July 15, 2016 13:43 WSPC/103-M3AS 1650040 ...szhang/research/p/2016b.pdfMathematical Models and Methods in Applied Sciences Vol. 26, No. 9 (2016) 1649–1669 c World Scientiﬁc Publishing

July 15, 2016 13:43 WSPC/103-M3AS 1650040

Mathematical Models and Methods in Applied SciencesVol. 26, No. 9 (2016) 1649–1669c© World Scientific Publishing CompanyDOI: 10.1142/S0218202516500408

Finite element approximations of symmetric tensorson simplicial grids in Rn: The lower order case

Jun Hu

LMAM, School of Mathematical Sciences, andBeijing International Center for Mathematical Research,

Peking University, Beijing 100871, P. R. [email protected]

Shangyou Zhang

Department of Mathematical Sciences, University of Delaware,Newark, DE 19716, USA

[email protected]

Received 6 May 2015Revised 11 March 2016Accepted 27 April 2016Published 18 July 2016

Communicated by D. Arnold

In this paper, we construct, in a unified fashion, lower order finite element subspacesof spaces of symmetric tensors with square-integrable divergence on a domain in anydimension. These subspaces are essentially the symmetric tensor finite element spacesof order k from [Finite element approximations of symmetric tensors on simplicial gridsin Rn: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each

(n − 1)-dimensional simplex, by(n+1)n

2face bubble functions in the symmetric tensor

finite element space of order n + 1 from [Finite element approximations of symmetrictensors on simplicial grids in Rn: The higher order case, J. Comput. Math. 33 (2015) 283–

296] when 1 ≤ k ≤ n − 1, and by(n−1)n

2face bubble functions in the symmetric tensor

finite element space of order n + 1 from [Finite element approximations of symmetrictensors on simplicial grids in R

n: The higher order case, J. Comput. Math. 33 (2015) 283–296] when k = n. These spaces can be used to approximate the symmetric matrix field ina mixed formulation problem where the other variable is approximated by discontinuouspiecewise Pk−1 polynomials. This in particular leads to first-order mixed elements onsimplicial grids with total degrees of freedom per element 18 plus 3 in 2D, 48 plus 6 in3D. The previous record of the degrees of freedom of first-order mixed elements is, 21plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified waywhich is completely different from those used in [D. Arnold, G. Awanou and R. Winther,Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008)1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, NumberMath. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in anyspace dimension. One example in this family is the Raviart–Thomas elements in onedimension, the second example is the mixed finite elements for linear elasticity in two

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http://dx.doi.org/10.1142/S0218202516500408

July 15, 2016 13:43 WSPC/103-M3AS 1650040

1650 J. Hu & S. Zhang

dimensions due to Arnold and Winther, the third example is the mixed finite elementsfor linear elasticity in three dimensions due to Arnold, Awanou and Winther.

Keywords: Mixed finite element; symmetric finite element; first-order system; simplicialgrid; inf–sup condition.

AMS Subject Classification: 65N30, 73C02

1. Introduction

The constructions, using polynomial-shape functions, of stable pairs of finite ele-ment spaces for approximating the pair of spaces H(div, Ω; S)×L2(Ω; Rn) in first-order systems are a long-standing, challenging and open problem, see for instance,Refs. 4 and 6. For mixed finite elements of linear elasticity, many mathematicianshave been working on this problem and compromised to weakly symmetric or com-posite elements, cf. Refs. 3, 7, 8, 34, 36, 37, 39–42 and 45. It is not until 2002that Arnold and Winther were able to propose the first family of mixed finiteelement spaces with polynomial-shape functions in two dimensions.10 Such a two-dimensional family was extended to a three-dimensional family of mixed elements,6

while the lowest order element with k = 2 was first proposed in Ref. 2. We referinterested readers to Refs. 2, 5, 6, 10, 12, 17–19, 11, 24, 31, 35, 43, 44, 9, 13, 20, 25,26, 27, 30 and 29, for recent progress on mixed finite elements for linear elasticity.See Ref. 14 for the DPG method for the linear elasticity problem.

In Refs. 32 and 33, Hu and Zhang proposed new ideas to design discrete stressspaces and analyze the discrete inf–sup condition. In particular, they were able toconstruct suitable H(div, Ω; S)-Pk space, namely, Σk,h defined in (4.1) below, withk ≥ 3 for 2D, and k ≥ 4 for 3D, finite element spaces for the stress discretizationin both two and three dimensions. In Ref 28, Hu constructed, in a unified fashion,suitable H(div, Ω; S)-Pk space with k ≥ n + 1, and proposed a set of degrees offreedom for the shape function space, in any dimension.

The purpose of this paper is to extend those elements in Ref. 28 to lower ordercases where 1 ≤ k ≤ n. Since it is, at moment, very difficult to prove that the pairof the space Σk,h and the L2-Pk−1 space, namely, Vk,h defined in (4.2), is stable, theΣk,h space has to be enriched by some higher order polynomials whose divergenceare in Vk,h. Thanks to Ref. 28, it suffices to control the piecewise rigid motion space.Hence, we only need to add, for each (n − 1)-dimensional simplex, (n+1)n

2 simplexbubble functions in Σn+1,h when 2 ≤ k ≤ n−1, and (n−1)n

2 simplex bubble functionsin Σn+1,h when k = n. This in particular leads to first-order mixed elements onsimplicial grids with total degrees of freedom per element 18 plus 3 in 2D, 48 plus 6in 3D. The previous record of the degrees of freedom of first-order mixed elementsis 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. These enriched bubblefunctions belong to the lowest order space from a family of Arnold–Winther mixedfinite elements in any space dimension which, together with the Vk,h space, form astable pair of spaces for first-order systems. Note that these spaces in this family ofArnold–Winther mixed finite elements are constructed in a unified and direct way

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Finite element approximations of symmetric tensors 1651

which is completely different from those used in Refs. 6 and 10. One example inthis family is the Raviart–Thomas elements in one dimension, the second exampleis the mixed finite elements for linear elasticity in two dimensions due to Arnoldand Winther,10 the third example is the mixed finite elements for linear elasticityin three dimensions due to Arnold, Awanou and Winther.6

We end this section by introducing first-order systems and related notations.We consider mixed finite element methods22 of first-order systems with symmetrictensors: find (σ, u) ∈ Σ × V := H(div, Ω; S) × L2(Ω; Rn), such that

(Aσ, τ) + (div τ, u) = 0 for all τ ∈ Σ,

(div σ, v) = (f, v) for all v ∈ V.(1.1)

Here the symmetric tensor space for the stress Σ is defined by

H(div, Ω; S) :=

τ =

τ11 · · · τ1n

......

