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    Nonlinear Analysis: Real World Applications 14 (2013) 11711181

    Contents lists available at SciVerse ScienceDirect

    Nonlinear Analysis: Real World Applications

    journal homepage: www.elsevier.com/locate/nonrwa

    Two-level defect-correction locally stabilized finite element method forthe steady NavierStokes equations

    Pengzhan Huang, Xinlong Feng , Haiyan SuCollege of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, PR China

    a r t i c l e i n f o

    Article history:

    Received 7 June 2011

    Accepted 15 September 2012

    Keywords:

    Defect-correction

    Two-level strategy

    NavierStokes equations

    Local Gauss integration

    Error estimate

    Finite element method

    a b s t r a c t

    This paper proposes a two-level defect-correction stabilized finite element method for the

    steady NavierStokes equations based on local Gauss integration. The method combines

    the two-level strategy with the defect-correction method under the assumption of the

    uniqueness condition. Both the simplified and the Newton scheme are proposed and

    analyzed. Moreover, the numerical illustrations agree completely with the theoretical

    expectations.

    2012 Elsevier Ltd. All rights reserved.

    1. Introduction

    The incompressible NavierStokes equations model Newtonian fluids, such as air flow at low speed, weather, bloodflow and water flow. The numerous works are devoted to this system, and the finite element methods and finite volumemethods are two classes of the most successful methods for the simulation. However, solving the NavierStokes equationsnumerically is still an important but challenging problem. The discretization of the NavierStokes equations by finiteelement methods andfinite volume methods maygenerallybring out two shortcomings: the violationof the discrete infsupcondition and spurious oscillations due to the domination of the nonlinear convection term.

    To increase the efficiency of a numerical method, an alternative idea is the two-level method. The basic idea of two-leveldiscretization strategy for solving nonlinear partial differential equations is to compute an initial approximation on a verycoarse mesh, then to solve a linear system on a fine mesh. Some details of the two-level method can be found in the worksof Xu [1,2], Layton [3], Layton and Lenferink [4], Layton and Tobiska [5], Zhang and He [6], Ervin et al. [7], He and Li [8], Heand Wang [9], Li [10], Shang [11] and Huang et al. [12].

    For solving nonlinear steady-state problems, the defect-correction method is an iterative improvement technique

    (see [13] for a survey of the technique). The basic idea of the defect-correction method is to solve artificial viscositynonlinear equations in the defect step and correct the residual by a linearized problem in the correction step for a few steps(e.g., see [14,15]). This method can increase the accuracy of the solution without refining the grid, so it has been successivelystudied in many references. Layton [16] initially investigated the defect-correction method for the incompressibleNavierStokes equations with high Reynolds number. In [17], Axelsson and Layton applied the defect-correction methodfor convectiondiffusion problems. Subsequently, Layton, Lee and Peterson also provided some further studies based on thedefect-correction method for NavierStokes equations (see [15] and the references therein). Recently, Kaya et al. [14] have

    This work is in part supported by the NSF of China (Nos. 11271313, 61163027, and 10901131), the China Postdoctoral Science Foundation (Nos.

    201104702, and 2012M512056), the Key Project of Chinese Ministry of Education (No. 212197) and the NSF of Xinjiang Province (No. 2010211B04). Corresponding author. Tel.: +86 9918582482.

    E-mail addresses: [email protected] (P. Huang), [email protected], [email protected] (X. Feng), [email protected] (H. Su).

    1468-1218/$ see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2012.09.008

    http://dx.doi.org/10.1016/j.nonrwa.2012.09.008http://www.elsevier.com/locate/nonrwahttp://www.elsevier.com/locate/nonrwamailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.nonrwa.2012.09.008http://dx.doi.org/10.1016/j.nonrwa.2012.09.008mailto:[email protected]:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/nonrwahttp://www.elsevier.com/locate/nonrwahttp://dx.doi.org/10.1016/j.nonrwa.2012.09.008
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    considered the synthesis of a subgrid stabilization method with defect-correction method for the stationary NavierStokesequations. Wang [18] presented a new defect-correction method for the NavierStokes equations. Besides, Liu and Hou [19]studied a two-level defect-correction method for the steady-state NavierStokes equations based on the spectral method.

