we have 1



−1 1



−1

|f2 (y) − f1 (y)|2 dy ≤ C ∥f ∥2 1 , 1+y H 2 (Γ1 ∪Γ2 ) |U1 (y, −1) − U1 (1, y)|2 dy = D(U1 |Γ1 , U1 |Γ2 ) ≤ C ∥U1 ∥2 1 ≤ C ∥U1 ∥2H 1 (S ) ≤ C ∥f1 ∥2 1 , 1+y H 2 (Γ1 ∪Γ2 ) H 2 (Γ1 )

and

∥U 3 ∥



2 1 H2

1

+ (Γ2 )

−1

|U3 (1, y)|2 dy ≈ ∥U3 ∥2 1 ≤ C ∥U3 ∥2H 1 (S ) ≤ C ∥f3 ∥2 1 . 1+y H 2 (Γ3 ) H 2 (Γ1 ∪Γ2 )

Hence, we obtain



1

−1

|g2 (y)|2 dy ≤ C ∥f ∥2 1 . 1+y H 2 (Γ1 ∪Γ2 ∪Γ3 )

(3.26)

Combining (3.20) and (3.25)–(3.26), we have

∥g 2 ∥

1 2 (Γ ) H00 2

≤ C ∥f ∥

1

.

(3.27)

1

.

(3.28)

H 2 (Γ1 ∪Γ2 ∪Γ3 )

Similarly, there holds

∥g 4 ∥

1 2 (Γ ) H00 4

≤ C ∥f ∥

H 2 (Γ1 ∪Γ3 ∪Γ4 )

Then (3.16) follows from (3.18) and (3.27)–(3.28).



4. Applications to the h-p FEM We would apply the extension operators RT and RS defined in (2.16) or (2.34) and (3.10) to construct a globally continuous and piecewise polynomial in the finite element space for the p and h-p FEM, which possesses the desired approximation properties and convergence rate of the high-order finite element method.  leads to the optimal  Let ∆h = Ωj , 1 ≤ j ≤ J be a quasi-uniform mesh over the domain Ω containing shape regular (curvilinear) triangles and quadrilateral elements Ωj , 1 ≤ j ≤ J. The finite element space for the p and h-p finite element solutions is defined as P ,1

SD (Ω ; ∆h ; M ) = S P (Ω ; ∆h ; M ) ∩ HD1 (Ω ) S P (Ω ; ∆h ; M ) = {ϕ | ϕ|Ωj ◦ Mj ∈ Ppκj (Ωst ), κ = 1 or 2, 1 ≤ j ≤ J } where Mj is a smooth mapping of T or S onto Ωj , and Ppκj (Ωst ) with κ = 1 or 2 is a set of polynomial of total degree ≤ pj if

Ωst is standard triangle T or a set of polynomials of separate degree ≤ pj if Ωst is standard square S, HD1 (Ω ) is a subspace of H 1 (Ω ) with functions vanishing on a portion of boundary ΓD . M = {M1 , M2 , . . . , MJ } and P = {p1 , p2 , . . . , pJ } denote the

union of the mapping and the distribution of element degrees. The construction of a desired continuous and piecewise polynomial in S P ,1 (Ω ; ∆h ) or S P ,1 (Ω ; ∆h ; M ) starts with a local approximation polynomial ϕj on each element Ωj based on the Chebyshev projection of u ∈ H k (Ωj ) the following proposition has been proved in [1–4,8]. Lemma 4.1. Let u ∈ H k (Ωj ), k > 1, where Ωj is a curved triangular or quadrilateral element of the mesh ∆h with size hj . Then there exists a polynomial ϕj ∈ Pp (Ωj ) such that for l = 0, 1 µj −l

∥u − ϕj ∥H l (Ωi ) ≤ C

hj

(pj + 1)k−l

∥u∥H k (Ωi )

(4.1)

with µj = min {pj + 1, k}, and u(Vl ) = ϕj (Vl ), Vl , 1 ≤ l ≤ 3 or 4 are the vertices of Ωi . We next need to adjust these local polynomials ϕj for constructing a globally continuous and piecewise polynomial in

