A new quadrilateral MINI-element for Stokes equations

DOI:10.1051/m2an/2013129 www.esaim-m2an.org

A NEW QUADRILATERAL MINI-ELEMENT FOR STOKES EQUATIONS

Oh-In Kwon

and Chunjae Park

Abstract. We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is dis- cretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done with the standardQ1-conforming elements enriched by one bubble a quadrilateral. The supercon- vergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniform meshes.

Mathematics Subject Classification. 65N30, 74S05, 76M10.

Received May 7, 2012. Revised August 5, 2013.

Published online June 30, 2014.

1. Introduction

In the finite element methods for incompressible Stokes problems, the pair of discrete velocity and pressure should fulfill the inf-sup condition for the stability, which is attributed to the analysis of Babu˘ska [2] and Brezzi [4]. On triangular meshes, the MINI-element pair [1] of “[P₁-conforming + bubble]²×P₁-continuous”

is one of the stable pairs of the lowest order, since the simplest pair of “[P₁-conforming ]²×P₀” is not stable.

The several pairs of “

[Q1-conforming + bubble]²+ bubble

×Q1-continuous” have been suggested as the MINI-element on quadrilateral meshes [3,7]. Their discrete velocities are the standard conforming spaces enriched by at least three bubbles a quadrilateral, while the triangular MINI-element is done by two bubbles a triangle.

In this paper, we will introduce a new stable MINI-element pair on quadrilateral meshes which is

“[Q1-conforming + bubble]²×P₁-midpoint-edge-continuous”. Compared to other quadrilateral MINI-elements, the discrete pressure has continuous averages over edges instead of the pointwise continuity. The number of bubbles for the velocity is then reduced to two a quadrilateral, similar to the triangular MINI-element. The proposed MINI-element pair is interpreted as a subspace of the stable “[Q2-conforming]²×piecewiseP₁” on quadrilateral meshes.

Regarding the finite element solutions with the MINI-elements, although the order of convergence in pressure is analyzed to beO(h) by the standard error analysis, the superconvergence of orderO(h^3/2) is fully understood on three-directional triangular meshes with the aid of the stabilized formulation [6]. We will analyze the similar superconvergence for the pressure of the proposed MINI-element on uniform rectangular meshes, as well as numerical tests on both uniform and non-uniform meshes.

In the following section, our MINI-element will be defined and proven to be stable through the inf-sup condition. Section3 will be assigned for the superconvergence in pressure, which consists of three subsections

Keywords and phrases.MINI-element, superconvergence.

1 Department of Mathematics, Konkuk University, 143-701 Seoul, Korea.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

for the stabilized formulation of the finite element method, the supercloseness of the bilinear interpolation and the main proof of the superconvergence, respectively. The paper will be closed with the numerical results in the final section.

Throughout the paper, standard notations for Sobolev spaces are employed andL²₀(Ω) denotes the space of all functions inL²(Ω) whose average overΩvanish. For a bounded setS⊂R², with its boundary∂S, · · · m,S

and| · · · |m,S will mean the norm and seminorm forH^m(S). We will denote by (·,·)S and<·,·>_∂S theL²(S) andL²(∂S) inner products, respectively. IfS=Ωor m= 0, they may be omitted in the indices.

2. A new quadrilateral MINI-element

Let Ωbe a simply connected polygonal domain and {T_h}_h>0 a regular family of triangulations ofΩ which consist of convex quadrilaterals. InT_h,his proportional to the maximum of diameters of quadrilaterals inT_h.

2.1. P

-midpoint-edge-continuous quadrilateral pressure

LetP_hbe a space of piecewise linear functions as follows:

P_h =

q_h∈L²(Ω) q_h|Q ∈span{1, x, y}for all quadrilateralsQ∈ Th ,

then theP₁-midpoint-edge-continuous quadrilateral finite element space [9] is defined by N C^h(Ω) ={q_h∈P_h q_his continuous at every midpoints of edges inTh}.

For each vertex V in T_h, a function ψ^V ∈ N C^h(Ω) is defined by its values at all midpointsm of edges inT_h such that

ψ^V(m) =

1, ifmbelongs to an edge which meetsV, 0, otherwise.

Then, if we fix an arbitrary vertexV₀ inTh, the following set is a basis forN C^h(Ω), {ψ^V∈ N C^h(Ω) V is a vertex inT_h, V=V₀}. Hence, the dimension ofN C^h(Ω) is the number of vertices inThsubtracted by one.

