# Fully Discrete Finite Element Methods for Two-Dimensional Bingham Flows

Authors: Cheng Fang; Yuan Li
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper presents fully discrete stabilized finite element methods for two-dimensional Bingham fluid flow based on the method of regularization. Motivated by the Brezzi-Pitkäranta stabilized finite element method, the equal-order piecewise linear finite element approximation is used for both the velocity and the pressure. Based on Euler semi-implicit scheme, a fully discrete scheme is introduced. It is shown that the proposed fully discrete stabilized finite element scheme results in theh1/2 error order for the velocity in the discrete norms corresponding to L2(0,T;H1(Ω)2)∩L∞(0,T;L2(Ω)2).


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## Body

## 1. Introduction

Bingham fluid is a rigid viscoplastic fluid which is a kind of non-Newtonian fluid. Many important substances are well approximated as Bingham fluid flow, such as the drilling muds, the cement slurries, and the coal slurries. A well understanding of Bingham fluid flow also is important to technology, engineering, and industrial applications, such as the heat transfer, the heavy crude oils, and underground natural resource recovery.The time-dependent Bingham flow problem is defined by(1)∂u∂t-div⁡σ+u·∇u=f,(2)div⁡u=0,for x∈Ω and t∈[0,T], where Ω⊂R2 is a bounded and convex domain and f denotes the density of external force. σ=-pI+gD(u)/|D(u)|+2μD(u) with D(u)=1/2[∇u+(∇u)T] for any u∈H1(Ω)2. μ>0 is the viscosity coefficient and g(t)∈L2(Ω) with g>0 in Ω. |D(u)| is the Frobenius norm of the tensor D(u) and is defined by (3)Du=∑1≤i,j≤2Diju21/2.The above equation should be completed by the appropriate initial and boundary conditions. For the sake of simplicity, we consider the following the initial and boundary conditions: (4)ux,t=0,forx∈∂Ω,t∈0,T,(5)ux,0=u0x,forx∈Ω.The numerical method for Bingham fluid flow is still a challenging task since Bingham fluid flow takes the form of nonlinear and nondifferentiable variational inequality problem. Generally, there exist two main numerical techniques: one based on the regularized method [1] and the other based on the augmented Lagrangian method [2]. Because the regularized method makes variational inequality problems into the equations, it is a powerful tool to deal with the variational inequality problems, such as the obstacle problem [3, 4], Navier-Stokes problems with subdifferential boundary conditions [5, 6], and etc.Although there have much works on the numerical simulation for steady and time-dependent Bingham fluid flow, such as [7–14] and references cited therein, the error estimates for the finite element approximation have not been studied as much [15, 16]. Roughly speaking, the convergence order for the velocity approximation is O(h1/2). Moreover, it appears to be difficult to improve this order for the general domain Ω⊂R2, which can be improved only for some special cases. For example, if Ω is a disc, Glowinski [1] improved the error estimate to O(hln⁡h1/2) in terms of the explicit formulation of the velocity. If the flow is axisymmetric, the optimal error estimate can be derived by Zhang [16] in terms of the accurate estimate to the nondifferentiable term. On the other hand, the solution of Bingham fluid flow is of the poor regularity; thus it does not lead us to employ the high-order finite element approximation, such as P2-P1 finite element for the approximation of the velocity and the pressure. In the development of the finite element approximation methods for the incompressible fluids, some kinds of lower-order mixed finite elements which do not satisfy the discrete inf-sup condition are work well by introducing a stabilized term in the discrete variational formulation. The first stabilized finite element method for Stokes problem is proposed by Brezzi-Pitkäranta [17] in the context of the P1-P1 triangular element. Subsequently, Hughes and Franca [18] derive a discrete Stokes formulation by introducing a pressure jump operator such that the stable condition holds for all mixed finite elements. In particular, the equal-order P1-P1 triangular element is very attractive and important in the scientific computation because it is computationally convenient. Therefore, many kinds of stabilized finite element are proposed, such as the locally stabilized method [19], the pressure projection stabilized method [20, 21], and the pressure stabilized method [22, 23]. Here, we will apply the Brezzi-Pitkäranta stabilized finite element method which has been used by Latché and Vola [15] to the steady Bingham fluid flow. Based on this stabilized methods, an augmented Lagrangian method has been studied by Zhang [14] to steady Bingham fluid flow in a lid-driven square cavity.In this paper, we extend Latché and Vola’s work to the two-dimensional time-dependent Bingham fluid flow and study the fully discrete stabilized finite element scheme by using the method of regularization. The time discretization is based on the backward Euler scheme with the time stepΔt. Denote the finite element spaces by Vh and Mh related to the Brezzi-Pitkäranta stabilized finite element for the approximation of the velocity and the pressure. Denote the fully discrete finite element approximation solution by (uεhn,pεhn)∈Vh×Mh. The main result derived in this paper is to prove the following error estimate for the fully discrete scheme: (6)utn-uεhn2+Δt∑n=1Nutn-uεhnV2≤cε+h+Δtε2/32,where N=T/Δt, tn=nΔt∈(0,T],(n=1,⋯,N), ε>0 is the regularized parameter, and c>0 is some positive constant independent of ε,h, and Δt.The rest of this paper is organized as follows. In Section2, we describe some preliminary knowledge about Bingham fluid flow and its regularized problems. In Section 3, we give the semidiscrete stabilized finite element approximation for the regularized problem. In Section 4, we present a fully discrete finite element approximation using Euler semi-implicit scheme and prove the error estimates (6).Throughout this paper, we use the symbolc to denote a positive constant whose value may change from place to place but that remains independent of h, Δt, and ε and that maybe depends on μ, Ω, and the norms of u, p, f, and g.