P ,1

SD (Ω ; ∆h ; M ) with polynomial extension operators RT and RS . Lemma 4.2. Let γh be an edge of an element Ωj with size hj which is a (curved) triangle or quadrilateral, and let ψ ∈ Pp0j (γh )

be a polynomial of degree ≤ pj defined on γh vanishing at the ending points of γh . Then there exists a polynomial Ψ ∈ Ppκj (Ωj ) such that Ψγh = ψ and Ψ∂ Ωj \γh = 0

∥Ψ ∥H 1 (Ωj ) ≤ C ∥ψ∥H 1/2 (γ 00

h)

with the constant C independent of hj and pj , where κ = 1 if Ωj is a triangle and κ = 2 if Ωj is a quadrilateral.

(4.2)

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B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

˜ = ψ ◦ Mj is a polynomial in PP0 (I ). By Theorem 2.2 or Theorem 3.3, there Proof. Since Mj−1 maps γh onto I = (−1, 1), ψ j ˜ [ψ]

is an extension Ψ˜ = RT quadrilateral, and

˜ [ψ]

∈ Pp1j (Ωst ) with Ωst = T if Ωj is a triangle or Ψ˜ = RS

∈ Pp2j (Ωst ) with Ωst = S if Ωj is a

˜ 1/2 ∥Ψ˜ ∥H 1 (Ωst ) ≤ C ∥ψ∥ (I ) H 00

with the constant C independent of pj . Note that 0



˜ 22 ≤ ∥ψ∥ L (I )

−1

˜ )|2 |ψ(ξ dξ + 1+ξ







1

˜ )|2 |ψ(ξ dξ , 1−ξ

0

˜ )|2 |ψ(ξ dτ + 1+ξ

0

which implies

˜ 2 1/2 ∥ψ∥

H00 (I )

˜ 2 1/2 + ≤ C |ψ| H (I )

−1

  ≤ C |ψ|2H 1/2 (γ ) + h

0

|ψ(τ )| dτ dist .(τ , ∂γh ) 2

γh

˜ )|2 |ψ(ξ dτ 1−ξ

1







≤ C ∥ψ∥2 1/2

H00 (γh )

with C independent of hj . Let Ψ = Ψ˜ ◦ Mj−1 . Then Ψ ∈ Ppκ (Ωst ) such that Ψγh = ψ and Ψ∂ Ωj \γh = 0, and by a simple scaling we obtain

˜ 1/2 ≤ C ∥ψ∥ 1/2 .  ∥Ψ ∥H 1 (Ωst ) ≤ C ∥Ψ˜ ∥H 1 (Ωst ) ≤ C ∥ψ∥ H (I ) H (γ ) 00

h

00

Theorem 4.3. Let ∆h = {Ωj , 1 ≤ j ≤ J } be quasi-uniform mesh with element size h = max1≤j≤J hj over Ω containing P ,1

(curvilinear) triangular and quadrilateral elements, and let SD (Ω ; ∆h ; M ) be the finite element space defined as above with  quasi-uniform degree p = min1≤j≤J pj . Then for u ∈ H k (Ω ) HD1 (Ω ) with k > 1 there exists a continuous and piecewise P ,1

polynomial ϕ ∈ SD (Ω ; ∆h ; M ) with p ≥ 0 such that

∥u − ϕ∥H 1 (Ω ) ≤ C

hµ−1

(p + 1)k−1

∥u∥H k (Ω ) ,

(4.3)

where µ = min {p + 1, k} and the constant C is independent of p, h and u. Proof. By Lemma 4.1 there exist ϕj ∈ Ppκj (Ωj ) such that ϕj = u at the vertices of Ωj and (4.1) holds. Suppose that a pair of neighboring elements Ωj and Ωi share a common edge γh and pi ≥ pj , then ψ = (ϕj − ϕi )|γh ∈ Ppi (γh ) vanishing at the endpoints of γh . By Lemma 4.2 there exists Ψ ∈ Pp1i (Ω2 ) or Pp22 (Ωi ) such that Ψ |γh = ψ and Ψ |∂ Ωi \γh = 0, and

∥Ψ ∥H 1 (Ωi ) ≤ C ∥ψ∥

1 2 (γ ) H00 h

.