Define an interpolationπ_h :H¹(Ω)∩C(Ω)→ N C^h(Ω) as π_hv=

V

v(V)

2 ψ^V, ∀v∈H¹(Ω)∩C(Ω), (2.1)

where the summation runs over all interior verticesV inT_h. Then, the interpolation error is estimated by the following, forv∈H²(Ω),

v−π_hv₀+h

⎛

⎝

Q∈Th

|v−π_hv|²_1,Q

⎞

⎠

1/2

≤ C h²|v|₂. (2.2)

It is noted thatN C^h(Ω) is a common subspace of other low order nonconforming spaces such as the Rotated Q₁ or DSSY finite element spaces [5,10].

2.2. A new MINI-element pair

Let Q = [−1,1]² be a reference rectangle and Q1,Q2 bilinear and biquadratic polynomial spaces on Q, respectively, that is,

Q₁= span{1,x,y,xy}, Q₂=Q₁ span

x²,y²,x²y, xy²,x²y² .

For each quadrilateralQ∈ Th, the canonical bijective bilinear map fromQ toQis denoted byF_Q and function spacesQ1(Q),Q2(Q) onQare defined as follows:

Q_k(Q) =

v◦F_Q⁻¹ v∈Q_k

, k= 1,2.

Then, the standard conforming finite element spacesQ_1,h,Q_2,hoverT_h are defined by Qk,h={v_h∈C(Ω) v_h|Q ∈Qk(Q) for all quadrilateralsQ∈ Th}, as well asQk,h,0=Qk,h∩H₀¹(Ω) fork= 1,2.

It is well-known that the pair of [Q2,h,0]²for velocity andP_h∩L²₀(Ω) for pressure satisfy the inf-sup condition for Stokes equations in the following proposition, whose proof is referred to Section 3.2, Chapter II in [8].

Proposition 2.1. Let p_h∈P_h∩L²₀(Ω). There existsv_h∈[Q2,h,0]² such that

Ω

p_h div v_h ds= ||p_h||²_0,Ω, |v_h|1,Ω≤ 1

β||p_h||0,Ω, for a positive constantβ which depends only on Ω.

In order to introduce a MINI-element pair, we choose a bubble function b_Q ∈ Q2(Q) for each quadrilateral Q∈ Th such that

b_Q◦F_Q=

x²−1 y²−1

∈Q₂, (2.3)

which vanishes on the boundary ofQ. Then, identifyingb_Q with its trivial extension intoC(Ω), we propose a new MINI-element pair for Stokes equations as follows:

X_h=

⎡

⎣Q_1,h,0

Q∈Th

span{b_Q}

⎤

⎦

2

, M_h=N C^h(Ω)∩L²₀(Ω).

The pairX_h×M_hturns out to be stable to solve Stokes equations in the following theorem.

Theorem 2.2 (Inf-sup condition). Let μ_h∈M_h. There existsv_h∈X_h such that

Ω

μ_h divv_h ds= ||μ_h||²_0,Ω, |v_h|_1,Ω≤ 1

β||μ_h||_0,Ω, for a positive constantβ which depends only on Ω.

Proof. Define an interpolation operatorΠ_h:Q2,h,0→Q1,h,0

Q∈Thspan{b_Q}such that Π_hv(V) =v(V),

Q

Π_hv ds=

Q

v ds, ∀v∈Q2,h,0, (2.4)

for all verticesVand quadrilateralsQin Th. Then, by Bramble lemma, we have

|v−Π_hv|1,Q≤Ch|v|2,Q, for all quadrilateralsQ∈ Th, (2.5) with some constantC regardless ofv, Qandh.

Applying the argument of Lemma A.6, Appendix A in [8] for triangles into quadrilaterals, the following inverse inequality is established with some constantC,

|v|2,Q≤Ch⁻¹|v|1,Q. (2.6)

From (2.5) and (2.6), the operatorΠ_his bounded by

|Π_hv|1,Ω≤C|v|1,Ω, ∀v∈Q2,h,0. (2.7) Now, ifμ_h∈M_h, by Proposition2.1, there existsw_h∈[Q_2,h,0]² such that

Ω

μ_h divw_h ds= ||μ_h||²_0,Ω, |w_h|_1,Ω≤ 1

β||μ_h||_0,Ω, (2.8)

for a positive constantβ which depends only onΩ. Forw_h= (w₁, w₂), definev_h∈X_hby v_h= (Π_hw₁, Π_hw₂).

Then, it will be shown

Ω

μ_h divw_h ds=

Ω

μ_h divv_h ds. (2.9)

We have, by integration by parts,

Ω

μ_h div (w_h−v_h) ds=

Q∈Th

∂Q

μ_h(w_h−v_h)·ndl−

Q∈Th

Q∇μ_h·(w_h−v_h) ds, (2.10) wherenis the unit vector which is outward normal to∂Q. Since∇μ_his a constant vector, by definition ofΠ_h in (2.4), the second term in the right hand side of (2.10) is null.