...

τn1 · · · τnn

∈ H(div, Ω; Rn×n)

∣∣∣∣∣∣∣∣∣τT = τ

, (1.2)

and the space for the vector displacement V is

L2(Ω; Rn) :=(u1, . . . , un)T

∣∣ ui ∈ L2(Ω), i = 1, . . . , n. (1.3)

This paper denotes by Hk(T ; X) the Sobolev space1 consisting of functions withdomain T ⊂ R

n, taking values in the finite-dimensional vector space X , and with allderivatives of order at most k square-integrable. For our purposes, the range spaceX will be either S, Rn, or R. Let ‖ · ‖k,T be the norm of Hk(T ), and S denote thespace of symmetric tensors, and H(div, T ; S) consist of square-integrable symmetricmatrix fields with square-integrable divergence. The H(div)-norm is defined by

‖τ‖2H(div,T ) := ‖τ‖2

0,T + ‖div τ‖20,T .

Let L2(T ; Rn) be the space of vector-valued functions which are square-integrable.Here, the compliance tensor A = A(x) : S → S, characterizing the properties of thematerial, is bounded and symmetric positive definite uniformly for x ∈ Ω.

The rest of the paper is organized as follows. In the next section, we presentsome preliminary results from Ref. 28; see also Refs. 32 and 33, for the cases n = 2and n = 3, respectively. In Sec. 3, based on these preliminary results, we propose afamily of Arnold–Winther mixed finite elements in any space dimension. In Sec. 4,we present lower order mixed finite elements and analyze the well-posedness of thediscrete problem and error estimates of the approximation solution. In Sec. 5, wepresent the first-order mixed elements. The paper ends with Sec. 6 which lists somenumerics.

2. Preliminary Results

Suppose that the domain Ω is subdivided by a family of shape regular simplicialgrids Th (with the grid size h). For any edge xixj of element K, i = j, let ti,j denote

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associated tangent vectors, which allow for us to introduce the following symmetricmatrices of rank one

Ti,j := ti,jtTi,j , 0 ≤ i < j ≤ n. (2.1)

For these matrices of rank one, we have the following result from Ref. 28; seealso Refs. 32 and 33, for the cases n = 2 and n = 3, respectively.

Lemma 2.1. The (n+1)n2 symmetric tensors Ti,j in (2.1) are linearly independent,

and form a basis of S.

With these symmetric matrices Ti,j of rank one, we define an H(div, K; S) bub-ble function space

ΣK,k,b :=∑

0≤i<j≤n

λiλjPk−2(K; R)Ti,j, (2.2)

where λi, i = 0, . . . , n, are the barycenter coordinates of element K. Define the fullH(div, K; S) bubble function space consisting of polynomials of degree ≤k:

Σ∂K,k,0 := τ ∈ Pk(K; S), τν|∂K = 0. (2.3)

Here ν is the normal vector of ∂K. We have the following result due to Ref. 28.

Lemma 2.2. It holds that

ΣK,k,b = Σ∂K,k,0. (2.4)

Let ΣK,b,h denote the sum of these H(div, K; S) bubble function spaces, namely,

Σk,b,h :=∑

K∈Th

ΣK,k,b. (2.5)

We need an important result concerning the divergence space of the bubblefunction space. To this end, we introduce the following rigid motion space on eachelement K:

R(K) := v ∈ H1(K; Rn), ε(v) := (∇v + ∇vT )/2 = 0. (2.6)

It follows from the definition that R(K) is a subspace of P1(K; Rn). For n = 1,R(K) is the constant function space over K. The dimension of R(K) is n(n+1)

2 . Fortwo dimensions, the rigid motion space R(K) is

R(K) :=(

a1

a2

)+ b

(−x2

x1

), a1, a2, b ∈ R

; (2.7)

for three dimensions, the rigid motion space R(K) reads

R(K) :=

a1

a2

a3

+ b1

−x2

x1

0

+ b2

−x3

0x1

+ b3

0−x3

x2

, ai, bi ∈ R, i = 1, 2, 3

.

(2.8)

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This allows for defining the orthogonal complement space of R(K) with respectto Pk−1(K; Rn) by

R⊥(K) := v ∈ Pk−1(K; Rn), (v, w)K = 0 for any w ∈ R(K), (2.9)

where the inner product (v, w)K over K reads (v, w)K =∫

K v · wdx. When k = 1we have R⊥(K) = 0.

Lemma 2.3. For any K ∈ Th, it holds that

div ΣK,k,b = R⊥(K). (2.10)

Proof. The proof can be found in Ref. 28; see also Refs. 32 and 33, for the casesn = 2 and n = 3, respectively.

We need a classical result and its variant.

Lemma 2.4. It holds the following Chu–Vandermonde combinatorial identity andits variant

n∑=0

C+1n+1C

k−1 =

n∑=0

Cn−n+1C

k−1 = Cn

n+k, (2.11)

andn∑

=0

C+1n+1C

k−1C

2+1 =

(n + 1)n2

Cnn+k−2, (2.12)

where the combinatorial number Cmn = n···(n−m+1)

m···1 for n ≥ m and Cmn = 0 for

n < m.

