Analysis of the small viscosity and large reaction coefficient in the computation of the generalized Stokes problem by a novel stabilized finite element method

Analysis of the small viscosity and large reaction coe ffi cient in the computation of the generalized Stokes problem

by a novel stabilized finite element method

Huo-Yuan Duan^a, Po-Wen Hsieh^b, Roger C. E. Tan^c, Suh-Yuh Yang^d,∗

aCollaborative Innovation Centre of Mathematics, School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

bDepartment of Applied Mathematics, Chung Yuan Christian University, Jhongli City, Taoyuan County 32023, Taiwan

cDepartment of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore

dDepartment of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan

Abstract

In this paper, we propose and analyze a novel stabilized finite element method (FEM) for the system of generalized Stokes equations arising from the time-discretization of transient Stokes problem. The system involves a small viscos- ity, which is proportional to the inverse of large Reynolds number, and a large reaction coefficient, which is the inverse of small time step. The proposed stabilized FEM employs theC⁰ piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence-free equation to define the stabilization terms. The stabilization parameters are fixed and element-independent, without a comparison of the viscosity, the reaction coefficient and the mesh size. Using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and theH¹-seminorm projection for pressure, instead of the usual nodal interpolants, we derive error estimates for the stabilized finite element approximations to velocity and pressure in theL² andH¹ norms and most importantly, we explicitly establish the dependence of error bounds on the viscosity, the reaction coefficient and the mesh size. Our analysis reveals that this stabilized FEM is particularly suitable for the generalized Stokes system with a small viscosity and a large reaction coefficient, which has never been achieved before in the error analysis of other stabilization methods in the literature. We then numerically confirm the effectiveness of the proposed stabilized FEM. Comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability.

Keywords: generalized Stokes problem, small viscosity, large reaction coefficient, stabilized finite element method, stabilization parameter

2000 MSC:65N12, 65N15, 65N30, 76M10

1. Introduction

LetΩ⊂R²be a bounded open convex polygonal domain with boundary∂Ω. In this paper, we propose and analyze a novel stabilized finite element method (FEM) for solving the following system of generalized Stokes equations with no-slip boundary condition:











σu−ν∆u+∇p = f inΩ,

∇ ·u = 0 inΩ, u = 0 on∂Ω,

(1)

∗Corresponding author. Tel.:+886-3-4227151 extension 65130; fax:+886-3-4257379.

Email addresses:[email protected](Huo-Yuan Duan),[email protected](Po-Wen Hsieh), [email protected](Roger C. E. Tan),[email protected](Suh-Yuh Yang)

whereu = (u1,u2)^> is the velocity field and p is the pressure; 0 < ν ≤ 1 is the kinematic viscosity which is proportional to the inverse of Reynolds numberRe;σ≥1 is the reaction coefficient; and f =(f1,f2)^> ∈(L²(Ω))²is a given source-like function. In principle, one of the parametersσandνin the generalized Stokes system (1) can be normalized as unity if we replace the velocity field with eitherσuorνu. However, in order to explicitly display the parametersσandνin our finite element error estimates and to serve as a base for the future study of more advanced incompressible Navier-Stokes system which contains an additional nonlinear velocity convection term and therefore the normalization cannot be applied, in this paper we keep the generalized Stokes system in its primitive form (1).

Typically, the generalized Stokes system (1) may arise from the time discretization of transient Stokes problem, where the reaction coefficientσis inversely proportional to the time stepδt. For problems involving chemical re- actions, a small time step, namely a largeσ, is needed in order to account for the stiffness due to the fast reaction [1, 2, 3]. In fact, the common finite element approach to the time-dependent PDEs is based on the so-called method of lines. That is, a semi-discrete formulation is formed by first approximating the spatial dependence using a certain finite element method, and the resulting ODE system is then discretized by applying implicit finite differences in time domain, such as the first-order backward Euler scheme or the second-order Crank-Nicolson scheme, to obtain a fully discrete problem [4, 5]. One can find that at each time level with an implicit time discretization, the resulting fully discrete problem is analogue to the finite element method applied directly to the generalized Stokes system (1).