    In the analysis and practice of employing finite element methods in solving the NavierStokes equations, the infsupcondition has played an important role because it ensures a stability and accuracy of the underlying numerical schemes.This condition is something that enforces a certain correlation between two finite element spaces so that they both have therequired properties when employed for the NavierStokes equations. However, due to the computational convenience and

    efficiency in practice, some mixed finite element pairs which do not satisfy the infsup condition are also popular. Hence,much attention has been paid to the study of the stabilized method for the NavierStokes and Stokes problems [2024].Recent studies have focused on stabilization of the lowest equal-order finite element pair P1P1 (linear functions) usingthe projection of the pressure onto the piecewise constant space [25]. This stabilization technique based on the two localGauss integrations technique does not require approximation of derivatives, specification of mesh-dependent parameters,and always leads to symmetric problems. In addition, no edge-based data structures and assembly are required. Therefore,this method is gaining more and more popularity in the computational fluid dynamics.

    It is well known that a few linearization methods are efficient for solving the stationary NavierStokes equations undersome strong uniqueness condition. Recently, He and Li [26] and Xu and He [27] have presented the three iterative methodsbased on the finite element discretization for the stationary NavierStokes equations, where the Newton and Oseen iterativemethods are the semi-implicit scheme for the nonlinear term, the Stokes iterative is the implicit/explicit scheme. Moreover,He and Li [28] and He [29] have given the Euler implicit/explicit scheme based on the mixed finite element to solve thestationary NavierStokes equations. The time iterative scheme is based on the Euler semi-implicit scheme which is implicit

    for the linear terms and the explicit for the nonlinear term.In this paper, we will combine the defect-correction method with the two-level discretization for solving the two-dimensional steady NavierStokes problem based on the locally stabilized finite element method with Gaussian quadraturerules. Both the simplified and the Newton scheme are proposed and analyzed. The remainder of this paper is organized asfollows. In Section 2, we introduce the studied equations, the notations and some well-known results used throughout thispaper. A stabilized finite element strategy is recalled in Section 3. The simplified and Newton two-level defect-correctionmethods are given in Sections 4 and 5, respectively. Then in Section 6, numerical experiments are shown to verify thetheoretical results completely. Finally, in Section 7, we conclude with a summary and possible extensions.

    2. Preliminaries

    Let be a bounded, convex and open subset ofR2 with a Lipschitz continuous boundary . We consider the steadyNavierStokes problem

    u + (u )u + p = f in ,

    div u = 0 in , (1)

    u = 0 on ,

    where u = (u1(x), u2(x)) represents the velocity vector, p = p(x) the pressure, f = (f1(x),f2(x)) the prescribed body forceand > 0 the viscosity.

    We shall introduce the following Hilbert spaces:

    X = H10 ()2, Y = L2()2, M = L20() =

    q L2() :

    q dx = 0

    .

    The spaces L2()m, m = 1, 2, are equipped with the L2-scalar product (, ) and L2-norm 0. The spaceX is endowed withthe usual scalar product (u, v) and the norm u0. Standard definitions are used for the Sobolev spaces W

    m,p(), with

    the norm m,p, m,p 0. We will write Hm

    () for Wm,2

    () and m for m,2. Next, let the closed subset V ofX begiven by

    V = {v X : div v = 0},

    and denote by H the closed subset ofY, i.e.,

    H = {v Y : div v = 0, v n| = 0}.

    We denote the Stokes operator byA = P, where Pisthe L2-orthogonal projection ofY onto Handset D(A) = H2()2 V.Moreover, we need a further assumption on provided in [30,26].

    We make a regularity assumption on the Stokes problem.(A). Assume that is smooth so that the unique solution (v, q) X M of the steady Stokes problem

    v + q = g in ,

    div v = 0 in ,

    v = 0 on ,

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    for any prescribed g Y exists and satisfies

    v2 + q1 c0g0, (2)

    where c0 > 0 is a constant depending on . Subsequently, c (with or without a subscript) will denote a positive constantwhich may stand for different values at its different occurrences.

    We remark that the validity of assumption (A) is known [31] if is ofC2 or if is a two-dimensional convex polygon.From the assumption (A), it is well known [32] that

    v2 c1Av0, v0 c2Av0, v D(A), (3)

    v0 c2v0, v X. (4)

    We define the continuous bilinear forms a(, ), d(, ) and a generalized bilinear form B((, ); (, )) onXX,X M and(X M) (X M), respectively, by

    a(u, v) = (u, v), u, v X,

    d(v, q) = (q, divv), v X, q M,

    and

    B((u,p); (v, q)) = a(u, v) d(v,p) + d(u, q), (u,p),(v, q) X M,

    and a trilinear form b(; , ) on X X X by

    b(u; v,w) = ((u )v,w) +1

    2((divu)v, w)

    =1

    2((u )v,w)

    1

    2((u )w, v), u, v , w X.