3. A Family of Arnold–Winther Mixed Elementsin Any Space Dimension

3.1. The lowest order Arnold–Winther mixed

elements in any space dimension

To define lower order mixed finite elements with k ≤ n, we extend the lowest orderArnold–Winther mixed elements in 2D10 and 3D2,6 to any space dimension. To thisend, we introduce the following divergence-free space for element K ∈ Th,

Σ3→n+1,DF (K; S) := τ ∈ Pn+1(K; S)\P2(K; S), div τ = 0, (3.1)

where

Pn+1(K; S)\P2(K; S) := τ ∈ Pn+1(K; S) and τ ∈ P2(K; S). (3.2)

For any τ ∈Pn+1(K; S)\P2(K; S), its divergence div τ is an n-dimensional vector-valued polynomial of degree ≤n. This implies that the dimension of div Pn+1(K; S)\P2(K; S) is

n

((2n)!n!n!

− (n + 1))

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which is equal to the number of the divergence-free constraints in (3.1). Since thedimension of the space Pn+1(K, S)\P2(K, S) is(

(2n + 1)!n!(n + 1)!

− (n + 2)!2!n!

)n(n + 1)

2,

it follows that the dimension of the space Σ3→n+1,DF (K; S) is((2n + 1)!n!(n + 1)!

− (n + 2)!2!n!

)n(n + 1)

2− n

(2n)!n!n!

+ n(n + 1). (3.3)

Then we can define the following enriched P2(K; S) space

P ∗2 (K; S) := P2(K; S) + Σ3→n+1,DF (K; S). (3.4)

The dimension of P ∗2 (K; S) is equal to the dimension of P2(K; S) plus the dimension

of Σ3→n+1,DF (K; S). Thanks to (3.3),

the dimension of P ∗2 (K; S) =

(2n + 1)!n!(n + 1)!

n(n + 1)2

− n(2n)!n!n!

+ n(n + 1). (3.5)

To present the degrees of freedom of P ∗2 (K; S), we define

M2(K) := τ ∈ P ∗2 (K; S), div τ = 0 and τν|∂K = 0, (3.6)

where ν is the normal vector of ∂K. For the space M2(K), we have the followingimportant result.

Lemma 3.1. The dimension of M2(K) is

(2n − 1)!n!(n − 1)!

n(n + 1)2

+n(n + 1)

2− n

(2n)!n!n!

. (3.7)

Proof. By Lemma 2.1, Ti,j, 0 ≤ i < j ≤ n, are linearly independent, which impliesthat the dimension of the bubble function space ΣK,n+1,b defined in (2.2) is

(2n − 1)!n!(n − 1)!

n(n + 1)2

. (3.8)

Since the dimension of R(K) is n(n+1)2 , the dimension of R⊥(K) (with respect to

Pn(K; Rn)) is

n(2n)!n!n!

− n(n + 1)2

. (3.9)

Thanks to Lemma 2.2, the H(div) bubble function space ΣK,n+1,b is the full H(div)bubble function space of symmetric tensor-valued polynomials of degree ≤n+1 overK. Then, the definitions of P ∗

2 (K; S) and M2(K) imply that M2(K) is identical tothe space of all the divergence-free functions in ΣK,n+1,b. Therefore, its dimensionof M2(K) is equal to

the dimension of ΣK,n+1,b − the dimension of R⊥(K)

which completes the proof.

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Before presenting the degrees of freedom of P ∗2 (K; S), we introduce a more

notation with 0 ≤ ≤ n which denotes an -dimensional simplex of K. For = 0, 0 is a vertex of K; for = 1, 1 is an edge of K.

Theorem 3.1. A matrix field τ ∈ P ∗2 (K; S) can be uniquely determined by the

following degrees of freedom:

(1) the mean moments of degree at most n − over , of tTl τνi, νT

i τνj, l =1, . . . , , i, j = 1, . . . , n− , (C2

n+1− + (n− ))Cn = (n−)(n++1)

2 Cn degrees of

freedom, for each -dimensional simplex of K, 0 ≤ ≤ n−1, with linearlyindependent tangential vectors t1, . . . , t, and n− linearly independent normalvectors ν1, . . . , νn−;

(2) the average of τ over K, n(n+1)2 degrees of freedom;

(3) the values of moments∫

K τ : θdx, θ ∈ M2(K), (2n−1)!n!(n−1)!

n(n+1)2 + n(n+1)

2 −n (2n)!n!n!

degrees of freedom.

Proof. We assume that all degrees of freedom vanish and show that τ = 0. Notethat the mean moment becomes the value of τ for a zero-dimensional simplex 0,namely, a vertex, of K. The first set of degrees of freedom implies that τν = 0 on∂K while the second set of degrees of freedom shows div τ = 0. Then the third setof degrees of freedom proves that τ = 0. Next we shall prove that the sum of thesedegrees of freedom is equal to the dimension of the space P ∗

2 (K, S). In fact the sumof the first set of degrees of freedom is

n−1∑=0

C+1n+1

(n − )(n + + 1)2

Cn,

we refer interested readers to Ref. 28 for a detailed proof of the numbers of degreesof freedom in the first set. By the Chu–Vandermonde combinatorial identity (2.11)and its variant (2.12), see more details from Ref. 28,

n−1∑=0

C+1n+1

(n − )(n + + 1)2

Cn =

(2n + 1)!n!(n + 1)!

n(n + 1)2

− (2n − 1)!n!(n − 1)!

n(n + 1)2

.

Hence the desired result follows from (3.5), and the sum of the second and thirdsets of degrees of freedom.