The usual mixed FEM for solving problem (1) is to discretize the following weak form by employing a pair of finite element spaces (Vh,Q_h) for the approximations ofuandp:Find(u,p)∈(H₀¹(Ω))²×L²₀(Ω)such that

σ(u,v)0+ν(∇u,∇v)0−(p,∇ ·v)0+γ(∇ ·u,q)0=(f,v)0, (2) for all(v,q) ∈ (H¹₀(Ω))²×L²₀(Ω),whereγ =±1 correspond to a symmetric method and a non-symmetric method.

However, it is now well known that the pair (V_h,Q_h) must satisfy the so-called inf-sup condition if stable and optimally accurate approximations are desired; see [6, 7, 8, 9]. Unfortunately, this condition prevents the use of standard equal orderC⁰interpolation spaces for velocity and pressure with respect to the same triangulation that are the most attractive from the viewpoint of implementation, or low order element pairs such as piecewise linear elements for velocity and piecewise constants for pressure.

In order to circumvent the inf-sup condition, a class of the so-called stabilized FEMs has been developed and intensively studied for almost thirty years, see, e.g., [10, 11, 12, 13, 14, 15, 16]. The stabilized FEMs are formed by adding to the discrete counterpart of the weak formulation (2) with the residuals of the partial differential equations.

An example for the derivation of such a stabilized FEM is based on the bubble condensation procedure (cf. [17, 18]).

This approach adopts a mixed finite element formulation but enriches theC⁰ piecewise P1 (or Q1) elements for velocity fielduwith suitable bubble functions and then give an expression of the bubble part of velocity in terms of its linear part, pressurepand source functionf. Such a bubble condensation procedure eventually leads to a stabilization method. A common feature for all stabilization methods is that some mesh-dependent stabilization parameters are involved with. The stabilization parameters play key roles in the method, not only enhancing the numerical stability

but also improving the accuracy in the finite element solutions. Nowadays, it is very popular to apply finite element stabilization techniques to produce better accuracy and stability in the finite element solution of convection-dominated convection-diffusion problems [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29], or to circumvent the inf-sup condition in solving saddle-point type problems such as the incompressible Stokes equations [1, 17, 30, 31, 32]; see also [2] and many references cited therein. However, it has been observed that the pressure instabilities may be caused in time step of the transient problems, as the time stepδt ≈ 1/σbecomes small compared to the spatial grid sizeh; see [1, 2, 3]. Therefore, in recent years, it has attracted a great deal of attention on the theoretical and computational studies of small time-step instabilities when implicit, finite difference time integration is applied in combination with finite element stabilization in the spatial semi-discretization.

In this paper, we will propose and analyze a new stabilized FEM for problem (1) with a small viscosityνand a large reaction coefficientσ. With novel stabilization parameters, we employ theC⁰piecewiseP1(orQ1) elements for both velocity field and pressure on the same mesh and use the residuals of the partial differential equations to define the stabilization terms. These additional stabilization terms include not only the term related to the momentum equation but also the divergence-free equation. Another feature, different from the stabilization methods, e.g., [17, 30], in the literature is that our stabilization parameters are fixed and element-independent, without the comparison among the viscosityν, the reaction coefficientσand the mesh sizeh. The third feature is that our error analysis is novel. We use the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the H¹-seminorm projection [9] for pressure, instead of the usual nodal interpolantsI_huandI_hp. This novel technique enables us to show that the proposed stabilized FEM usingC⁰ piecewise equal-order finite elements, such asP1-P1

andQ1-Q1elements, is particularly suitable for problem (1) with a smallνand a largeσ.