    With the above notations, the variational formulation of problem (1) reads as follows: find (u,p) X M such that forall (v, q) X M,

    B((u,p); (v, q)) + b(u; u, v) = (f,v). (5)

    It is easy to verify that b and B satisfy the following important properties [8,26]:

    b(u; v,w) = b(u; w, v),

    |b(u; v,w)| + |b(v; u, w)| + |b(w; u, v)| c3u0Av0w0, (6)|b(u; v,w)| Nu0v0w0, (7)

    and

    |B((u,p); (v, q))| c(u0 + p0)(v0 + q0), (u,p),(v, q) X M,

    sup(v,q)XM

    |B((u,p); (v, q))|

    v0 + q0 0(u0 + p0), (u,p) X M,

    where N = supu,v,wX|b(u,v,w)|

    u0v0w0, and 0 is a positive constant that depends only on .

    The following existence and uniqueness of the solution of(5) are classical results [26].

    Theorem 2.1. Given f X, there exists at least a solution pair (u,p) X M which satisfies (5) and

    u0 1f1, f1 = sup

    vX

    (f, v)

    v0. (8)

    And if and f satisfy the following uniqueness condition:

    1 2Nf1 > 0, (9)

    then the solution pair (u,p) of problem (5) is unique.

    Furthermore, we have the following regularity result [26,9].

    Theorem 2.2. Assume that (A) holds and f Y . Then the solution pair (u,p) D(A) (H1() M) of problem (5) satisfiesthe following regularity

    Au0 + p1 cf0. (10)

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    3. A stabilized finite element method

    Let H be a real positive parameter tending to 0. The finite element subspace XH MH ofX M is characterized by KH, apartitioning of into triangles K with the mesh size H, assumed to be uniformly regular in the usual sense. Moreover,the fine mesh partition Kh can be thought of as generated from KH by a mesh refinement process. Also, we introducefinite element space Xh Mh based on Kh. Furthermore, let K be a finite element partition with mesh size . Here, = h or H and H h.

    Then we define

    X =

    u C0()2 X : u|K P1(K)2, K K

    ,

    M =

    q C0() M : q|K P1(K), K K

    ,

    where P1(K) represents the space of linear functions on K.

    Note that the lowest equal-order pair X M does not satisfy the discrete infsup condition

    supvX

    d(v, q)

    v0 1q0, q M,

    where the constant 1 > 0 is independent of. In order to fulfill this condition, a stabilized generalized bilinear term isused [10,33]:

    B((u,p); (v, q)) = B((u,p); (v, q)) + G(p, q),

    where G(p, q) can be defined by

    G(p, q) =

    KK

    K,2

    pq dx

    K,1

    pq dx

    , p, q M,

    where

    K,ig(x)dx indicates a local Gauss integral over K that is exact for polynomials of degree i, i = 1, 2. In particular, the

    trial function p M must be projected to piecewise constant space W defined below when i = 1 for any q M. Let

    be a L2-projection operator, which is defined by

    (p, q) = (p, q), p L2(), q W. (11)

    Here W

    L2() denotes the piecewise constant space associated with the triangulation K

    . The following properties ofthe projection operator can be proved [33]:

    p0 c5p0, p L2(), (12)

    p p0 c6p1, p H1(). (13)

    As a result of(11), the bilinear form G(, ) can be expressed as

    G(p, q) = (p p, q q), p, q M. (14)

    Now, the corresponding discrete variational formulation of(5) for the NavierStokes equations is recast: Find (u,p) X M such that

    B((u,p); (v, q)) + b(u; u, v) = (f, v) (v, q) X M. (15)

    We define the subspace V ofX given by

    V = {v X : d(v, q) = 0, q M}.

    Next, let P : Y V denote the L2-orthogonal projection defined by

    (Pu, v) = (u, v), u Y, v V.

    Moreover, a discrete analogue A = P of the Stokes operator A is defined through the condition that (u, v) =(u, v) for all u, v X. From [26], we know that the trilinear form b satisfies the following estimate:

    |b(u; v, w)| c4| log |1/2u0v0w0, (16)

    for all u, v Vh, w Y.

    The following theorem establishes the weak coercivity of the bilinear form B((, ); (, )) for the finite element pairX M .

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    Theorem 3.1 ([10,33]). For all (u,p),(v, q) X M, the bilinear form B((, ); (, )) satisfies the continuity property

    |B((u,p); (v, q))| c7(u0 + p0)(v0 + q0), (17)

    and the weak coercivity property

    sup(v,q)XM

    |B((u,p); (v, q))|

    v0 + q0 2(u0 + p0), (18)

    where 2 > 0 is independent of .