Denote by m_E the midpoint of edgeE ∈ T_h. Sincew_h−v_h is continuous on interior edges and vanishes on boundary edges, the first term is rewritten as:

Q∈Th

∂Q

μ_h(w_h−v_h)·ndl=

Q∈Th

E∈∂Q

E

μ_h−μ_h(m_E)

(w_h−v_h)·ndl. (2.11) We note thatμ_h−μ_h(m_E) andw_h−v_h are the linear and quadratic functions onE vanishing at the midpoint and two endpoints of E, respectively. Thus, all integrals over edges in (2.11) vanish, since the integrands are odd and cubic.

This means (2.9) and completes the proof with (2.7), (2.8).

Let (u, p) ∈ [H₀¹(Ω)]²×L²₀(Ω) be the solution of the variational form of a Stokes equation such that, for a source functionf ∈[L²(Ω)]²,

(∇u,∇v)−(p,divv) + (q,divu) = (f,v), ∀(v, q)∈

H₀¹(Ω)₂

×L²₀(Ω), (2.12) and (u_h, p_h)∈X_h×M_h be the solution of the following discrete variational problem,

(∇u_h,∇v_h)−(p_h,divv_h) + (q_h,divu_h) = (f,v_h), ∀(vh, q_h)∈X_h×M_h. (2.13) If (u, p) belongs to [H²(Ω)]²×H¹(Ω), through the inf-sup condition in Theorem2.2, the approximation error is estimated by

|v−v_h|₁+p−p_h0≤Ch(|v|₂+|p|₁), (2.14) for a constantC which depends only onΩand the regularity ofT_h.

While the order of convergence in pressure isO(h) as in (2.14) on general quadrilateral meshes, the remaining of the paper will be devoted to the analysis of the superconvergence of orderO(h^3/2) on uniform rectangular meshes.

3. Superconvergence in pressure

The superconvergence in pressure is well understood [6] for the MINI-element of Arnoldet al. [1] on three- directional triangular meshes. In this section, following the arguments in [6] with the attention on the midpoint- edge-continuous pressure instead of the continuous one, we will prove that the proposed MINI-element has the same property. Throughout the section, we assume that T_h consists of uniform rectangles, whose edges are all parallel to the coordinate axes.

For a functionv_h inX_h, the bilinear and bubble parts of v_hare denoted byv_L,v_b, respectively, so that:

v_h=v_L+v_b, v_L∈[Q1,h,0]², v_b|Q ∈[span{b_Q}]² for allQ∈ Th. We note that they are orthogonal to each other inH¹inner product since:

(∇v_L,∇v_b)_Q = 0, for each rectangleQ∈ T_h. (3.1)

3.1. Stabilized variational form

We will show that if (u_h, p_h)∈X_h×M_hsatisfy (2.13), the pair of the bilinear partu_Lofu_hand the pressure p_his a solution of a stabilized variational problem.

Set a bilinear formB_h onH₀¹(Ω)×L²₀(Ω) as:

B_h(u, p;v, q) = (∇u,∇v)−(p,divv) + (q,divu) +

Q∈Th

(−Δu+∇p, τ_Q∇q)_Q, (3.2)

whereτ_Q is the bubble functionb_Q in (2.3) multiplied by a constant for each rectangleQ∈ Th such that τ_Q=

Qb_Q ds

Q|∇b_Q|² ds b_Q. We note thatτ_Q is a nonnegative function which satisfies,

(∇τ_Q,∇τ_Q)_Q = (τ_Q,1)_Q=α²h²(1,1)_Q, τ_Q∞,Q=γ²h², (3.3) with some fixed positive constantsαandγ. From (3.3), we have an identity,

τ_Q^1/2∇q_h

0,Q=αh∇q_h0,Q, (3.4)

for all rectanglesQ∈ Th and piecewise linear pressuresq_h∈M_h.