We denote by Σ∗2,h the space of symmetric tensor fields that belong piecewise

to P ∗2 (K; S), and with the continuity conditions induced by the degrees of freedom

defined in Theorem 3.1. In particular, for τ ∈ Σ∗2,h, the normal components τν are

continuous across all the internal (n − 1)-dimensional simplices n−1 and, henceΣ∗

2,h ⊂ H(div, Ω; S).

Remark 3.1. For n = 2, Σ∗2,h is the lowest order Arnold–Winther element stress

space proposed in Ref. 10; for n = 3, Σ∗2,h is the discrete stress space defined in

Ref. 2, see also Ref. 6.

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To define a family of first-order mixed elements, we need a family of simplifiedlowest order mixed elements, which is defined by

P ∗2 (K; S) := τ ∈ P ∗

2 (K; S), div τ ∈ R(K). (3.10)

Since the dimension of R(K) is n(n+1)2 , the equation gives n(n+1)

2 constraints on

P ∗2 (K; S). Hence the dimension of P ∗

2 (K; S) is

(2n + 1)!n!(n + 1)!

n(n + 1)2

− n(2n)!n!n!

+n(n + 1)

2.

A complete set of degrees of freedom for P ∗2 (K; S) is obtained by removing the

n(n+1)2 average values over K for P ∗

2 (K; S). The global space Σ∗2,h is defined in a

similar way as Σ∗2,h.

Remark 3.2. For n = 2, 3, Σ∗2,h are the simplified lowest order element stress

spaces in Refs. 10 and 6, respectively.

3.2. Higher order Arnold–Winther mixed elements

To define Arnold–Winther mixed elements of order k > 2, we introduce the followingdivergence-free space for element K ∈ Th,

Σk+1→k+n−1,DF (K; S) := τ ∈ Pk+n−1(K; S)\Pk(K; S), div τ = 0. (3.11)

Here Pk+n−1(K; S)\Pk(K; S) and Pk+n−2(K; R)\Pk−1(K; R) are defined ina similar way as Pn+1(K; S)\P2(K; S) defined in (3.2). The dimension ofΣk+1→k+n−1,DF (K; S) can be analyzed by a similar argument as that forΣ3→n+1,DF (K; S) in the previous subsection. In fact, since the dimension of thespace Pk+n−2(K; R)\Pk−1(K; R) is

((k + 2n − 2))!n!(k + n − 2)!

− (n + k − 1)!n!(k − 1)!

,

the number of the divergence-free constraints imposed in (3.11) is

n

(((k + 2n − 2))!n!(k + n − 2)!

− (n + k − 1)!n!(k − 1)!

).

In addition, the dimension of the space Pk+n−1(K; S)\Pk(K; S) is((k + 2n − 1)!n!(k + n − 1)!

− (n + k)!k!n!

)n(n + 1)

2.

It follows that the dimension of the space Σk+1→k+n−1,DF (K; S) is((k + 2n − 1)!n!(k + n − 1)!

− (n + k)!k!n!

)n(n + 1)

2− n

(((k + 2n − 2))!n!(k + n − 2)!

− (n + k − 1)!n!(k − 1)!

).

(3.12)

Then, we define the following enriched Pk(K; S) space

P ∗k (K; S) := Pk(K; S) + Σk+1→k+n−1,DF (K; S). (3.13)

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It follows that the dimension of P ∗k (K; S) is equal to the sum of the dimension of

Pk(K; S) and the dimension of Σk+1→k+n−1,DF (K; S), namely,

(k + 2n − 1)!n!(k + n − 1)!

n(n + 1)2

− n

((k + 2n − 2)!n!(k + n − 2)!

− (n + k − 1)!n!(k − 1)!

). (3.14)

To present the degrees of freedom of P ∗k (K; S), we define

Mk(K) := τ ∈ P ∗k (K; S), div τ = 0 and τν|∂K = 0, (3.15)

where ν is the normal vector of ∂K. For the space Mk(K), we have the followingimportant result.

Lemma 3.2. The dimension of Mk(K) is

(k + 2n − 3)!n!(k + n − 3)!

n(n + 1)2

+n(n + 1)

2− n

(k + 2n − 2)!n!(k + n − 2)!

. (3.16)

Proof. The dimension of the space ΣK,k+n−1,b reads

(k + 2n − 3)!n!(k + n − 3)!

n(n + 1)2

. (3.17)

Since the dimension of R(K) is n(n+1)2 , the dimension of R⊥(K) (with respect to

Pk+n−2(K; Rn)) is

n(k + 2n− 2)!n!(k + n − 2)!

− n(n + 1)2

. (3.18)

It follows from the definition of P ∗k (K; S) and Lemma 2.2 that Mk(K) contains

all divergence-free tensor-value functions of ΣK,k+n−1,b. Then the desired resultfollows from Lemma 2.3.

Theorem 3.2. A matrix field τ ∈ P ∗k (K; S) can be uniquely determined by the

following degrees of freedom:

(1) the mean moments of degree at most k + n − − 2 over , of tTl τνi, νT

i τνj ,

l = 1, . . . , , i, j = 1, . . . , n−, (C2n+1−+(n−))C

k+n−2 = (n−)(n++1)2 C

k+n−2

degrees of freedom, for each -dimensional simplex of K, 0 ≤ ≤ n−1, with

linearly independent tangential vectors t1, . . . , t, and n− linearly independentnormal vectors ν1, . . . , νn−;

(2) the values∫

K τ : θdx for any θ ∈ ε(Pk−1(K; Rn)), nCnn+k−1 −

n(n+1)2 degrees of

freedom;(3) the values

∫K τ : θdx for any θ∈Mk(K), (k+2n−3)!

n!(k+n−3)!n(n+1)

2 + n(n+1)2 −n (k+2n−2)!

n!(k+n−2)!

degrees of freedom.

Proof. We assume that all degrees of freedom vanish and show that τ = 0. Notethat the mean moment become the value of τ for a zero-dimensional simplex 0,namely, a vertex, of K. The first set of degrees of freedom implies that τν = 0 on

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∂K while the second set of degrees of freedom shows div τ = 0. Then the third setof degrees of freedom proves that τ = 0.