Let ϕ˜ j = ϕj on Ωj and ϕ˜ i = ϕi + Ψ on Ωi . Therefore ϕ˜ i |γh = ϕi |γh + ψ = ϕ1 |γh , and the continuity on γh is realized. Furthermore

∥u − ϕ˜ i ∥H 1 (Ωi ) ≤ ∥u − ϕi ∥H 1 (Ωi ) + ∥Ψ ∥H 1 (Ωi ) ≤ ∥u − ϕi ∥H 1 (Ωi ) + C ∥ψ∥ ≤ ∥u − ϕi ∥H 1 (Ω2 ) + C (∥u − ϕj ∥ 1

1 2 (γ ) H00 h

+ ∥ u − ϕi ∥ 1

1 2 (γ ) H00 h

1 2 (γ ) H00 h

).

(4.4)

−s1

2 According to [17,19] H00 (γ ) = H s1 (γ ), H0s (γ ) θ,2 with θ = 2s−s1 for 0 < s1 < 12 < s < 1. By the exactness of θ -exponent of interpolation spaces and the trace theorem, we have for k > 1 and for 0 < s1 < 21 < s < min{1, k − 12 }

∥u − ϕj ∥

1 2 (γ ) H00





θ ≤ ∥u − ϕj ∥1H−θ s1 (γ ) ∥u − ϕj ∥H s (γ )

≤ ∥u − ϕj ∥

s− 1 2 s−s1 1

s + H 1 2 (Ωj )

∥u − ϕ j ∥

Similarly there holds

∥u − ϕi ∥H 1/2 (γ ) ≤ C (pi + 1)−(k−1) ∥u∥H k (Ωi ) 00

1 −s 1 2 s−s1 1 s+ H 2 (Ωj )

≤ C (pj + 1)−(k−1) ∥u∥H k (Ωj ) .

(4.5)

B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

269

which together with (4.4) implies

  ∥u − ϕ˜ i ∥H 1 (Ωi ) ≤ C ∥ψ∥H 1/2 (γ ) ≤ C (pi + 1)−(k−1) ∥u∥H k (Ωi ) + (pj + 1)−(k−1) ∥u∥H k (Ωj ) .

(4.6)

00

The adjusting can be carried out on each pair of element and on the elements with a side on ΓD , which results in a P ,1 pull-back polynomial ϕ ∈ SD (Ω ; ∆h ; M ) with ϕ|Ωj = ϕ˜ j , 1 ≤ j ≤ J such that

∥u − ϕ∥H 1 (Ω ) ≤

J 

∥u − ϕ˜ j ∥H 1 (Ωj ) ≤ C

j =1

J  j=1

µ j −1

hµ−1

hj

∥u∥H k (Ωj ) ≤ C ∥u∥H k (Ω ) (pj + 1)k−1 (p + 1)k−1

with µ = min1≤j≤J µj = min{p + 1, k} for u ∈ H k (Ω ) ∩ HD1 (Ω ) with k > 1.



Consider the elliptic boundary value problem

 −∆ u + u = f u |ΓD = 0,

in Ω ⊂ R2 , ∂u | = g, ∂ n ΓN

(4.7)

where Ω is an open set in R2 with its boundary ∂ Ω = ΓD B(u, v) = F (v),



ΓN . The variational equation is to seek u(x) ∈ HD1 (Ω ) such that

∀v ∈ HD1 (Ω ),

(4.8)

where B is a bilinear form on HD1 (Ω ) × HD1 (Ω ) and F is a linear functional on HD1 (Ω ), given by B(u, v) =

 Ω

(∇ u · ∇v + u v) dx,

F (v) =

 Ω

f v dx +

 ΓN

g v ds.