## 2. Bingham Fluid Flow and Regularized Problem

In what follows,Ω⊂R2 denotes a bounded domain, and its boundary is denoted by ∂Ω. We employ the standard notation Hl(Ω) and ·l, l≥0, for the Sobolev spaces of all functions having square integrable derivatives up to order l in Ω and the standard Sobolev norm. When l=0, we will write L2(Ω) and · instead of H0(Ω) and ·0, respectively. Let X be a Banach space. Denote by X′ the dual space of X and by ·,·X the dual product between X and X′. Denote by Bochner spaces Lp(0,T;X),1≤p<+∞ the linear spaces of measurable functions from the interval [0,T] into X such that (7)∫0TutXpdt<+∞.If p=+∞, the functions in L∞(0,T;X) are required to satisfy (8)supt∈0,TutX<+∞.For the mathematical setting, the function spaces below are frequently used in this paper: (9)V=H01Ω2,Vσ=v∈V;div⁡v=0inΩ,H=v∈L2Ω2;div⁡v=0,v·n=0on∂Ω,M=L02Ω=q∈L2Ω;∫Ωqdx=0.The norm in V is defined by (10)vV=∫Ω∇v2dx1/2.We introduce the bilinear forms (11)au,v=μ∫ΩDu:Dvdx,∀u,v∈V,dv,q=∫Ωqdiv⁡vdx,∀v∈V,q∈M,and the trilinear form (12)bu,v,w=∫Ωu·∇v·wdx+12∫Ωdiv⁡uv·wdx=12∫Ωu·∇v·wdx-12∫Ωu·∇w·vdx,∀u,v,w∈V.Then (13)bu,v,w=-bu,w,v,∀u,v,w∈V.Moreover, there exists some positive constant c such that (14)bu,v,w≤cuVvVwV.It follows from Korn’s inequality that (a(v,v))1/2 is equivalent to the norm in V; i.e., there exist two constants κ1>0 and κ2>0 such that (15)κ1μvV2≤av,v≤κ2μvV2,∀v∈V.Supposef∈L2(0,T;L2(Ω)2), g∈L2(0,T;L2(Ω)) with g(t)>0 a.e. in Ω, the initial condition u0∈Vσ. The variational formulation of Bingham fluid flow (1), (2), (4), and (5) takes the following form of the variational inequality problem of the second kind: find (u(t),p(t))∈V×M such that (16)ut,v-u+au,v-u+bu,u,v-u-dv-u,p+jv-ju≥f,v-u,du,q=0,for any (v,q)∈V×M and t∈(0,T), where j(η)=∫Ω‍g|∇η|dx for any η∈H1(Ω)2. An alternate variational formulation of (1), (2), (4), and (5) reads as follows: (17)ut,v-u+au,v-u+bu,u,v-u+jv-ju≥f,v-u,for any v∈Vσ and t∈(0,T).Now, we introduce Stokes operatorA [24, 25]. Denote by P the orthogonal projection P:L2(Ω)2→H. Stokes operator A is defined by (18)A:DA⊂H⟶H,A=-PΔ,DA=H2Ω2∩Vσ.Its inverse A-1 is self-adjoint and compact from H to H.Since Bingham fluid flow takes the form of nondifferentiable variational inequality problem due toj(η), it is natural to approximate j(η) by a family of functions jε(η) which is convex and differentiable, where ε>0 is small parameter. Take jε(η) in the form (19)jεη=∫Ωgε2+∇η2dx,∀η∈H1Ω2,then it satisfies (20)jεη-jη≤ε∫Ωgdx,∀η∈H1Ω2,and (21)grad⁡jεη,ξ=limh→01hjεη+hξ-jεη=∫Ωg∇η·∇ξε2+∇η2dx,∀ξ,η∈H1Ω2.The regularized problem of (16) is (22)uεt,v-uε+auε,v-uε+buε,uε,v-uε-dv-uε,pε+jεv-jεuε≥f,v-uε,∀v∈V,duε,q=0,∀q∈M,with the initial condition uε(x,0)=u0(x) for x∈Ω. Moreover, (17) is equivalent to (23)uεt,v-uε+auε,v-uε+buε,uε,v-uε+jεv-jεuε≥f,v-uε,∀v∈Vσ.Sincejε(η) is differentiable, according to (21) and after a simple calculation, the regularized problems (22) and (23) also take the following forms: (24)uεt,v+auε,v+buε,uε,v-dv,pε+βuε,v=f,v,∀v∈V,duε,q=0,∀q∈M,and (25)uεt,v+auε,v+buε,uε,v+βuε,v=f,v,∀v∈Vσ.where (26)βη,ξ=∫Ωg∇η·∇ξε2+∇η2dx,∀ξ,η∈H1Ω2.Moreover, β· satisfies the monotone property: (27)βξ-βη,ξ-η≥0,∀ξ,η∈H1Ω2.Settingv=uε in (25), using (13) and (β(uε),uε)≥0, we get (28)12ddtuεt2+μκ1uεtV2=f,uε≤μκ12uεtV2+12μκ1ft2.Then (29)ddtuεt2+μκ1uεtV2≤1μκ1ft2.Integrating the above inequality from 0 to s<T and from 0 to T with respect to t, respectively, we conclude (30)supt∈0,Tuεt2+μκ1∫0TuεtV2dt≤u02+1μκ1∫0Tft2dt.The following theorem gives the error betweenu and uε with respect to ε.Theorem 1.