Lemma 3.1. If (u_h, p_h)∈X_h×M_h satisfies (2.13), then(u_L, p_h)∈[Q1,h,0]²×M_h does B_h(u_L, p_h;v_L, q_h) = (f,v_L) +

Q∈Th

(f, τ_Q∇q_h)_Q, ∀(v_L, q_h)∈[Q_1,h,0]²×M_h. (3.5)

Proof. Let (v_L, q_h)∈ [Q_1,h,0]²×M_h, then for any bubble v_b, (2.13) establishes that, from the orthogonality in (3.1),

(∇u_L,∇v_L)−(p_h,divv_L) + (q_h,divu_L) = (f,v_L+v_b)−(∇u_b,∇v_b) + (p_h,divv_b)−(q_h,divu_b)

= (f,v_L) +

Q∈Th

(f − ∇p_h,v_b)_Q−(∇u_b,∇v_b) +

Q∈Th

(∇q_h,u_b)_Q. (3.6)

We note that ∇q_h is piecewisely constant and u_b|_Q = τ_Q(c₁, c₂) for some constants c₁, c₂. If we choose a bubblev_b such thatv_b|_Q =τ_Q∇q_h for each rectangleQ∈ T_h, then the following identity comes from (3.3),

(∇u_b,∇v_b) =

Q∈Th

(∇q_h,u_b).

Thus, the last two terms in (3.6) vanish and (3.6) implies (3.5), sinceΔu_L= 0.

Although the bilinear formB_h in (3.2) is not uniformly coercive on [Q1,h,0]²×M_h, we have a stability forB_h in the following lemma. Define a triple norm as

|||(v, q)|||B =

⎛

⎝|v|²₁+||q||²₀+

Q∈Th

τ_Q^1/2∇q²

0,Q

⎞

⎠

1/2

. (3.7)

Lemma 3.2. If (v_L, q_h)∈[Q1,h,0]²×M_h, there existsw_L∈[Q1,h,0]² such that

B_h(v_L, q_h;w_L, q_h)≥β_B |||(vL, q_h)|||B |||(wL, q_h)|||B, (3.8) whereβ_B is a constant which depends only on Ω.

Proof. By the inf-sup condition in Theorem2.2, there existsz_h∈X_hsuch that (−q_h,divz_h) =||q_h||²₀, |z_h|1≤ 1

β||q_h||0, (3.9)

with some constantβ regardless ofh.

For the bubble partz_b, we expand, from (3.4),

|(qh,divz_b)|=

Q∈Th

(∇q_h,z_b)_Q ≤

Q∈Th

h∇q_h0,Q|z_b|1,Q

=α⁻¹

Q∈Th

τ_Q^1/2∇q_h

0,Q|z_b|1,Q

≤α⁻¹

⎛

⎝

Q∈Th

τ_Q^1/2∇q_h²

0,Q

⎞

⎠

1/2⎛

⎝

Q∈Th

|z_b|²_1,Q

⎞

⎠

1/2

≤β²

2 |z_b|²₁+ 1 2α²β²

⎛

⎝

Q∈Th

τ_Q^1/2∇q_h²

0,Q

⎞

⎠,

(3.10)

as well as for the bilinear partz_L,

|(∇v_L,∇z_L)| ≤ |v_L|1|z_L|1≤ β²

2 |z_L|²₁+ 1

2β²|v_L|²₁. (3.11)

Then, by the orthogonality (3.1), we combine (3.9)–(3.11) to get the following:

B_h(v_L, q_h;z_L,0)≥ 1

2||q_h||²₀−σ

⎛

⎝|v_L|²₁+

Q∈Th

τ_Q^1/2∇q_h²

0,Q

⎞

⎠, (3.12)

for a constantσ= max

1/(2α²β²),1/(2β²) .

If we setw_L=v_L+ 2/(1 + 2σ)z_L, then, from (3.12),w_L satisfies that

B_h(v_L, q_h;w_L, q_h) =B_h(v_L, q_h;v_L, q_h) + 2/(1 + 2σ)B_h(v_L, q_h;z_L,0)

≥1/(1 + 2σ)|||(v_L, q_h)|||²_B. From (3.9), the triple norm of (w_L, q_h) is bounded by

|||(w_L, q_h)|||B ≤

1 + 2/(β+ 2βσ)

|||(v_L, q_h)|||B. Thus,w_Lsatisfies (3.8) for the constantβ_B= 1/(1 + 2σ)

1 + 2/(β+ 2βσ)₋₁

which depends only onΩ.

3.2. Supercloseness

LetQ = [−1,1]² be the reference rectangle andE ={(x, y)∈Q x = 1} an edge ofQ. If g∈H¹(Q), for a vector fieldV = (x+ 1,0) onQ, we have the following:

2

Eg² dl=

Qdiv (g²V) ds=

Q

g²+ 2g∇g·V ds.