We will derive error estimates in theL² andH¹norms for the proposed stabilized FEM using theC⁰ piecewise P1-P₁(orQ1-Q₁) elements. Most importantly, we will explicitly establish the dependence of the error bounds on the parametersν,σandh. Roughly speaking, we prove that, leaving aside the term “kuk₂+kpk₁” of regularity norm of the exact solution, theH¹-norm error bound of the velocity is inversely proportional to the square root of the reaction coefficientσ, theL²-norm error bound behaves like the multiplication of theH¹-norm error bound by the mesh size h, and that the L²-norm error bound of the pressure is in the same order ofh+ √

ν ifσh² ν. Moreover, we will compare numerically the effectiveness of the newly proposed stabilized FEM with three existing stabilization methods in the literature, including the Barrenechea-Blasco stabilized FEM [30], the Barrenechea-Valentin stabilized FEM [17], and the Bochev-Gunzburger-Lehoucq stabilized FEM [1]. Through a series of numerical experiments, we find that in addition to offer rather sharp error estimates, the proposed stabilized FEM can achieve better accuracy and stability when compared with the Barrenechea-Blasco method and the Bochev-Gunzburger-Lehoucq method, while it is comparable with the Barrenechea-Valentin method whenσh²νand 0< ν1.

We remark that the div-div stabilization term related to the divergence-free equation in the finite element formu- lation was first introduced in [33]; see also, e.g., [32, 34, 35]. It has been proved in [36] that the div-div stabilization term results in better error bounds of finite element solutions for the generalized Stokes equations (1). However, in

that method, the finite element pair (V_h,Q_h) has to satisfy the inf-sup condition, the analysis focuses on the small reaction coefficientσ∈[0,1], and the error bounds for the velocity are measured in the viscosity √

ν-weightedH¹- seminorm. Therefore, the use of equal orderC⁰ finite element spaces for velocity and pressure is prevented and the poor convergence may be caused by small viscosity νand large reaction coefficient σ. In contrast, in this paper, the newly proposed stabilized FEM is an unusual stabilization method inspired by the works of Franca-Farhat [22]

and Barrenechea-Valentin [17]. It employs theC⁰piecewise equal-order finite elements aiming at dealing with small viscosity and large reaction. Our analysis reveals that the viscosityνand the reaction constant σrespectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights; see Theorem 5 and Theorem 6 in Section 3. In other words, up to the regularity norm

“kuk₂+kpk₁” of the exact solution, the proposed method is particularly suitable for the generalized Stokes system (1) with a small viscosityνand a large reaction coefficientσ. Recently, it has also been shown in [37] that the finite element formulation including the div-div stabilization term is more effective for convection-dominated problems, such as the Oseen equations with a small viscosity.

In summary, the main goal of this paper are three aspects. The first is to develop a new unusual stabilization method in order to obtain better error bounds. The second is to provide a new analysis to prove the robustness of the unusual stabilization methods with respect to the small viscosityνand the large reactionσ. The third is to provide numerical experiments to further demonstrate the robustness of the unusual type stabilization methods, including the proposed stabilized FEM and the Barrenechea-Valentin stabilized FEM, for the generalized Stokes equations with small viscosity and large reaction. These three aspects will be successfully realized in this paper.

The remainder of this paper is organized as follows. In Section 2, we introduce the new stabilized FEM and present three existing stabilized FEMs in the literature. In Section 3, error estimates of the newly proposed method are derived. The dependence of error bounds on the parametersν,σandhare explicitly established. In Section 4, several numerical examples are presented, including an example on the unstructured meshes and the time-dependent lid-driven cavity problem. Finally, in Section 5, summary and conclusions are given.

2. The new stabilized finite element method

We use the standard notation and definitions for the Sobolev spacesH^m(Ω) for nonnegative integersm(cf. [7, 9, 38, 39]). The associated inner product and norm are denoted by (·,·)mandk · k_m, respectively. As usual, we define

L²₀(Ω) = {q∈L²(Ω) : Z

Ωq=0}, H₀¹(Ω) = {v∈H¹(Ω) : v=0 on∂Ω}.