Suppose thatu∈L∞(0,T;L2(Ω)2)∩L2(0,T;Vσ) and uε∈L∞(0,T;L2(Ω)2)∩L2(0,T;Vσ) are the solutions of (17) and (23), respectively, then there holds (31)u-uεL∞0,T;L2Ω22+u-uεL20,T;Vσ2≤cε.Proof.
Takingv=uε in (17) and v=u in (23) and adding them, we obtain (32)12ddtu-uε2+μκ1u-uεV2≤bu,u,uε-u-buε,uε,uε-u+juε-jεuε+jεu-ju.By using (13) and Young’s inequality, we have (33)bu,u,uε-u-buε,uε,uε-u=bu,u-uε,uε-u+bu-uε,uε,uε-u=bu-uε,uε,uε-u≤cuεVu-uεVu-uε≤μκ12u-uεV2+cu-uε2uεV2.From (20) and Hölder’s inequality, we have (34)juε-jεuε+jεu-ju≤2ε∫Ωgdx≤2εΩ1/2g.With the above two estimates, (32) reduces to (35)ddtu-uε2+μκ1u-uεV2≤cuεV2u-uε2+cεg.Therefore, the error (31) can be derived from Gronwall’s inequality, u(0)-uε(0)=0, and (30).


## 3. Semidiscrete Stabilized Finite Element Approximation

In this section, we will discuss the semidiscrete Brezzi-Pitkäranta stabilized finite element approximation for the regularized problem (22). To verify the existence and uniqueness of the solution to the discrete problem, the weakly coercive property for a generalized bilinear form is established.LetTh be a family of quasi-uniform triangular partition of Ω. The corresponding ordered triangles are denoted by K1,K2,⋯,Kn. Let hi=diam⁡(Ki), i=1,⋯,n, and h=max⁡{h1,h2,⋯,hn}. For every K∈Th, let Pr(K) denote the space of the polynomials on K of degree at most r. For simplicity, we consider the conforming finite element spaces Vh and Mh defined by (36)Vh=vh∈CΩ¯2,vhK∈P1K2,∀K∈Th∩V,Mh=qh∈CΩ¯,qhK∈P1K,∀K∈Th∩M.Denote byIh and Jh the L2 orthogonal projection operators onto Vh and Mh, respectively, which satisfy (37)v-Ihv+hv-IhvV≤chivi,∀v∈H2Ω2∩V,i=1,2,(38)q-Jhq≤chq1,∀q∈H1Ω∩M.The following inverse inequalities will be used in this section: (39)∇qhKi≤chi-1qhKi,∇qh≤ch-1qh.The semidiscrete Brezzi-Pitkäranta stabilized finite element approximation of (22) reads as follows: find (uεh(t),pεh(t))∈Vh×Mh such that (40)uεht,vh-uεh+auεh,vh-uεh+buεh,uεh,vh-uεh-dvh-uεh,pεh+jεvh-jεuεh≥f,vh-uεh,duεh,qh+αChpεh,qh=0,for any (vh,qh)∈Vh×Mh, where α>0 is the stable parameter, and the stable term Ch(·,·) is defined by (41)Chph,qh=∑i=1nhi2∫Ki∇ph·∇qhdx,∀ph,qh∈Mh.The discrete initial condition uεh0=u0h=Ihu0. Moreover, an alternative to (40) is (42)uεht,vh+auεh,vh+buεh,uεh,vh-dvh,pεh+βuεh,vh=f,vh,∀vh∈Vhduεh,qh+αChpεh,qh=0,∀qh∈Mh.Define a mesh-dependent norm[·]h on Mh by (43)qhh2=Chqh,qh,∀qh∈Mh.Then, it holds that Ch(ph,qh)≤[ph]h[qh]h for all ph, qh∈Mh and (44)dv,qh≤chvqhh,∀v∈V,qh∈Mh,which is derived by Latché and Vola [15]. Moreover, Ch(p,q) also is defined for any couple of functions p, q∈H1(Ω) and satisfies (45)qh≤chq1,∀q∈H1Ω.Introduce the generalized bilinear