    Theorem 3.2 ([10]). Let the exact solution (u,p) be in D(A) (H1() M). Then, (u,p) satisfies the following stability anderror estimate:

    u0 f1, (19)

    u u0 + ((u u)0 + p p0) c82(u2 + p1). (20)

    4. Simplified two-level defect-correction stabilized finite element approximation

    The simplified two-level defect-correction method based on local Gauss integration we study is as follows.

    Step I. Solve a defectNavierStokesproblem on coarse mesh, i.e., find (uH,pH) XHMH such thatfor all (v, q) XHMH

    BH((uH,pH); (v, q)) +H

    a(uH, v) + b(uH; uH, v) = (f,v), (21)

    where > 0 is an artificial viscosity parameter which is used as a stabilizing factor in this algorithm.

    Step II. Solve a correction NavierStokes problem on fine mesh, i.e., find (uh,ph) XhMh such that for all(v, q) XhMh

    Bh((uh,ph); (v, q)) +

    h

    a(uh, v) + b(uH; uH, v) = (f, v) +

    h

    a(uH,v). (22)

    A similar argument to that used in [12] yields the following theorem which establishes the stability of the above scheme.

    Theorem 4.1. Under the assumptions of Theorem 2.1, (uH,pH) and (uh,ph) defined by scheme (21) and (22), respectively, satisfy

    uH0

    1

    H +

    f1, AHuH0 c

    1

    H +

    f0, (23)

    uh0

    1

    h + +

    1

    H +

    f1. (24)

    In order to derive error estimates, we need the Galerkin projection (R, Q) : X M X M with respect to theStokes problem as follows:

    B((R(u,p), Q(u,p)); (v, q)) = B((u,p); (v, q)), (v, q) X M, (25)

    for each (u,p) X M. The following property is classical [25,34,5,10,6]

    R(u,p) u0 + ((R(u,p) u)0 + Q(u,p) p0) c92(u2 + p1), (26)

    for all (u,p) D(A) (H1() M).

    Theorem 4.2. Under the assumptions of Theorem 2.1, (uH,pH) defined by scheme (21) satisfies

    u uH0 + H((u uH)0 + p pH0) c(H2 + H). (27)

    Proof. Setting (eH, H) = (RH(u,p) uH, QH(u,p) pH), subtracting (21) from (5) and using (25), we have

    BH((eH, H); (v, q)) + b(u uH; u, v) + b(uH; u uH, v) 1Ha(uH, v) = 0. (28)

    Taking (v, q) = (eH, H) in (28), we arrive at

    eH20 + G(H, H) + b(eH; u, eH) |b(u RH(u,p); u, eH)|

    + |b(uH; u RH(u,p), eH)| + |1Ha(uH, eH)|. (29)

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    Next, we deduce from (6)(8), (14), (23) and (26) that

    eH20 + G(H, H) |b(eH; u, eH)| eH

    20 Nu0eH

    20

    (1 N2f1)eH20, (30)

    and

    |b(u RH(u,p); u, eH)| + |b(uH; u RH(u,p), eH)| + |1Ha(uH, eH)|

    c3(Au0 + AHuH0)eH0u RH(u,p)0 + HuH0eH0

    (cH2(u2 + p1) + H(H + )1f1)eH0. (31)

    Combining this inequality with (10) and (30) and uniqueness condition (9) yields,

    eH0 c(H2 + H). (32)

    Finally, it follows from (4), (18), (26) and (28)that

    u uH0 eH0 + u RH(u,p)0 c2eH0 + cH2 c(H2 + H),

    (u uH)0 eH0 + (u RH(u,p))0 c(H + H),

    p pH0 p QH(u,p)0 + H0 cH + 12 (N(eH0

    + (u RH(u,p))0)(u0 + uH0) + HuH0)

    c(H + H). (33)Thus, (27) follows.

    Theorem 4.3. Under the assumptions of Theorem 2.1, the simplified two-level defect-correction stabilized finite element solution(uh,ph) defined by scheme (22) satisfies

    (u uh)0 + p ph0 c(h + H

    2 + (H + )). (34)

    Proof. We derive from (5), (22) and (25) that for all (v, q) Xh Mh

    Bh((eh, h); (v, q)) + b(u uH; u, v) + b(u; u uH, v) + b(uH u; u uH, v)

    + 1h(a(uH, v) a(uh,v)) = 0, (35)

    where eh = Rh(u,p) uh and h = Qh(u,p) p

    h.