Thus, ifEis an edge of a rectangleQinTh andg∈H¹(Q), then the following local trace theorem holds, with a constantC which depends only the shape ofQ,

g_0,E≤C

h⁻¹g²_0,Q+g_0,Q|g|_1,Q1/2

. (3.13)

Especially, if r_h is a linear function onQin Th, since|r_h|_1,Q≤Ch⁻¹r_h0,Q, we have

r_h0,E ≤Ch^−1/2r_h0,Q. (3.14)

We will establish two superclosenesses in the following two lemmas, which are the principal ingredients for the proof of the superconvergence of the pressurep_h in (2.13). For any continuous functionv∈C(Ω), denote byv_I the standard bilinear interpolation ofv intoQ_1,hsuch that

v_I(V) =v(V) for all verticesV inTh. Ifv= (v₁, v₂)∈[C(Ω)]², then v_I = (v_1,I, v_2,I).

Lemma 3.3. Let u∈[H³(Ω)]². Then, for all w_h∈[Q1,h]²,

Ω∇(u−u_I)· ∇w_h ds≤ Ch²|u|3|w_h|1, whereC is a constant which depends only on Ω.

Proof. Letu, u_I, w be the first components ofu,u_I andw_h, respectively. It is sufficient to prove

Ω

(u−u_I)_xw_xds≤ Ch²|u|₃ |w|₁. (3.15) For each rectangleQin Th, define a linear functionalΨ onH³(Q) by the following:

Ψ(v) =

Q

(v−v_I)_xw_x ds, ∀v∈H³(Q).

Then, we have

|Ψ(v)| ≤Ch|v|_2,Q|w|_1,Q. (3.16)

Sincew_xis a function ofy, by simple calculation with the definition ofv_I, we have Ψ

x²

=Ψ y²

=Ψ(xy) = 0. (3.17)

Let u_xx, u_xy, u_yy be the respective averages ofu_xx, u_xy, u_yy overQ and ϕa quadratic function in P₂(Q) such that

ϕ= 1 2

u_xxx²+u_yyy²

+u_xy xy. (3.18)

Then, by Poincar´e Lemma, we establish

|u−ϕ|_2,Q=

u_xx−u_xx²_0,Q+ 2u_xy−u_xy²_0,Q+u_yy−u_yy²_0,Q1/2

≤Ch|u|_3,Q. (3.19) From (3.16), (3.17), (3.19), we have

|Ψ(u)|=|Ψ(u−ϕ)| ≤Ch|u−ϕ|_2,Q|w|_1,Q≤Ch²|u|_3,Q|w|_1,Q,

which means (3.15).

The following proposition is a frequently used characteristic for the jump of a midpoint-edge-continuous function across an edge in Th.

Proposition 3.4. Let r_h ∈ N C^h(Ω) and Q₁, Q₂ be two adjacent rectangles in Th which share an edge E. If g∈H¹(Q₁∪Q₂), then

E

g(r_h|Q1−r_h|Q2) dl ≤ C

2 k=1

|g|_1,Qkτ_Q^1/2∇r_h

0,Qk

, (3.20)

whereC is a constant which depends only the shape of rectangles in Th.

Proof. Sincer_h is continuous at the midpoint ofE, there is an averager_h ofr_h overE such that

E

r_h|Q1 dl=

E

r_h|Q2 dl=

E

r_h dl.

We split the left hand side of (3.20) into

Eg(r_h|Q1−r_h|Q2) dl ≤

Eg(r_h|Q1−r_h) dl +

Eg(r_h|Q2 −r_h) dl .

For each k = 1,2, denote by g_k the average of g over Q_k, then we have, from (3.4) and the local trace Theorem (3.13),

E

g(r_h|Qk−r_h) dl =

E

(g−g_k)(r_h|Qk−r_h) dl

≤ g−g_k0,E r_h|Qk−r_h0,E

≤Ch^1/2|g|_1,Qkh^1/2|r_h|_1,Qk ≤ C|g|_1,Qkτ_Q^1/2∇r_h

0,Qk

,

whereC is a constant which depends only on the shape ofQ_k. It means (3.20).

Lemma 3.5. Let u∈[H³(Ω)]². Then, for all r_h∈M_h,

Ω

r_h div (u−u_I) ds ≤ Ch^3/2||u||₃

⎛

⎝||r_h||²₀+

Q∈Th

τ_Q^1/2∇r_h²

0,Q

⎞

⎠

1/2

, whereC is a constant which depends only on Ω.