formsB(·,·;·,·) and Bh(·,·;·,·) on (V,M)×(V,M) defined by, respectively, (46)Bw,r;v,q=aw,v-dv,r+dw,q,∀w,r,v,q∈V×M,Bhw,r;v,q=Bw,r;v,q+αChr,q,∀w,r,v,q∈V×M.Then the solution (uεh,pεh) of (40) or (42) satisfies (47)uεht,vh-uεh+Bhuεh,pεh;vh-uεh,qh-pεh+buεh,uεh,vh-uεh+jεvh-jεuεh≥f,vh-uεh,∀vh,qh∈Vh,Mhor (48)uεht,vh+Bhuεh,pεh;vh,qh+buεh,uεh,vh+βuεh,vh=f,vh,∀vh,qh∈Vh,Mh.First, we establish the following stable theorem to verify the existence and uniqueness of the solution to the discrete problem.Theorem 2.
For anyα>0, there exist two positive constants β1 and β2 independent of h such that Bh(·,·;·,·) on (Vh,Mh)×(Vh,Mh) satisfies (49)Bhwh,rh;vh,qh≤β1whV+rhvhV+qh,and (50)β2whV+rh≤supvh,qh∈Vh×MhBhwh,rh;vh,qhvhV+qh.Proof.
A similar results for (49)-(50) have been shown in [26]. For completeness, we supply its full proof. In terms of (39), it is easily to check that (51)Bhwh,rh;vh,qh=awh,vh-dvh,rh+dwh,qh+αChrh,qh≤μκ2whVvhV+vhVrh+whVqh+α∑i=1nhi2∇rhKi21/2∑i=1nhi2∇qhKi21/2≤μκ2whVvhV+vhVrh+whVqh+cαrh·qh≤β1whV+rhvhV+qh,which completes the proof of (49). Since the operator div is an isomorphism from V to M, then for every rh∈Mh⊂M, there exists a unique ϕ∈V such that (52)div⁡ϕ=rh,ϕV≤C1rh,dϕ,rh=rh2,where C1>0 is independent of h. Setting ϕh=Ihϕ and (vh,qh)=(-βϕh,0), where β>0 is a positive constant determined latter, and using Hölder’s inequality, Young’s inequality, (44), we obtain (53)Bhwh,rh;-βϕh,0=-βawh,ϕh+βdϕh,rh=-βawh,ϕh+βdϕh-ϕ,rh+βdϕ,rh≥-βμκ2whVϕhV-cβhϕ-ϕh·rhh+βrh2≥-βμκ2whVϕV-cβϕVrhh+βrh2≥-βC1μκ2whVrh-βC2rhhrh+βrh2≥β2rh2-C12μ2κ22βwhV2-C22βrhh2.As a result of the above inequality, taking (vh,qh)=(wh-βϕh,rh), it yields (54)Bhwh,rh;wh-βϕh,rh=Bhwh,rh;wh,rh+Bhwh,rh;-βϕh,0≥μκ1-C12μ2κ22βwhV2+α-C22βrhh2+β2rh2.Choosing β=min⁡κ1/2μC12κ22,α/2C22, we have (55)Bhwh,rh;wh-βϕh,rh≥μκ12whV2+β2rh2≥C3whV+rh2,where C3=min⁡μκ1/2,β/2. On the other hand, (56)vhV+qh=wh-βϕhV+rh≤whV+βϕV+rh≤whV+βC1+1rh≤C4whV+rh,where C4=1+C1β, which together with (55) leads to (57)Bhwh,rh;vh,qh≥C3C4whV+rhvhV+qh.We complete the proof of (50) with β2=C3/C4.It follows from Theorem2 that the discrete problem (47) shows a unique solution (uεh,pεh)∈(Vh,Mh) satisfying (58)supt∈0,Tuεht2+μκ1∫0TuεhtV2dt≤u0V2+1μκ1∫0Tft2dt.In order to derive the error estimates for the finite element approximation(uεh,pεh), we make the following regularity assumptions of the solution (uε,pε) to (22):(R1)uε∈L∞(0,T;V)∩L2(0,T;D(A)), pε∈L∞(0,T;M)∩L2(0,T;H1(Ω)).(R2)uεt∈L∞(0,T;L2(Ω)2)∩L2(0,T;Vσ),pεt∈L2(0,T;L2(Ω)).(R3)∫0T∫Ωgε2|∇uεt|2/(ε2+|∇uε|2)3/2dxdt≤C.The semidiscrete finite element approximation solution(uεh,pεh)∈Vh×Mh is of the following error estimate.Theorem 3.