Next we shall prove that the sum of these degrees of freedom is equal to thedimension of the space P ∗

k (K; S). In fact, it follows from the Chu–Vandermondecombinatorial identity (2.11) and its variant (2.12) that the number of degrees inthe first set is

n−1∑=0

C+1n+1

(n − )(n + + 1)2

Cn+k−2 =

n(n + 1)2

(Cnk+2n−1 − Cn

k+2n−3); (3.19)

we refer interested readers to Ref. 28 for a detailed proof of the numbers of degreesof freedom in the first set. The desired result follows from (3.14) and (3.16).

We denote by Σ∗k,h the space of symmetric tensor fields that belong piecewise

to P ∗k (K; S), and with the continuity conditions induced by the degrees of freedom

defined in Theorem 3.2. In particular, for τ ∈ Σ∗k,h, the normal components τν are

continuous across all the internal (n − 1)-dimensional simplices n−1 and, henceΣ∗

k,h ⊂ H(div, Ω; S).

Remark 3.3. For n = 2, Σ∗k,h is the higher order mixed element stress spaces in

Ref. 10; for n = 3, Σ∗k,h is the higher order mixed element stress spaces in Ref. 6.

4. A Family of Lower Order Mixed Elements

4.1. Mixed methods

For 2 ≤ k ≤ n, we follow the idea of Refs. 28, 32 and 33 to define the followingdiscrete stress space:

Σk,h := σ ∈ H(div, Ω; S), σ = σc + σb, σc ∈ H1(Ω; S),

σc|K ∈ Pk(K; S), σb|K ∈ ΣK,k,b, ∀K ∈ Th, (4.1)

which is an H(div) bubble enrichment of the H1 space

Σk,h := τ ∈ H1(Ω; S), τ |K ∈ Pk(K; S), ∀K ∈ Th.

For τ ∈ Σk,h, the degrees of freedom on any element K are: the mean moments ofdegree at most k − − 1 over each -dimensional simplex of K, 0 ≤ ≤ n of τ .A standard argument is able to prove that these degrees of freedom are unisolvent.

In order to get a stable pair of spaces, we take the discrete displacement spaceas the space of discontinuous piecewise vector-valued polynomials of degree ≤k−1,namely,

Vk,h := v ∈ L2(Ω; Rn), v|K ∈ Pk−1(K; Rn) for all K ∈ Th. (4.2)

Unfortunately, we cannot establish the stability of the pair of spaces Σk,h and Vk,h.We have to enrich the discrete stress space by some bubble functions. More precisely,

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the displacement space Vk,h can be decomposed as

Vk,h = V Rk,h ⊕ V R⊥

k,h ,

where

V Rk,h := v ∈ L2(Ω; Rn), v|K ∈ R(K) for all K ∈ Th

and

V R⊥k,h := v ∈ L2(Ω; Rn), v|K ∈ R⊥(K) for all K ∈ Th.

By Lemma 2.3,

div Σk,b,h = V R⊥k,h ,

where Σk,b,h is defined in (2.5). That is to say, to get a stable pair of spaces, we onlyneed to enrich the discrete stress space Σk,h by some higher order bubble functionswhich are able to control the piecewise rigid motions in V R

k,h. We shall select thesebubble functions from the lowest order Arnold–Winther element stress space Σ∗

2,h

defined in the previous section. To this end, given an (n−1)-dimensional simplex F

of Th, let ωF := K−∪K+ denote the union of two elements that share F . We recallthat R(ωF ) is the rigid motion space over ωF while R(ωF )|F is the restriction onF . Further we let (R(ωF )|F )⊥ denote the orthogonal complement space of R(ωF )|Fwith respect to P1(F ; Rn) which allows to define the following (n− 1)-dimensionalsimplex H(div) bubble function space:

B1F :=

τ ∈ Σ∗

2,h, τ = 0 on Ω\ωF ,

∫F

τν · pds = 0 for any p ∈ (R(ωF )|F )⊥,

the averages of τ over both K− and K+ vanish,

the values of∫

K

τ : θdx vanish for any θ ∈ M2(K), K = K− and K+

.

(4.3)

Here ν is the normal vector of F . We also need a subspace of B1F defined by

B2F :=

τ ∈ B

1F ,

∫F

τν · pds = 0 for any p ∈ P0(F ; Rn)

. (4.4)

Hence we define the following enriched stress space

Σ+k,h = Σk,h +

∑F

B1F for 2 ≤ k ≤ n − 1; (4.5)

and

Σ+k,h = Σk,h +

∑F

B2F for k = n. (4.6)

Lemma 4.1. The space Σ+k,h is a direct sum of the spaces Σk,h and ΣF B1

F for2 ≤ k ≤ n − 1, and it is a direct sum of the spaces Σk,h and ΣF B2

F for k = n.

Proof. We only prove the first part of the theorem since the proof of the secondpart is similar. In fact, on the one hand, given F , it follows from the definition of

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B1F that for any τ ∈ B1

F it vanishes on the following degrees of freedom:

• the mean moments of degree at most n− over , of tTl τνi, νT

i τνj , l = 1, . . . , ,

i, j = 1, . . . , n − , (C2n+1− + (n − ))C

n = (n−)(n++1)2 C

n degrees of freedom,for each -dimensional simplex of K, 0 ≤ ≤ n − 2, with linearly indepen-dent tangential vectors t1, . . . , t, and n− linearly independent normal vectorsν1, . . . , νn−,

for any K ∈ Th. On the other hand, for any τ ∈ Pk(K; S) with 2 ≤ k ≤ n − 1, ifit vanishes on the above degrees of freedom, τν = 0 on ∂K where ν is the normalvector of ∂K; see Ref. 28 for more details. This indicates that τν = 0 on F whichimplies the first part of the theorem.