Proof. Letu, u_I be the first components of u,u_I, respectively. It is enough to show that

Ωr_h (u−u_I)_xds ≤ Ch^3/2||u||3

⎛

⎝||r_h||²₀+

Q∈Th

τ_Q^1/2∇r_h²

0,Q

⎞

⎠

1/2

. (3.21)

For each rectangleQin T_h of widthh, define a linear functionalΨ onH³(Q) by Ψ(v) =

Q

r_h(v−v_I)_x−h²

12(v_xxr_h)_x

ds, ∀v∈H³(Q). (3.22)

Since|r_h|_1,Q≤Ch⁻¹r_h0,Q, we estimate|Ψ(v)|as

|Ψ(v)| ≤Chr_h0,Q|v|_2,Q+h² 12

|v|_3,Qr_h0,Q+|v|_2,Q|r_h|_1,Q

(3.23a)

≤Chr_h0,Q

|v|_2,Q+h|v|_3,Q

. (3.23b)

We note that the integrand in (3.22) vanishes, ifvbelongs to span{1, x, y, xy, y²}. By definition ofv_I, we have Ψ(x²) = 2

Q

r_h(x−α)−h² 12(r_h)_x

ds, (3.24)

where αis thex-coordinate of the center of Q. Since the integral in (3.24) vanishes for the linear functionr_h, Ψ(ϕ) = 0 for all quadratic functions ϕ∈P₂(Q). Then, by similar arguments in (3.18) and (3.19), the following estimation is obtained from (3.23a):

|Ψ(u)|=|Ψ(u−ϕ)| ≤Chr_h0,Q

|u−ϕ|_2,Q+h|u|_3,Q

≤Ch²r_h0,Q|u|_3,Q. (3.25) Next, the second term in the integrand forΨ(u) in (3.22) satisfies:

Q∈Th

Q

(u_xxr_h)_xds=

Q∈Th

∂Q

u_xxr_hn_x dl (3.26a)

=

E

E

u_xx(r_h|_QL−r_h|_QR) dl+

∂Ω

u_xxr_hn_xdl, (3.26b)

whereE runs over all interior vertical edges andQ_L sharesE withQ_Rin its right side.

By Proposition3.4above, the first term in (3.26a) and (3.26b) is estimated by

E

E

u_xx(r_h|_QL−r_h|_QR) dl

≤C|u|_3,Ω

⎛

⎝

Q∈Th

τ_Q^1/2∇r_h²

0,Q

⎞

⎠

1/2

. (3.27)

From the global trace theorem foru_xxand the local one for r_h in (3.14), the second term is done as

∂Ω

u_xxr_hn_x dl≤ u_xx0,∂Ωr_h0,∂Ω ≤Ch^−1/2u_3,Ωr_h0,Ω. (3.28)

We establish (3.21) from (3.22) and (3.25)–(3.28).

3.3. Superconvergence

If the solution (u, p) in (2.12) has more regularity (u, p)∈[H²(Ω)]²×H¹(Ω), then, since −Δu+∇p=f, (u, p) satisfies the following:

B_h(u, p;v, q) = (f,v) +

Q∈Th

(f, τ_Q∇q)_Q, ∀(v, q)∈[H₀¹(Ω)]²×M_h. (3.29) For the finite element solution (u_h, p_h)∈X_h×M_h in (2.13), we have the following orthogonality, from (3.29) and Lemma3.1,

B_h(u−u_L, p−p_h;v_L, q_h) = 0, ∀(vL, q_h)∈[Q1,h,0]²×M_h, (3.30) whereu_L is the bilinear part ofu_h.

We reach at the superconvergence of the pressurep_h∈M_hin the following theorem.

Theorem 3.6. Let (u, p)∈[H₀¹(Ω)]²×L²₀(Ω),(u_h, p_h)∈X_h×M_h be the solutions of (2.12),(2.13), respec- tively. If the solution (u, p)belongs to[H³(Ω)]²×H²(Ω), then

||p−p_h||0 ≤ Ch^3/2(||u||3+|p|2), for some constant C which depends only onΩ.

Proof. Letp_i∈ N C^h(Ω) be the standard interpolation of p∈H²(Ω) by (2.1) and definep_I ∈M_has p_I =p_i−

Ω

p_i ds.

Then, from (2.2), the interpolation error is estimated by

p−p_I0+h

⎛

⎝

Q∈Th

|p−p_I|²_1,Q

⎞

⎠

1/2

≤ C h²|p|₂, (3.31)

for some constantCwhich depends only onΩ.