(4.9)

P ,1

The finite element solution of the h-p version uhp ∈ SD (Ω ; ∆; M ) is such that B(uhp , v) = F (v),

∀v ∈ SDP ,1 (Ω ; ∆; M). P ,1

Since the finite element solution uhp is the projection of u on SD (Ω ; ∆; M ), there holds

∥u − uhp ∥H 1 (Ω ) ≤ C

inf

χ∈SDP ,1 (Ω ;∆h ;M)

∥u − χ∥H 1 (Ω ) ≤ C ∥u − ϕ∥H 1 (Ω ) ,

which leads to the optimal convergence of the finite element solution of the h-p version. Theorem 4.4. Let ∆h = {Ωj , 1 ≤ j ≤ J } be quasi-uniform mesh with element size h = max1≤j≤J hj over Ω containing P ,1 (curved) triangular and quadrilateral elements, and let SD (Ω ; ∆h ; M ) be the finite element space defined as above with quasiuniform degree p = min1≤j≤J pj . If u ∈ H k (Ω ) ∩ HD1 (Ω ) with k > 1 is a solution of the elliptic boundary problem (4.7), and P ,1

uhp ∈ SD (Ω ; ∆h ; M ) is the finite element solution. Then

∥u − uhp ∥H 1 (Ω ) ≤ C

hµ−1

(p + 1)k−1

∥u∥H k (Ω ) ,

(4.10)

where µ = min {p + 1, k} and the constant C is independent of p, h and u. Remark 4.1. The polynomial extensions RT and RS play important roles in error analysis for the high-order FEM for smooth functions in H k (Ω ) with k > 1. The arguments can be applied to general mesh ∆h composed of triangular, quadrilateral and curvilinear elements, and general distribution P of element degrees. Remark 4.2. For the singular solutions of r γ logν r-type the above approach to construct a continuous and piecewise P ,1 polynomial ϕ ∈ SD (Ω ; ∆h ; M ) by adjusting individual Chebyshev projection ϕj on neighboring elements with the extension operators RT and RS will not lead to the optimal convergence since one needs optimal error estimation for ∥u −ϕj ∥ s+ 1 with s + 12 > 1, but it is not available for singular solutions. It is known that the error in Chebyshev projection H

2

(Ωj )

ϕj is optimal in H 1 (Ωj )-norm, but not in H l (Ωj )-norm with l ̸= 1. It has been an open question since the early 1980s. Very recently, a new approach to construct a continuous and piecewise polynomial with the best approximation property was proposed in [20], which combines Chebyshev projection on Ωj and the polynomial extension ET and ES to construct a local Chebyshev projection–interpolation ϕ˜ j on each element Ωj such that ϕ˜ j has the optimal error bound H 1 (Ωj )-norm and ϕ˜ j is equal to the Chebyshev projection–interpolation on ∂ Ωj . A simple assembly of ϕ˜ j yields a continuous and piecewise polynomial possessing the optimal convergence. For the details of analysis of local Chebyshev projection–interpolation and the application to the p and h-p version of finite element method, we refer to [20,21].

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B. Guo, J. Zhang / Journal of Computational and Applied Mathematics 261 (2014) 249–270

5. Concluding remark The polynomial extensions in two dimensions have been of great research interest for developing the theory of the highorder FEM in different periods of the past decades. In this paper we have improved the construction and estimation of the polynomial extensions. On a triangle T we further developed the polynomial extension of the convolution type by explicitly constructing the operators RT and ET which lift a polynomial on one side γ and all sides of T , and provide constructive 1

1

2 (γ ) → H 1 (T ) and the continuity of ET : H 2 (∂ T ) → H 1 (T ). On a square and vigorous proof for the continuity of RT : H00 1

2 S we adopted a new approach to construct continuous polynomial extension operators RS : H00 (γ ) → H 1 (S ) and ES : 1

H 2 (∂ S ) → H 1 (S ), namely, using spectral solutions of eigenvalue problem and two-point boundary value problem on an interval I = (−1, 1), which lift a polynomial on one side and all sides of S to the interior of S. These extensions play a vital role for constructing a continuous and piecewise polynomial ϕ with the optimal bound of approximation error for smooth and singular solutions, which is the bottle-neck of the optimal convergence of the p and h-p FEM. The approach and techniques developed in this paper can be generalized to polynomial extensions on tetrahedrons T , hexahedrons D and triangular prisms Λ in three dimensions [9,10,6]. Actually, many of them have been adopted, and many remain challenging issues. For example, the extension operator RΛ from a square face F to the interior of a prism Λ and the extension operator EΛ from all faces of Λ to the interior of Λ were addressed several years ago [10], but the continuity of the 1

1

2 (∂ Λ) → H 1 (Λ) have not been proved. Consequently, the optimal convergence operators RΛ : H 2 (F ) → H 1 (Λ) and EΛ : H00 of the p and h-p finite element method for problems on polyhedral domains with singular solutions of r γ logν r-type on meshes containing prism elements has not been established yet.