It follows from the definitions of Vk,h (Pk−1 polynomials) and Σ+k,h (enriched Pk

polynomials) that

div Σ+k,h ⊂ Vk,h.

This, in turn, leads to a strong divergence-free space:

Zh := τh ∈ Σ+k,h | (div τh, v) = 0 for all v ∈ Vk,h

= τh ∈ Σ+k,h | div τh = 0 pointwise. (4.7)

The mixed finite element approximation of problem (1.1) reads: find (σh, uh) ∈Σ+

k,h × Vk,h such that(Aσh, τ) + (div τ, uh) = 0 for all τ ∈ Σ+

k,h,

(div σh, v) = (f, v) for all v ∈ Vk,h.(4.8)

4.2. Stability analysis and error estimates

The convergence of the finite element solution follows the stability and the standardapproximation property. So we consider first the well-posedness of the discreteproblem (4.8). By the standard theory, we only need to prove the following twoconditions, based on their counterpart at the continuous level.

(1) K-ellipticity. There exists a constant C > 0, independent of the meshsize h

such that

(Aτ, τ) ≥ C‖τ‖2H(div) for all τ ∈ Zh, (4.9)

where Zh is the divergence-free space defined in (4.7).(2) Discrete B-B condition. There exists a positive constant C > 0 independent of

the meshsize h, such that

inf0=v∈Vk,h

sup0=τ∈Σ+

k,h

(div τ, v)‖τ‖H(div)‖v‖0

≥ C. (4.10)

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Theorem 4.1. For the discrete problem (4.8), the K-ellipticity (4.9) and the dis-crete B-B condition (4.10) hold uniformly. Consequently, the discrete mixed problem(4.8) has a unique solution (σh, uh) ∈ Σ+

k,h × Vk,h.

Proof. The K-ellipticity immediately follows from the fact that div Σ+k,h ⊂ Vk,h.

To prove the discrete B-B condition (4.10), for any vh ∈ Vk,h, we will construct aninterpolation operator Πh : H1(Ω; S) → Σ+

k,h such that, for any τ ∈ H1(Ω; S),∫K

div(τ − Πhτ) · vdx = 0 for any v ∈ Vk,h, (4.11)

for any K ∈ Th. Further, if τ ∈ Hk+1(Ω; S), it holds

‖τ − Πhτ‖0 + h‖div(τ − Πhτ)‖0 ≤ Chk+1|τ |k+1. (4.12)

We only show the above result for the cases 2 ≤ k ≤ n−1 since the proof for the casek = n is similar. First let Ih : H1(Ω; S) → Σk,h be a Scott–Zhang38 interpolationoperator such that

‖τ − Ihτ‖0 + h‖∇Ihτ‖0 ≤ Ch‖∇τ‖0. (4.13)

Since Ih preserves symmetric Pk functions locally,

‖τ − Ihτ‖0 + h‖∇(τ − Ihτ)‖0 ≤ Chk+1|τ |k+1, (4.14)

provided that τ ∈ Hk+1(Ω; S). See Ref. 28 for more details.Second, these enriched bubble functions in

∑F B1

F on the (n − 1)-dimensionalsimplices F allow for defining a correction δF

h ∈ B1F such that∫

F

δFh ν · pds =

∫F

(τ − Ihτ)ν · pds for any p ∈ R(K)|F . (4.15)

For these corrections δFh , we have

‖δFh ‖0,ωF + h‖div δF

h ‖0,ωF ≤ C(‖τ − Ihτ‖0,ωF + h‖∇(τ − Ihτ)‖0,ωF ). (4.16)

Finally, we take

Π1hτ = Ihτ +

∑F

δFh . (4.17)

We get a partial-divergence matching property of Π1hτ : for any p ∈ R(K), as the

symmetric gradient ε(p) = 0,∫K

(div Π1hτ − div τ) · pdx =

∫∂K

(Π1hτ − τ)ν · pds = 0. (4.18)

Next we make a correction δKh ∈ ΣK,k,b on each element K such that∫

K

div δKh · pdx =

∫K

div(τ − Π1hτ) · pdx for any p ∈ R⊥(K). (4.19)

The existence of δKh follows from Lemma 2.3, which also implies that

‖div δKh ‖0,K ≤ ‖div(τ − Π1

hτ)‖0,K . (4.20)

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In addition, the δKh can be selected such that

‖δKh ‖0,K = min‖δK‖0,K , div δK = Π⊥

K div(τ − Π1hτ), δK ∈ ΣK,k,b, (4.21)

where Π⊥K : L2(K, Rn) → R⊥(K) denotes the L2 projection operator. It follows

that ‖div δKh ‖0,K defines a norm for it. Then, a scaling argument proves

‖δKh ‖0,K ≤ Ch‖div δK

h ‖0,K ≤ Ch‖div(τ − Π1hτ)‖0,K . (4.22)

Finally, we define

Πhτ = Π1hτ +

∑K

δKh . (4.23)

Hence Eq. (4.11) follows from (4.18) and (4.19). The estimate (4.12) follows from(4.14), (4.16), (4.20) and (4.22). In addition, by (4.13), (4.20) and (4.22),

‖τ − Πhτ‖0 + h‖∇Πhτ‖0 ≤ Ch‖∇τ‖0. (4.24)

In the sequel, we use the interpolation operator Πh to prove the discrete inf–supcondition (4.10). Indeed, by the stability of the continuous formulation, there is aτ ∈ H1(Ω; S) such that,

div τ = vh and ‖τ‖1 ≤ C‖vh‖0.

In this paper, we only consider the domain such that the above stability holds. Werefer interested readers to Ref. 23 for the classical result which states it is true forLipschitz domains in Rn; see Ref. 21 for more refined results.