    Using (6), (7), (18) and (27), it follows from (35) thateh0 + h0

    12

    2c3Au0u uH0 + N(u uH)

    20 + h(uH0 + u

    h0)

    c(H2 + H + h + 2). (36)

    Thanks to (26) and (36), it follows that

    (u uh)0 + p ph0 (u Rh(u,p))0 + p Qh(u,p)0 + eh0 + h0

    c9h(u2 + p1) + c(2 + H2 + H + h)

    c(h + 2 + H2 + H + h). (37)

    Hence, we finish the proof of the theorem.

    Remark 4.1. Setting = 0 in (21) and (22), we get the general simplified two-level scheme. By Theorems 4.2 and 4.3, we

    obtain

    u uH0 + H((u uH)0 + p pH0) cH2,

    and

    (u uh)0 + p ph0 c(h + H

    2).

    These results are given by He and Li [8] and Li [10]. Hence, our work can be looked as a generalization of theirs.

    Remark 4.2. From Theorem 3.2, for the usual stabilized finite element solution (uh,ph), which involves solving one largeNavierStokes problem on a fine mesh with mesh size h, we have the following error estimate:

    (u uh)0 + p ph0 ch.

    Furthermore, if we choose and H such that = O(H) and h = O(H2) for the simplified two-level defect-correction

    stabilized finite element solution, then we get the convergence rate of same order as the usual stabilized finite elementmethod from Theorem 4.3. However, our method is more efficient than the one-level scheme.

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    5. Newton two-level defect-correction stabilized finite element approximation

    The Newton two-level defect-correction method based on local Gauss integration we study is as follows.

    Step I. Solve a defect NavierStokes problem on a coarse mesh, i.e., find (uH,pH) XH MH by (21).

    Step II. Solve a correction NavierStokes problem on a fine mesh, i.e., find (uh,ph) Xh Mh such that for all(v, q) Xh Mh

    Bh((uh,ph); (v, q)) +h

    a(uh, v) + b(uH; uh, v) + b(uh; uH, v) = (f, v) +

    h

    a(uH, v) + b(uH; uH,v). (38)

    Next, we will study the stability of the Newton two-level defect-correction stabilized finite element. Also, a similarargument to that used in [12] yields the following theorem which establishes the stability of the above scheme.

    Theorem 5.1. Assume that f X. (uh,ph) defined by scheme (38) satisfies

    uh0

    1

    H+ hh++ 1

    h+ HH+

    f1. (39)

    Theorem 5.2. Under the assumptions of Theorem 5.1, the Newton two-level defect-correction stabilized finite element solution(uh,ph) defined by scheme (38) satisfies

    (u uh)0 + p ph0 c(h + | log h|

    1/2(H3 + 2H + H2)). (40)

    Proof. Subtracting (38) from (5), using (25) and taking (v, q) = (eh, h), we get

    Bh((eh, h); (eh, h)) + b(u Rh(u,p); u, eh) + b(Rh(u,p); u Rh(u,p), eh) + b(eh; uH, eh)

    b(Rh(u,p) uH; eh, Rh(u,p) uH) + 1h(a(uH, eh) a(u

    h, eh)) = 0, (41)

    where eh = Rh(u,p) uh and h = Qh(u,p) p

    h.

    Using (7), (14), (16), (23) and (27), it follows from (41) that

    eh0 N(u Rh(u,p))0(u0 + Rh(u,p)0) + Nf1(H + )1eh0

    + c4| log h|1/2(Rh(u,p) uH)0Rh(u,p) uH0 + h(uH0 + u

    h0). (42)

    Thanks to (23), (26), (27) and (39) and the stability condition, it follows

    eh0 ch + c| log h|1/2(H3 + H2 + 2H). (43)

    Applying (18) and (41), we have

    h0 12 (N(u Rh(u,p))0(u0 + Rh(u,p)0) + Neh0uH0

    + c4| log h|1/2(Rh(u,p) uH)0Rh(u,p) uH0 + h(uH0 + u

    h0))

    c(h + | log h|1/2(H3 + 2H + H2)). (44)

    Combining this inequality with (26) and (43), we finish the proof of the theorem.

    Remark 5.1. If we choose and Hsuch that = O(H) and h = O(| log h|1/2H3) for the Newton two-level defect-correctionstabilized finite element solution, then we get the convergence rate of the same order as the usual stabilized finite elementmethod from Theorem 5.2. However, our method is more efficient than the one-level scheme.

    Remark 5.2. Since it is valid that

    1

    H +