Acknowledgments The work of first author was supported partially by NSERC of Canada under Grant OGP0046726, and partially by Shanghai University under Leading Academic Discipline Project of Shanghai Municipal Education Commission (J50101). The work of second author was supported partially by National Natural Science Foundation of China under Grant 11261026, and partially by NSERC of Canada under Grant OGP0046726 while working in University of Manitoba. References [1] I. Babuška, B.Q. Guo, Direct and inverse approximation theorems of the p version of finite element method in the framework of weighted Besov spaces, part 1: approximability of functions in weighted Besov spaces, SIAM J. Numer. Anal. 39 (2002) 1512–1538. [2] I. Babuška, B.Q. Guo, Direct and inverse approximation theorems of the p-version of the finite element method in the framework of weighted Besov spaces, part 2: optimal convergence of the p-version of the finite element method, Math. Models Methods Appl. Sci. 12 (2002) 689–719. [3] I. Babuška, M. Suri, The optimal convergence rate of the p-version of the finite element method, SIAM J. Numer. Anal. 24 (1991) 750–776. [4] I. Babuška, M. Suri, The h-p version of the finite element method with quasiuniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987) 199–238. [5] I. Babuška, M. Szabó, N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal. 18 (1981) 515–545. [6] R. Muñoz-Sola, Polynomial liftings on a tetrahedron and applications to the h-p version of the finite element method in three dimensions, SIAM J. Numer. Anal. 34 (1997) 282–314. [7] B.Q. Guo, Approximation theory for the p-version of the finite element method in three dimensions, part 2: convergence of the p-version, SIAM J. Numer. Anal. 47 (2009) 2578–2611. [8] B.Q. Guo, W. Sun, The optimal convergence of the h-p version of the finite element method with quasiuniform meshes, SIAM J. Numer. Anal. 45 (2007) 698–730. [9] B.Q. Guo, L.J. Yi, Local Chebyshev projection-interpolation operator and application to the h-p version of the finite element method in three dimensions, Math. Models Methods Appl. Sci. 23 (2013) 777–801. [10] B.Q. Guo, J.M. Zhang, Stable and compatible polynomial extensions and application to the p and h-p finite element method, SIAM J. Numer Anal. 47 (2009) 1195–1225. [11] I. Babuška, A. Craig, J. Mandel, J. Pitkäranta, Efficient preconditioning for the p version finite element method in two dimensions, SIAM J. Numer. Anal. 28 (1991) 624–661. [12] B.Q. Guo, W. Cao, A preconditioner for the h-p version of the finite element method in two dimensions, Numer. Math. 75 (1996) 59–77. [13] G.H. Hardy, T.E. Littlewood, G. Polya, Inequalities, Cambridge University Press, Cambidge, 1934. [14] N. Heuer, F. Leydecker, An extension theorem for polynomials on triangles, Calcolo 45 (2008) 69–85. [15] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Publishing Inc., Boston, London, Melboune, 1985. [16] C. Canuto, D. Funaro, The Schwarz algorithm for spectral methods, SIAM J. Numer Anal. 25 (1988) 3–38. [17] J. Bergh, J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1976. [18] C. Bernardi, Y. Maday, in: P.G. Ciarlet, J.L. Lions (Eds.), Spectral Methods, in: Handbook of Numerical Analysis, vol. 5, North-Holland, Amsterdam, 1997. [19] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, New York, 1972. [20] B.Q. Guo, I. Babuška, Local Jacobi operators and applications to the p-version of the finite element method in two dimensions, ICES Report 08-11, Univ. of Texas, 2008, and SIAM J. Numer. Anal. 48, 2010, pp. 147–163. [21] B.Q. Guo, L.J. Yi, Local Chebyshev operator and application to the h-p version of the finite element method for problems with nonhomogeneous Dirichlet boundary conditions in two dimensions, Preprint.