Suppose that the assumptions(R1) and(R2) are satisfied. For any α>0, if (uε,pε) and (uεh,pεh) are the solutions of (24) and (48), respectively, then they satisfy (59)uε-uεhL∞0,T;L2Ω2+uε-uεhL20,T;V≤ch1/2.Proof.
Problem (24) takes the following form: (60)uεt,v+Buε,pε;v,q+buε,uε,v+βuε,v=f,v,∀v,q∈V×M.Subtracting (60) from (48) and taking vh=Ihuε-uεh and qh=Jhpε-pεh in the resulting equation, we get (61)uεt-uεht,Ihuε-uεh+Bhuε-uεh,pε-pεh;Ihuε-uεh,Jhpε-pεh+buε,uε,Ihuε-uεh-buεh,uεh,Ihuε-uεh+βuε-βuεh,Ihuε-uεh=αChpε,Jhpε-pεh.Since Ih is the L2 orthogonal projection operator, the first term in (61) can be rewritten as (62)uεt-uεht,Ihuε-uεh=12ddtuε-uεh2+uεt-uεht,Ihuε-uε=12ddtuε-uεh2+uεt,Ihuε-uε.It follows from (27) that (63)βuε-βuεh,Ihuε-uεh=βuε-βuεh,uε-uεh+βuε-βuεh,Ihuε-uε≥βuε-βuεh,Ihuε-uε.Then, substituting (62) and (63) into (61), it yields (64)12ddtuε-uεh2+μκ1Ihuε-uεhV2+αJhpε-pεhh2≤uεt,uε-Ihuε+BhIhuε-uε,Jhpε-pε;Ihuε-uεh,Jhpε-pεh+buεh,uεh,Ihuε-uεh-buε,uε,Ihuε-uεh-βuε-βuεh,Ihuε-uε+αChpε,Jhpε-pεh.All terms in the right hand side of (64) can be estimated in terms of Hölder’s inequality, Young’s inequality, (37), (38), (44), and (45). The term (uεt,uε-Ihuε) is bounded by (65)uεt,uε-Ihuε≤uεtuε-Ihuε≤ch2uεtAuε.The second term in the right hand side of (64) satisfies (66)BhIhuε-uε,Jhpε-pε;Ihuε-uεh,Jhpε-pεh=aIhuε-uε,Ihuε-uεh-dIhuε-uεh,Jhpε-pε+dIhuε-uε,Jhpε-pεh+αChJhpε-pε,Jhpε-pεh≤μκ2Ihuε-uεVIhuε-uεhV+Ihuε-uεhVJhpε-pε+ch-1Ihuε-uεJhpε-pεhh+αJhpε-pεhJhpε-pεhh≤μκ18Ihuε-uεhV2+α4Jhpε-pεhh2+cIhuε-uεV2+cJhpε-pε2+ch-2α-1Ihuε-uε2+αJhpε-pεh2≤μκ18Ihuε-uεhV2+α4Jhpε-pεhh2+ch2Auε2+ch2pε12.We rewrite b(uεh,uεh,Ihuε-uεh)-b(uε,uε,Ihuε-uεh) as (67)buεh,uεh,Ihuε-uεh-buε,uε,Ihuε-uεh=buεh,Ihuε-uε,Ihuε-uεh-bIhuε-uεh,uε,Ihuε-uεh+bIhuε-uε,uε,Ihuε-uεh.Then three terms in the right hand side of (67) are bounded, respectively, by (68)buεh,Ihuε-uε,Ihuε-uεh≤cuεhVIhuε-uεVIhuε-uεhV≤μκ18Ihuε-uεhV2+cuεhV2Ihuε-uεV2≤μκ18Ihuε-uεhV2+ch2uεhV2Auε2.(69)bIhuε-uε,uε,Ihuε-uεh≤cuεVIhuε-uεVIhuε-uεhV≤μκ18Ihuε-uεhV2+cuεV2Ihuε-uεV2≤μκ18Ihuε-uεhV2+ch2uεV2Auε2.(70)bIhuε-uεh,uε,Ihuε-uεh≤cIhuε-uεhVuεVIhuε-uεh≤μκ18Ihuε-uεhV2+cuεV2Ihuε-uεh2≤μκ18Ihuε-uεhV2+cuεV2uε-uεh2+ch4uεV2Auε2.Finally, we estimate (β(uε)-β(uεh),Ihuε-uε) by (71)βuε-βuεh,Ihuε-uε=∫Ωg∇uεε2+∇uε2-∇uεhε2+∇uεh2·∇Ihuε-uεdx≤2∫Ωg∇Ihuε-uεdx≤2guε-IhuεV≤chgAuε.By using (45), we have (72)αChpε,Jhpε-pεh≤αpεhJhpε-pεhh≤α4Jhpε-pεhh2+α-1pεh2≤α4Jhpε-pεhh2+ch2pε12.Substituting these estimates (65)–(72) into (64) gives (73)ddtuε-uεh2+μκ1uε-uεhV2+αpε-pεhh2≤ch2uεtAuε+ch2Auε2+ch2pε12+ch2uεhV2Auε2+ch2uεV2Auε2+cuεV2uε-uεh2+ch4uεV2Auε2+chgAuε.Therefore, (59) follows from Gronwall inequality,(R1),(R2), and (58).