Let{T_h}_0<h≤1be a family of triangulations ofΩ. A triangulationT_hofΩinto elementsKconsisting of triangles (or quadrilaterals) is performed in the usual way; the intersection of any two elements is a vertex, or an edge or empty, andΩ =∪_K∈ThK. For each triangulation the subscripth∈(0,1] refers to the level of refinement of the triangulation.

In particular, the mesh size parameterhis defined ash=max{hK :K∈ T_h}, wherehKdenotes the diameter of element

K. We always assume that the family{T_h}_0<h≤1of triangulations is shape regular [6, 7, 38, 39]. As usual, (·,·)m,K and k · k_m,Kdenote the associated inner product and norm inH^m(K), respectively, whereKis a given element inT_h.

Define the finite element spaceV_h=V₁×V₁for velocityu, whereV₁⊆H₀¹(Ω) denotes the space ofC⁰piecewise linear (or bilinear) finite elements over the triangulationT_h. The standard interpolation theory [7, 38] ensures that if u∈(H²(Ω)∩H¹₀(Ω))², then there exists an interpolantI_hu∈ V_hsuch that

ku− I_huk_s,K ≤Ch^2−s_K kuk_2,K ∀0≤s≤2, ∀K∈ T_h, (3) whereCis a positive constant independent of KandhK. Also, we define theC⁰piecewise linear (or bilinear) finite element spaceQ_h ⊆H¹(Ω)∩L²₀(Ω) for pressure pwith the interpolation property: ifp∈H²(Ω)∩L²₀(Ω), then there exists an interpolationI_hp∈ Q_hsuch that

kp− I_hpks,K ≤Ch^2−s_K kpk2,K ∀0≤s≤2, ∀K∈ T_h. (4) We remark that in this paper we useCto denote a generic positive constant, possibly different at different occurrences, which is always independent ofhand other parameters introduced.

Since the pair of finite element spaces (Vh,Q_h) does not verify the inf-sup condition, the corresponding discrete counterpart of the weak formulation (2) is generally unstable. Next, we are going to design a stabilization method for solving the generalized Stoke system (1). In what follows, for retaining the property of symmetry of the associated bilinear form, we only consider the case ofγ =−1 in the weak form (2). We propose the following new stabilized FEM for the generalized Stokes system (1):Find(u_h,ph)∈ V_h× Q_hsuch that

B((uh,ph),(vh,qh))=L((vh,qh)) ∀(vh,qh)∈ V_h× Q_h, (5) where the bilinear formB(·,·) and the linear formL(·) are, respectively, defined as follows:

B((u_h,p_h),(v_h,q_h)) = σ(u_h,v_h)₀+ν(∇u_h,∇v_h)₀−(p_h,∇ ·v_h)₀−(∇ ·u_h,q_h)₀

−X

K∈T_h

h²

σh²+12ν(σuh−ν∆uh+∇ph, σvh−ν∆vh+∇qh)0,K

+X

K∈T_h

12ν

σh²+12ν(∇ ·u_h,∇ ·v_h)_0,K, (6)

L((vh,qh)) = (f,vh)0− X

K∈T_h

h²

σh²+12ν(f, σvh−ν∆vh+∇qh)_0,K. (7) Note that in (6) and (7), we have∆v_h|_K =0for allK∈ T_handv_h∈ V_h. However, we still retain the terms therein for the clarity of presentation.

Here we remark that the stabilization terms in (6) include not only the term related to the momentum equation but also the continuity equation. This is quite different from the stabilization methods given in [1, 17, 30]; see also (13), (17) and (21) below. Furthermore, the test function in the momentum stabilization includes the reaction termσv_h which is different again from that of [1, 30]. Secondly, the stabilization parametersh²/(σh²+12ν) and 12ν/(σh²+12ν) in (6) are fixed and element-independent, without the comparison amongν,σandh.