It follows from (4.11) and (4.24) that

div Πhτ = vh and ‖τ − Πhτ‖0 + h‖∇Πhτ‖0 ≤ Ch‖vh‖0, (4.25)

which shows (4.10) and completes the proof.

Theorem 4.2. Let (σ, u) ∈ Σ × V be the exact solution of problem (1.1) and(τh, uh) ∈ Σ+

k,h × Vk,h the finite element solution of (4.8). Then, for 2 ≤ k ≤ n,

‖σ − σh‖H(div) + ‖u − uh‖0 ≤ Chk(‖σ‖k+1 + ‖u‖k), (4.26)

and

‖σ − σh‖0 ≤ Chk+1|σ|k+1. (4.27)

Proof. The estimate (4.27) follows from the stability of the elements in Theorem4.1 and the standard theory of mixed finite element methods.15,16 Let the interpo-lation operator Πh be defined in (4.11). It follows

div Πhσ = div σh. (4.28)

This leads to

(A(σh − Πhσ), σh − Πhσ) = (A(σh − σ), σh − Πhσ)

+ (A(σ − Πhσ), σh − Πhσ)

= (A(σ − Πhσ), σh − Πhσ).

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Hence

‖σh − Πhσ‖0 ≤ C‖σ − Πhσ‖0.

Then the estimate (5.8) follows from the triangle inequality and (4.12).

5. First-Order Mixed Elements

In order to get first-order mixed elements, we propose to take the following discretedisplacement space

V1,h := v ∈ L2(Ω; Rn), v|K ∈ R(K) for any K ∈ Th. (5.1)

To design the space for the stress, we define

Σ1,h := τ ∈ H1(Ω; S), τ |K ∈ P1(K, S) for any K ∈ Th. (5.2)

Since the pair (Σ1,h, V1,h) is unstable, we propose to enrich Σ1,h by some (n − 1)-dimensional simplex bubble function spaces. Given an (n− 1)-dimensional simplexF of Th, we define

BF :=

τ ∈ Σ∗2,h, τ = 0 on Ω\ωF ,

∫F

τν · pds = 0 for any p ∈ (R(ωF )|F )⊥,

the values of∫

K

τ : θdx vanish for any θ ∈ M2(K), K = K− and K+

.

(5.3)

This allows for defining the following enriched stress space

Σ+1,h = Σ1,h +

∑F

BF . (5.4)

For this enriched space Σ+1,h, the number of degrees of freedom on each simplex

is 18 and 48 for n = 2, 3, respectively, which are the simplest conforming mixedelements so far. A similar argument of Lemma 4.1 shows that Σ+

1,h is a direct sumof Σ1,h and ΣF BF .

The mixed finite element approximation of problem (1.1) reads: find (σh, uh) ∈Σ+

1,h × V1,h such that(Aσh, τ) + (div τ, uh) = 0 for all τ ∈ Σ+

1,h,

(div σh, v) = (f, v) for all v ∈ V1,h.(5.5)

It follows from div Σ+1,h ⊂ V1,h that div τ = 0 for any τ ∈ Zh, which implies the

above K-ellipticity condition (4.9). A similar proof of Theorem 4.1 shows the dis-crete inf–sup condition (4.10). In particular, there exists an interpolation operatorΠh : H1(Ω, S) → Σ+

1,h such that

‖τ − Πhτ‖0 + h‖div(τ − Πhτ)‖ ≤ hk‖τ‖k, k = 1, 2, (5.6)

and∫K

div(τ − Πhτ) : pdx =∫

∂K

(τ − Πhτ)ν · pds = 0 for any p ∈ R(K), (5.7)

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for any K ∈ Th. A summary of these results leads to the error estimates in thefollowing theorem.

Theorem 5.1. Let (σ, u) ∈ Σ × V be the exact solution of problem (1.1) and(τh, uh) ∈ Σ+

1,h × V1,h the finite element solution of (5.5). Then,

‖σ − σh‖H(div) + ‖u − uh‖0 ≤ Ch(‖div σ‖1 + ‖u‖1) (5.8)

and

‖σ − σh‖0 ≤ Ch2‖σ‖2. (5.9)

6. Numerical Test

We compute a 2D pure displacement problem on the unit square Ω = [0, 1]2 witha homogeneous boundary condition that u ≡ 0 on ∂Ω. In the computation, we letthe compliance tensor in (1.1):

Aσ =12µ

(σ − λ

2µ + nλtr(σ)δ

), n = 2,

where δ =(

1 0

0 1

), and µ = 1/2 and λ = 1 are the Lame constants. Let the exact

solution be

u =

(ex−yx(1 − x)y(1 − y)

sin(πx) sin(πy)

). (6.1)

The true stress function σ and the load function f are defined by the equationsin (1.1), for the given solution u.

In the computation, the level one grid consists of two right triangles, obtainedby cutting the unit square with a north-east line. Each grid is refined into a half-sized grid uniformly, to get a higher level grid. In all the computation, the discretesystems of equations are solved by Matlab backslash solver.

We use the bubble enriched P2 symmetric stress finite element with P1 discon-tinuous displacement finite element, k = 2 in (4.2) and in (4.6), and k = 2 in (4.1).That is, three P3 bubbles are enriched each edge. In Table 1, the errors and theconvergence order in various norms are listed for the true solution (6.1). The opti-mal order of convergence is observed for both displacement and stress, see Table 1,as shown in the theorem.

Table 1. The errors, eh = σ − σh, and the order of conver-gence, by the 2D k = 2 element in (4.6) and (4.2), for (6.1).

‖u − uh‖0 rate ‖eh‖0 rate ‖div eh‖0 hn

1 0.27452 0.0 1.24637 0.0 6.97007772 0.02 0.07432 1.9 0.18054 2.8 2.13781130 1.73 0.01959 1.9 0.02429 2.9 0.57734125 1.94 0.00497 2.0 0.00314 2.9 0.14709450 2.05 0.00125 2.0 0.00040 3.0 0.03694721 2.0

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Table 2. The errors, eh = σ − σh, and the order of conver-

gence, by the Arnold–Winther 21/3 element, for (6.1).