Forp_I and the bilinear interpolantu_I ofu, the orthogonality in (3.30) is rewritten as

B_h(u−u_I, p−p_I;v_L, q_h) =B_h(u_L−u_I, p_h−p_I;v_L, q_h). (3.32) Then, there exists (w_L, r_h)∈[Q1,h,0]²×M_h such that, by (3.32) and Lemma3.2,

|||(uL−u_I, p_h−p_I)|||B|||(wL, r_h)|||B ≤ 1

β B_h(u_L−u_I, p_h−p_I;w_L, r_h) (3.33a)

= 1

β B_h(u−u_I, p−p_I;w_L, r_h), (3.33b) with some constantβ, since (u_L−u_I, p_h−p_I)∈[Q1,h,0]²×M_h.

From the definition ofB_h, we have

B_h(u−u_I, p−p_I;w_L, r_h) = (∇(u−u_I)· ∇w_L) +

r_h,div (u−u_I)

−(p−p_I,divw_L) +

Q∈Th

(∇(p−p_I), τ_Q∇r_h)_Q−

Q∈Th

(Δ(u−u_I), τ_Q∇r_h)_Q. (3.34)

We apply the superclosenesses in Lemmas3.3and3.5for the first two terms in the right hand side of (3.34) to get,

∇(u−u_I)· ∇w_L

≤Ch²|u|₃ |||(w_L, r_h)|||_B, (3.35a) r_h,div (u−u_I)

≤Ch^3/2u₃ |||(w_L, r_h)|||_B. (3.35b) The third term is simply done by (3.31) into

|(p−p_I,divw_L)| ≤Ch²|p|₂ |||(w_L, r_h)|||_B. (3.36) With (3.3) and (3.31), the fourth term is estimated by

Q∈Th

∇(p−p_I), τ_Q∇r_h

Q

=

Q∈Th

τ_Q^1/2∇(p−p_I), τ_Q^1/2∇r_h

Q

≤γCh²|p|₂|||(w_L, r_h)|||_B.

(3.37)

For the last term in the right hand side of (3.34), letΔu∈[P_h]² be a piecewise constant interpolation ofΔu

such that

Q

Δuds=

Q

Δuds, for allQinTh, (3.38)

whose interpolation error is estimated by

Δu−Δu

0≤Ch|u|₃. (3.39)

Then, by (3.3) and (3.38), we have, (Δ(u−u_I), τ_Q∇r_h)_Q=

Δu−Δu, τ_Q∇r_h

Q+

Δu, τ_Q∇r_h

Q (3.40a)

= τ_Q^1/2

Δu−Δu

, τ_Q^1/2∇r_h

Q+α²h²(Δu,∇r_h)_Q, (3.40b) sinceΔu_I vanishes and∇r_h is a constant vector.

By (3.3) and (3.39), the sum of the first terms in (3.40a) and (3.40b) for allQ∈ Th is estimated by

Q∈Th

τ_Q^1/2

Δu−Δu

, τ_Q^1/2∇r_h

Q

≤γCh²|u|₃|||(w_L, r_h)|||_B. (3.41) For the second term, through integration by parts and since divuvanishes, we have:

Q∈Th

(Δu,∇r_h)_Q =

Q∈Th

< Δu·n, r_h>_∂Q −

Q∈Th

(divΔu, r_h)_Q

=

Q∈Th

< Δu·n, r_h>_∂Q . Applying Proposition3.4for the edges in the interior ofΩ, we obtain

Q∈Th

(Δu,∇r_h)_Q

≤C|Δu|₁|||(w_L, r_h)|||_B+|< Δu·n, r_h>_∂Ω|. (3.42)

Table 1. Error table for uniform rectangular meshes (M1).

Mesh ||u−uh||0 ||∇u− ∇uh||0 ||p−ph||0

Value Order Value Order Value Order

16×16 2.2187E-3 1.8359E-1 5.7334E-2

32×32 5.2254E-4 2.0861 8.8997E-2 1.0447 2.3560E-2 1.2830 64×64 1.2736E-4 2.0367 4.4001E-2 1.0162 9.6639E-3 1.2857 128×128 3.1456E-5 2.0175 2.1889E-2 1.0074 3.5357E-3 1.4506 256×256 7.8178E-6 2.0085 1.0917E-2 1.0035 1.1435E-3 1.6285 512×512 1.9491E-6 2.0039 5.4530E-3 1.0015 3.5626E-4 1.6825 1024×1024 4.8663E-7 2.0019 2.7253E-3 1.0007 1.1460E-4 1.6363

Table 2. Error table for non-uniform bisection mesh (M2).

mesh ||u−uh||0 ||∇u− ∇uh||0 ||p−ph||0

value order value order value order

16×16 2.6092E-3 1.9955E-1 6.1404E-2

32×32 6.1353E-4 2.0884 9.6495E-2 1.0482 2.4388E-2 1.3322 64×64 1.4974E-4 2.0347 4.7694E-2 1.0167 9.8453E-3 1.3087 128×128 3.7029E-5 2.0157 2.3729E-2 1.0072 3.5883E-3 1.4561 256×256 9.2103E-6 2.0073 1.1837E-2 1.0033 1.1704E-3 1.6164 512×512 2.2979E-6 2.0029 5.9127E-3 1.0014 3.7041E-4 1.6598 1024×1024 5.7649E-7 1.9949 2.9551E-3 1.0006 1.2099E-4 1.6142

Figure 1. 32×32 Uniform rectangular mesh (M1).