We next define the induced energy norm denoted byk · k_honV_h× Q_h, which is related to the bilinear formB(·,·) introduced in (6),

k(v_h,q_h)k²_h=νk∇v_hk²₀+ 12σν

σh²+12νkv_hk²₀+ h²

σh²+12νk∇q_hk²₀+ 12ν

σh²+12νk∇ ·v_hk²₀. (8) SinceR

Ωqh =0 for allqh ∈ Q_h, one can easily verify thatk · k_his indeed a norm onV_h× Q_h. With this energy norm, we immediately have the following estimates of the continuous bilinear formB(·,·) on (Vh× Q_h)×(Vh× Q_h):

Lemma 1.The bilinear form B(·,·)is weakly coercive on(Vh× Q_h)×(Vh× Q_h), that is, inf

0,(vh,qh)∈Vh×Qh

sup

0,(χ_h,ϕ_h)∈V_h×Q_h

B((vh,qh),(χ_h, ϕh)) k(vh,qh)khk(χ_h, ϕh)kh

≥ 1, (9)

inf

0,(χ_h,ϕh)∈Vh×Qh

sup

0,(vh,q_h)∈V_h×Q_h

B((vh,qh),(χ_h, ϕh)) k(vh,qh)khk(χ_h, ϕh)kh

≥ 1. (10)

Proof.Given (v_h,qh)∈ V_h× Q_h, we have from (6) that B((v_h,qh),(v_h,−qh)) = σkv_hk²₀+νk∇v_hk²₀− h²

σh²+12ν(σv_h+∇qh, σv_h− ∇qh)0+ 12ν

σh²+12νk∇ ·v_hk²₀

= νk∇v_hk²₀+ 12σν

σh²+12νkv_hk²₀+ h²

σh²+12νk∇qhk²₀+ 12ν

σh²+12νk∇ ·v_hk²₀

= k(vh,qh)k²_h,

which combining with the factk(vh,−qh)kh=k(vh,qh)khimplies sup

0,(χ_h,ϕh)∈Vh×Qh

B((vh,qh),(χ_h, ϕh)) k(χ_h, ϕh)kh

≥ k(v_h,qh)kh.

It clearly leads to (9). The estimate (10) holds by the symmetry property of the bilinear formB(·,·). 2

Notice that the weak coercivity (9)-(10) of the bilinear formB(·,·) ensures the unique solvability of the stabilized FEM (5); see, e.g., [6, 7, 8]. We also remark that the stabilized FEM (5) is a consistent formulation, since (5) is satisfied when the finite element solution (u_h,p_h) is replaced by the exact solution (u,p) of the generalized Stokes system (1). As a consequence, we have the following orthogonality property:

B((u,p)−(uh,ph),(vh,qh))=0 ∀(vh,qh)∈ V_h× Q_h. (11) In the rest of this section, we give a brief review of three existing stabilized FEMs in the literature which are particularly designed for the generalized Stokes system (1) with a large reaction coefficientσ.

• The Barrenechea-Blasco stabilized FEM:In [30], Barrenechea and Blasco developed the following stabilized FEM:Find(u_h,ph)∈ V_h× Q_hsuch that

BBB((u_h,ph),(v_h,qh))=LBB((v_h,qh)) ∀(v_h,qh)∈ V_h× Q_h, (12)

where the bilinear formBBB(·,·) and the linear formLBB(·) are, respectively, defined as follows:

BBB((uh,ph),(vh,qh)) = σ(uh,vh)0+ν(∇uh,∇vh)0−(∇ ·vh,ph)0+(∇ ·uh,qh)0

−X

K∈Th

τ_K(σu_h−ν∆u_h+∇p_h,−ν∆v_h− ∇q_h)_0,K, (13) LBB((vh,qh)) = (f,vh)0−X

K∈T_h

τK(f,−ν∆vh− ∇qh)0,K, (14) τK is an element-dependent stabilization parameter given by

τK=αh²_K

ν (15)

andαis a positive constant such that

τK<min (8

ν, 3 4σ

)

and α < 1 4C²_inv. HereCinv>0 is the constant in the local inverse inequality [30]:

k∆vhk_0,K≤Cinvh⁻¹_Kk∇vhk_0,K ∀K∈ T_h, vh∈V1.