## 4. Fully Discrete Scheme

In this section, we present a fully discrete scheme to approximate the solution(uε,pε) of the regularized problem. The time discretization is based on the backward Euler scheme. Let N be a positive integer. We set the time step Δt=T/N and the discrete time tn=nΔt, n=0,1,⋯,N. Then the fully discrete finite element approximation of (22) is given as follows: for n=1,⋯,N and the iterative initial condition uεh0=Ihu0, find uεhn∈Vh and pεhn∈Mh such that (74)dtuεhn,vh-uεhn+Bhuεhn,pεhn;vh-uεhn,qh-pεhn+buεhn-1,uεhn,vh-uεhn+jεvh-jεuεhn≥fn,vh-uεhn,∀vh∈Vh,qh∈Mh,where (75)dtuεhn=uεhn-uεhn-1Δt,fn=ftn,gn=gtn.It is easy to check that scheme (74) is unconditional stable.Theorem 4.
For anyα>0, the solution uεhn of (74) satisfies (76)uεhm2+μκ1Δt∑n=1muεhnV2≤u02+1μκ1Δt∑n=1Nfn2,1≤m≤N.Proof.
Taking(vh,qh)=(0,0) and (vh,qh)=(2uεhn,2pεhn) in (74) yields (77)dtuεhn,uεhn+μκ1uεhnV2+αpεhnh2+jεuεhn=fn,uεhn≤μκ12uεhnV2+12μκ1fn2.That is, (78)2dtuεhn,uεhn+μκ1uεhnV2≤1μκ1fn2.From (79)a-b,2a=a2-b2+a-b2,∀a,b∈R,we get (80)uεhn2-uεhn-12+uεhn-uεhn-12+μκ1ΔtuεhnV2≤Δtμκ1fn2.Summing (80) from n=1 to n=m≤N leads to (76).
Denote(81)en=uεtn-uεhn,zn=uεtn-Ihuεtn,rn=pεtn-pεhn,qn=pεtn-Jhpεtn.Now, we begin to prove the following error estimate between(uε(tn),pε(tn)) and (uεhn,pεhn) in the discrete norms corresponding to L2(0,T;V)∩L∞(0,T;L2(Ω)2).Theorem 5.
Suppose that the initial dataf∈H1(0,T;L2(Ω)2), g∈H1(0,T;L2(Ω)), and u0∈Vσ. Under the assumptions(R1)–(R3), there exists some τ0>0 such that when Δt<τ0, we have (82)max1≤n≤Nen2+μκ1Δt∑n=1NenV2+αΔt∑n=1Nrnh2≤ch+Δtε2/32.Proof.
Settingvh=Ihuε(tn) and qh=Jhpε(tn) in (74), we get (83)dtuεhn,en-zn+Bhuεhn,pεhn;en-zn,rn-qn+buεhn-1,uεhn,en-zn+jεIhuεtn-jεuεhn≥fn,en-zn.By taking v=uεhn,q=pεhn and t=tn in (22), we obtain (84)uεttn,-en+dtuεtn,en-dtuεtn,en+buεtn,uεtn,-en+Bhuεtn,pεtn;-en,-rn+jεuεhn-jεuεtn≥fn,-en-αChpεtn,rn.Taking the sum of (83) and (84) yields (85)dten,en+Bhen,rn;en,rn≤-dtuεhn,zn-uεttn-dtuεtn,en-Bhuεhn,pεhn;zn,qn+buεhn-1,uεhn,en-buεhn-1,uεhn,zn-buεtn,uεtn,en+jεIhuεtn-jεuεtn+fn,zn+αChpεtn,rn.Since Ih is the L2 orthogonal projection, then -(dtuεhn,zn)=0. Moreover, there holds (86)jεIhuεtn-jεuεtn≤cznVgn.Noting -uεhn=en-uε(tn) and -pεhn=rn-pε(tn), we have (87)-Bhuεhn,pεhn;zn,qn=Bhen,rn;zn,qn-Bhuεtn,pεtn;zn,qn.Using (44) and Young inequality, Bh(en,rn;zn,qn) is bounded by (88)Bhen,rn;zn,qn=aen,zn-dzn,rn+den,qn+αChrn,qn≤μκ2enVznV+ch-1znrnh+enVqn+αrnhqnh≤δenV2+ηαrnh2+cznV2+ch-2α-1zn2+cqn2+cαqnh2,where δ>0 and η>0 are two small constants determined later. We