Then, the global trace theorem forΔuand the local one in (3.14) forr_h establish

< Δu·n, r_h>_∂Ω≤Ch^−1/2Δu₁ |||(w_L, r_h)|||_B. (3.43) Combining (3.33a)–(3.37) and (3.40a)–(3.43), we have

|||(u_L−u_I, p_h−p_I)|||B ≤Ch^3/2(||u||3+|p|2),

which completes the proof through (3.31) and the definition of the triple norm in (3.7).

4. Numerical results

For the test problem, we chose the stream functionφonΩ= (0,1)² such that φ(x, y) =s(x)s(y), fors(t) = sin (2πt)

t²−t ,

(a) 8×8 initial non-uniform mesh (b) 32×32 non-uniform mesh by bisecting

Figure 2.Non-uniform meshes by recursive bisecting from 8×8 mesh (M2).

(a) 8×8 zigzag mesh (b) 32×32 zigzag mesh

Figure 3. Zigzag meshes (M3).

Table 3. Error table for zigzag meshes (M3).

Mesh ||u−uh||0 ||∇u− ∇uh||0 ||p−ph||0

Value Order Value Order Value Order

16×16 5.0339E-3 2.8134E-1 1.4795E-1

32×32 1.2634E-3 1.9944 1.4068E-1 0.9999 7.5569E-2 0.9692 64×64 3.1487E-4 2.0044 7.0199E-2 1.0029 3.7558E-2 1.0087 128×128 7.8388E-5 2.0061 3.5027E-2 1.0030 1.8444E-2 1.0260 256×256 1.9551E-5 2.0034 1.7489E-2 1.0020 9.0836E-3 1.0218 512×512 4.8839E-6 2.0011 8.7357E-3 1.0014 4.4880E-3 1.0172 1024×1024 1.1883E-6 2.0391 4.3573E-3 1.0035 2.1831E-3 1.0397

and the velocityuand pressurepas

u(x, y) =curl φ, p(x, y) = sin(2πx)

1

25−10 tan²y + 3 10

· Solving (2.13) with the source functionf =−Δu+∇p, we obtained (u_h, p_h)∈X_h×M_h.

The numerical results are shown in Tables1and2for uniform rectangular(M1) and non-uniform bisection(M2) meshes as depicted in Figures1and2, respectively. The non-uniform bisection meshes are obtained by recursive bisecting from the 8×8 initial non-uniform mesh in Figure2a.

We can observe the expected order of convergence for the velocity in (2.14). For the pressure, the supercon- vergence of more order than 3/2 analyzed in Theorem3.6appears in uniform rectangular and even non-uniform bisection meshes.

For the zigzag meshes (M3) as in Figure3, the numerical results in Table3does not show superconvergence in pressure any more.

Acknowledgements. This research was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2009509). This paper also resulted from the Konkuk University research support program.

References

[1] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations.CALCOLO21(1984) 337–344.

[2] I. Babu˘ska, The finite element method with Lagrange multipliers.Numer. Math.20(1973) 179–192.

[3] W. Bai, The quadrilateral ‘Mini’ finite element for the Stokes problem.Comput. Methods Appl. Mech. Eng.143(1997) 41–47.

[4] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers.

RAIRO Anal. Numer. R28(1974) 129–151.

[5] J. Douglas Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems.RAIRO: M2AN33(1999) 747–770.

[6] H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized low order finite element discretization of the Stokes Problem.Math. Comput.80(2011) 697–722.

[7] L.P. Franca, S.P. Oliveira and M. Sarkis, Continuous Q1-Q1 Stokes elements stabilized with non-conforming null edge average velocity functions.Math. Models Meth. Appl. Sci.17(2007) 439–459.

[8] V. Girault and P.A. Raviart,Finite element methods for the Navier-Stokes equations: Theory and Algorithms. Springer-Verlag, New York (1986).

[9] C. Park and D. Sheen,P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J.

Numer. Anal.41(2003) 624–640.

[10] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element.Numer. Methods Partial Differ. Eq.8(1992) 97–111.