In practical computations, we can takeCinvbeing any positive constant sincek∆vhk_0,K=0 forvh∈V1.

• The Barrenechea-Valentin stabilized FEM:Inspired by the work of [22, 25], Barrenechea and Valentin [17]

introduced the following unusual stabilized FEM:Find(uh,ph)∈ V_h× Q_hsuch that

BBV((u_h,ph),(v_h,qh))=LBV((v_h,qh)) ∀(v_h,qh)∈ V_h× Q_h, (16) where the bilinear formBBV(·,·) and the linear formLBV(·) are, respectively, defined as follows:

BBV((u_h,ph),(v_h,qh)) = σ(u_h,v_h)₀+ν(∇u_h,∇v_h)₀−(∇ ·v_h,ph)₀+(∇ ·u_h,qh)₀

−X

K∈T_h

τK(σuh−ν∆uh+∇ph, σvh−ν∆vh− ∇qh)0,K, (17) LBV((v_h,qh)) = (f,v_h)₀−X

K∈Th

τ_K(f, σv_h−ν∆v_h− ∇qh)_0,K, (18) τK is an element-dependent stabilization parameter given by

τ_K = h²_K

σh²_Kξ(λK)+(4ν/m1) (19)

and

λ_K= 4ν

m1σh²_K, m₁=min{1/3,C₁},

C1h²_Kk∆vhk²_0,K≤ k∇vhk²_0,K ∀vh∈V1, ξ(λ)=max{λ,1}.

In practical computations, we takem₁ = 1/3 sincek∆v_hk_0,K = 0 forv_h ∈ V₁. The “unusual” feature of this stabilization method is the subtraction of a termP

K∈T_hτ_K(σu_h, σv_h)_0,K fromσ(u_h,v_h)0 of the standard mixed FEM for the generalized Stokes system (1).

• The Bochev-Gunzburger-Lehoucq stabilized FEM:In [1], the authors considered the following stabilized FEM:Find(u_h,ph)∈ V_h× Q_hsuch that

B_BGL((u_h,p_h),(v_h,q_h))=L_BGL((v_h,q_h)) ∀(v_h,q_h)∈ V_h× Q_h, (20) where the bilinear formBBGL(·,·) and the linear formLBGL(·) are, respectively, defined as follows:

BBGL((uh,ph),(vh,qh)) = σ(uh,vh)0+ν(∇uh,∇vh)0−(∇ ·vh,ph)0−(∇ ·uh,qh)0

−X

K∈Th

τ_K(σu_h−ν∆u_h+∇p_h,−γν∆v_h+∇q_h)_0,K, (21) LBGL((vh,qh)) = (f,vh)0−X

K∈T_h

τK(f,−γν∆vh+∇qh)0,K, (22)

γ=−1,0,1 andτKis an element-dependent stabilization parameter given by

τK =δh²_K (23)

andδ >0 is a parameter independent ofhKthat needs to be carefully chosen.

We conclude this section with the following two remarks.

Remark 1.The choice of the first stabilization parameterh²/(σh²+12ν) in the newly proposed stabilized FEM (5) is inspired by our recent work [40] on the scalar reaction-convection-diffusion equation and the element stabilization parameterτ_K described in (19). One can verify that ifσh²_K ν, then the element stabilization parameterτ_Kin (19) can be expressed as

τK = h²_K σh²_K+12ν,

which amounts to the first stabilization parameterh²/(σh²+12ν) in (6) providedhK ≥Ch.

Remark 2.All the above three stabilization methods, (12), (16) and (20), produce a non-symmetric algebraic system.