‖u − uh‖0 rate ‖eh‖0 rate ‖div eh‖0 hn

1 0.30554 0.0 1.58058 0.0 10.31991249 0.02 0.22589 0.4 0.89927 0.8 6.81340378 0.63 0.10922 1.0 0.25584 1.8 3.61633797 0.94 0.05354 1.0 0.06633 1.9 1.83690959 1.05 0.02661 1.0 0.01674 2.0 0.92212628 1.0

As a comparison, we also test the Arnold–Winther element from Ref. 10, whichhas a same degree of freedom as ours, 21, on each element. But the displacementin that element is approximated by the rigid-motion space only, instead of the fullP1 space, i.e. 3 dof versus 6 dof on each triangle. The total degrees of freedom forthe stress for the new element are 3|V| + 3|E| + 3|K|, where |V|, |E| and |K| arethe numbers of vertices, edges and elements of Th, respectively, while those for theArnold–Winther element are 3|V| + 4|E|. Since the three bubble functions on eachelement can be easily condensed, these two elements almost have the same com-plexity for solving. The errors and the orders of convergence are listed in Table 2.Because the new element uses the full P1 displacement space, the order of con-vergence is one higher than that of the Arnold–Winther element. Also as the newelement includes the full P2 stress space, the order of convergence of stress is oneorder higher, see the data in Tables 1 and 2.

Appendix A. The Basis Functions of Σk,h in Two Dimensions

Let x0x1x2 =: K ∈ Th with three edges Ei and corresponding three barycentricvariables λi. Here λi is a linear function which vanishes on edge Ei and assumesa nodal value 1 at the opposite vertex xi. Given Ei = −−−−−−→xi−1xi+1, its two endpointsare xi−1 and xi+1, which allows for defining its k − 1 interior nodal points by

xEi,j =j

kxi−1 +

k − j

kxi+1, j = 1, . . . , k − 1. (A.1)

We also define (k−1)(k−2)2 nodal points inside K by

xK,l,m =l

kx0 +

m

kx1 +

k − l − m

kx2, 1 ≤ l, m and l + m ≤ k − 1. (A.2)

Then the nodes for the Lagrange element of order k is

XK = xi, i = 0, 1, 2 ∪ xEi,j , i = 0, 1, 2, j = 1, . . . , k − 1∪ xK,l,m, 1 ≤ l, m and l + m ≤ k − 1.

Given node xEi,j on edge Ei, j = 1, . . . , k−1, let φEi,j ∈ Pk(K; R) be its associatednodal basis function of the Lagrange element of order k such that

φEi,j(xEi,j) = 1 and φEi,j(x′) = 0 for any x′ ∈ XK other than xEi,j. (A.3)

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Let ni = 〈ni,1, ni,2〉T and n⊥i = 〈−ni,2, ni,1〉T be normal and tangent vectors on

edge Ei, respectively. We define a matrix of rank one by

TEi = n⊥i n⊥

i

T. (A.4)

With these (k− 1)-edge bubble functions φEi,j on each edge and the matrix TEi ofrank one, we can define exactly (k − 1) stress functions τEi,j by

τEi,j = φEi,jTEi , j = 1, 2, . . . , k − 1, i = 0, 1, 2. (A.5)

By the definition, we have

τEi,j · nl|El= 0, i, l = 0, 1, 2, j = 1, . . . , k − 1, (A.6)

which implies that they are H(div) bubble functions on element K.Given E we need a basis which takes TE for S. To this end, we let

TE,1 = nEnTE and TE,2 =

12(n⊥

EnTE + nE(n⊥

E)T ).

It is straightforward to see that TE , TE,1 and TE,2 are linearly independent andtherefore form a basis of S. The canonical basis of S reads

T1 =

(1 0

0 0

), T2 =

(0 11 0

), and T3 =

(0 00 1

). (A.7)

Let XE denote all interior nodes, defined in (A.1), of all the edges, XK denote allinterior nodes, defined in (A.2), of all the elements, and XV denote all the verticesof Th. Define the Lagrange element space of order k by

Ph := H1(Ω; R) ∩ v ∈ L2(Ω; R), v|K ∈ Pk(K; R), ∀K ∈ Th.

Given node x ∈ XV ∪ XE ∪ XK, let φx ∈ Ph be its associated nodal basis function,which is similarly defined as φEi,j in (A.3).

The basis functions of Σk,h can be classified into four classes:

(1) Vertex-based basis functions: Given vertex x ∈ XV, its three associated basisfunctions of Σk,h read

τV,x,i = φxTi, i = 1, 2, 3.

(2) Volume-based basis functions: Given node x ∈ XK inside K, its three associatedbasis functions of Σk,h read

τK,x,i = φxTi, i = 1, 2, 3.

(3) Edge-based basis functions with nonzero fluxes: Given node x ∈ XE on edge E,its two associated basis functions with nonzero fluxes of Σk,h read

τ(nb)E,x,i = φxTE,i, i = 1, 2.

(4) Edge-based bubble functions: Given node x ∈ XE on edge E which is shared byelements K1 and K2, its bubble functions in Σk,h read

τ(b)E,x,i = φx|KiTE , i = 1, 2.

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It is straightforward to see that these functions defined in the above four termsform a basis of Σk,h, which are very easy to construct.

Acknowledgment

The first author was supported by the NSFC Projects 11271035, 91430213 and11421101.

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July 15, 2016 13:43 WSPC/103-M3AS 1650040 ...szhang/research/p/2016b.pdfMathematical Models and Methods in Applied Sciences Vol. 26, No. 9 (2016) 1649–1669 c World Scientiﬁc Publishing