estimate Bh(uε(tn),pε(tn);zn,qn) by (89)Bhuεtn,pεtn;zn,qn=auεtn,zn-dzn,pεtn+duεtn,qn+αChpεtn,qn≤cAuεtn+pεtn1zn+uεtnVqn+αpεtnhqnh.We rewrite b(uεhn-1,uεhn,en)-b(uεhn-1,uεhn,zn)-b(uε(tn),uε(tn),en) as (90)buεhn-1,uεhn,en-buεhn-1,uεhn,zn-buεtn,uεtn,en=buεtn-1,uεhn,en-ben-1,uεhn,en+ben-1,uεhn,zn-buεtn-1,uεhn,zn-buεtn,uεtn,en=ben-1,uεhn,zn-ben-1,uεhn,en+buεtn-1,uεtn,en-buεtn,uεtn,en+buεtn-1,en,zn-buεtn-1,uεtn,zn=ben-1,uεhn,zn-ben-1,uεhn,en-b∫tn-1tnuεttdt,uεtn,en+buεtn-1,en,zn-buεtn-1,uεtn,zn=ben-1,uεtn,en+ben-1,en,zn+ben-1,uεtn,zn-b∫tn-1tnuεttdt,uεtn,en+buεtn-1,en,zn-buεtn-1,uεtn,zn.All terms in the right hand side of (90) are estimated, respectively, by (91)ben-1,uεtn,en≤cen-1·Auεtn·enV,(92)ben-1,en,zn≤cen-1·znL∞enV,(93)ben-1,uεtn,zn≤cen-1·Auεtn·znV,(94)buεtn-1,en,zn≤cuεtn-1VznVenV,(95)buεtn-1,uεtn,zn≤cuεtn-1VAuεtn·zn,and (96)b∫tn-1tnuεttdt,uεtn,en≤∫tn-1tnuεttdt·∇uεtnL3enL6≤c∫tn-1tnuεttdtAuεtnenV≤cΔtuεttL∞0,T;L2Ω2Auεtn·enV≤δenV2+cΔt2Auεtn2.The last term (fn,zn)+αCh(pε(tn),rn) in (85) can be easily bounded by (97)fn,zn+αChpεtn,rn≤ηαrnh2+cαpεtnh2+fnzn.Combining (86)–(97) into (85) leads to (98)dten,en+μκ1enV2+αrnh2≤-uεttn-dtuεn,en+2δenV2+2ηαrnh2+cznV2+ch-2α-1zn2+cqn2+cαqnh2+cAuεtn+pεtn1zn+uεtnVqn+αpεtnhqnh+znVgn+cαpεtnh2+fnzn+cen-1Auεtn+znL∞enV+cen-1AuεtnznV+cuεtn-1VznVenV+cuεtn-1VAuεtnzn+cΔt2Auεtn2.In terms of (37), (38), and (45), it is easy to check (99)zn+hznV≤ch2Auεtn,znL∞≤chAuεtn,qn+qnh+pεtnh≤chpεtn1.Then, using Young’s inequality, we get (100)dten,en+μκ1enV2+αrnh2≤-uεttn-dtuεtn,en+4δenV2+2ηαrnh2+chAuεtn2+pεtn12+chuεtnV2+chgn2+ch2fn2+c1+h2Auεtn2en-12+chuεtn-1V2Auεtn2+cΔt2Auεtn2.Choosing δ=μκ1/8 and η=1/4 gives (101)dten,en+μκ12enV2+α2rnh2≤-uεttn-dtuεtn,en+chAuεtn2+pεtn12+chuεtnV2+chgn2+ch2fn2+c1+h2Auεtn2en-12+chuεtn-1V2Auεtn2+cΔt2Auεtn2.Multiplying (101) by 2Δt and summing for n=1,⋯,m≤N yield (102)2∑n=1men-en-1,en+μκ1Δt∑n=1menV2+αΔt∑n=1mrnh2≤2Δt∑n=1Nuεttn-dtuεtn,en+chΔt∑n=1mAuεtn2+Δt∑n=1mpεtn12+chΔt∑n=1muεhtnV2+chΔt∑n=1Ngn2+ch2Δt∑n=1Nfn2+chmax1≤n≤muεtnV2Δt∑n=1mAuεtn2+c1+h2Δt∑n=1mAuεtn2en-12+cΔt2Δt∑n=1mAuεtn2,where we use(A1),(R1), and (76). Since (103)2∑n=1men-en-1,en=em2-e02+∑n=1men-en-12,then we obtain (104)em2+μκ1Δt∑n=1menV2+αΔt∑n=1mrnh2≤2Δt∑n=1Nuεttn-dtuεtn,en+cΔt2+h+c1+h2Δt∑n=1mAuεtn2en-12,where we use e0=u0-Ihu0≤chu0V. Thus, it follows from the discrete Gronwall’s inequality (cf. [27, 28]) that there exists some τ0>0 such that when Δt<τ0, (105)em2+μκ1Δt∑n=1menV2+αΔt∑n=1mrnh2≤cΔt2+h+2Δt∑n=1Nuεttn-dtuεtn,en.Next, we estimate Δt∑n=1Nuεttn-dtuεtn,en. Since (uε(t),pε(t)) solves (24), we have (106)uεttn-dtuεtn,enΔt=uεttnΔt-uεtn+uεtn-1,en=∫tn-1tnuεttn-uεtt,endt=∫tn-1tnauεt-uεtn,endt-∫tn-1tnft-ftn,endt+∫tn-1tnβuεt-βuεtn,endt+∫tn-1tnbuεt,uεt,en-buεtn,uεtn,endt-∫tn-1tnden,pεt-pεtndt.Under the assumptions(R2) and(R3), there holds [16] (107)∑n=1N∫tn-1tnauεt-uεtn,endt-∑n=1N∫tn-1tnft-ftn,endt+∑n=1N∫tn-1tnβuεt-βuεtn,endt≤δΔt∑n=1NenV2+cΔtε2/32.where δ>0 is some sufficiently small constant. Under the assumption(R2), the last two terms in (106) can be estimated by (108)∑n=1N∫tn-1tn-den,pεt-pεtndt=∑n=1N∫tn-1tndiv⁡en,∫ttnpεssdsdt≤∑n=1N∫tn-1tnenV∫ttnpεssdsdt≤∑n=1NenVΔt∫tn-1tnpεttdt≤δΔt∑n=1NenV2+cΔt2pεtL20,T;L2Ω.and (109)∑n=1N∫tn-1tnbuεt,uεt,en-buεtn,uεtn,endt≤∑n=1N∫tn-1tnbuεt,-∫ttnuεssds,en+b-∫ttnuεssds,uεtn,endt≤∑n=1N∫tn-1tnuεtV+uεtnV∫ttnuεssdsVenVdt≤c∑n=1N∫tn-1tn∫tn-1tnuεssVdsenVdt=c∑n=1NenVΔt∫tn-1tnuεttVdt≤δΔt∑n=1NenV2+cΔt2uεtL20,T;V.Then, substituting (106)–(109) into (105), for sufficiently small δ>0 we obtain (82).Remark 6.
The termΔt∑n=1N[pεn-pεhn]h2 derived in Theorem 5 does not provide any convergence result for the finite element approximation of the pressure pε. The reason is that the mesh-dependent norm [·]h behaves like h·1.Theorem 7.
Under the assumptions in Theorem5, let (u,p) and (uεhn,pεhn) be the solution of (16) and (74), then (110)utn-uεhn2+Δt∑n=1Nutn-uεhnV2≤cε+h+Δtε2/32.Proof.
From the triangle inequality, Theorems1 and 5, we get (111)utn-uεhn2≤cutn-uεtn2+uεtn-uεhn2≤cε+h+Δtε2/32.By a similar method, we have (112)Δt∑n=1Nutn-uεhnV2≤cΔt∑n=1Nutn-uεtnV2+cΔt∑n=1Nuεtn-uεhnV2≤cΔt∑n=1Nutn-uεtnV2+ch+cΔtε2/32.The first term Δt∑n=1Nutn-uεtnV2 in the right hand side of the above inequality is bounded by (113)Δt∑n=1Nutn-uεtnV2≤∑n=1N∫tn-1tnutn-uεtnV2-ut-uεtV2dt+∫0Tut-uεtV2dt≤∑n=1N∫tn-1tn∫ttnddsus-uεsV2dsdt+cε≤cΔt∑n=1N∫tn-1tnau-uε,ut-uεtdt+cε=cΔt∫0Tau-uε,ut-uεtdt+cε≤cΔtu-uεL20,T,Vut-uεtL20,T,V+cε≤cΔtε1/2+ε≤cΔt2+ε,which together with (111) completes the proof of (110).Remark 8.
Bingham fluid flow in cylindrical pipes is actually the parabolic variational inequality problem of the second kind. Zhang [16] has studied a fully discrete finite element scheme for Bingham flow fluid in cylindrical pipes, where he showed that (114)utn-uεhn2+Δt∑n=1Nutn-uεhnV2≤cε+h+Δtε2/32.Thus, the error estimate (110) derived in Theorem 7 extends Zhang’s results in [16].Remark 9.
If we chooseΔt=O(ε4/3) and ε=O(h3/2) in (110), then we obtain the following error estimate which is nearly optimal for general domain: (115)utn-uεhn2+Δt∑n=1Nutn-uεhnV2≤ch.



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