The Barrenechea-Valentin method (16) also can be converted into a symmetric one by simply replacingqh in the right-hand sides of (17) and (18) with−qh. Note that our stabilization method (5) is symmetrical.

3. Error estimates of the new stabilized finite element method

We now proceed to estimate the error of the finite element solution (uh,ph) of the newly proposed stabilized FEM (5). Letg∈(L²(Ω))²be a given source function. Consider the following auxiliary boundary value problem:

( σw−ν∆w = g inΩ,

w = 0 on∂Ω. (24)

The corresponding stabilized FEM on the spaceV_his defined as follows:

Findw_h∈ V_h such that B_aux(w_h,v_h)=L_aux(v_h) ∀v_h ∈ V_h, (25)

where the bilinear formBaux(·,·) and the linear formLaux(·) are, respectively, given by B_aux(w_h,v_h) = σ(w_h,v_h)₀+ν(∇w_h,∇v_h)₀−X

K∈T_h

h²

σh²+12ν(σw_h−ν∆w_h, σv_h−ν∆v_h)_0,K, (26) Laux(v_h) = (g,v_h)0−X

K∈T_h

h²

σh²+12ν(g, σv_h−ν∆v_h)_0,K. (27) The unique solvability of the stabilized FEM (25) is ensured by the strong coercivity of the bilinear formBaux(·,·),

Baux(vh,vh)=νk∇vhk²₀+ 12νσ

σh²+12νkvhk²₀ ∀vh∈ V_h. (28) Moreover, the convergence of the stabilized finite element solutionw_h of (25) can be proved using the similar tech- niques developed by Franca and Farhat in [22] for a scalar reaction-diffusion problem; see also [40]. We thus state without proof the following lemma:

Lemma 2.Letw∈(H₀¹(Ω)∩H²(Ω))²be the solution of the auxiliary boundary value problem (24). Then the unique stabilized C⁰piecewise linear (or bilinear) finite element solutionw_hof (25) converges towin the following way:

k∇(w−wh)k²₀+ 12σ

σh²+12νkw−whk²₀ ≤Ch²kwk²₂. (29) In what follows, we shall present and analyze two finite element solution-projections, respectively associated with velocityuand pressure p. These finite element solution-projections will be used as the intermediate finite element interpolations in the derivation of the error bounds of our stabilization method (5). First, letu∈(H₀¹(Ω)∩H²(Ω))²be the exact solution of problem (1). Then we define a source functiongby

g:=σu−ν∆u. (30)

Note thatu obviously solves the auxiliary boundary value problem (24) with this given source function g. As a consequence of Lemma 2, we have following results:

Lemma 3.Letu∈(H₀¹(Ω)∩H²(Ω))²be the solution of problem (1) andwh∈ V_hthe unique solution of (25) associated with the source functionggiven in (30). Then there exists a constant C>0independent ofν,σand h such that

ku−w_hk₁ ≤ Chkuk₂, (31)

ku−whk₀ ≤ Ch rh²

12+ ν

σ kuk2. (32)

Proof.Sinceu∈(H₀¹(Ω)∩H²(Ω))²andg:=σu−ν∆u, the exact solutionwof the auxiliary boundary value problem (24) is then given byw=u. From Lemma 2, we have

k∇(u−w_h)k²₀+ 12σ

σh²+12νku−w_hk²₀≤Ch²kuk²₂. (33) Combining (33) with the following Poincar´e-Friedrichs inequality [7, 9],

kvk0≤Cp fk∇vk0 ∀v∈(H₀¹(Ω))²,

we can easily obtain the estimate (31). The estimate (32) is a direct consequence of (33). 2

We remark that in Lemma 3, sincew_h is the stabilized finite element solution of problem (25) with the source functiong:=σu−ν∆u, we have for allv_h∈ V_hthat

σ(wh,vh)0+ν(∇wh,∇vh)0−X

K∈T_h

h²

σh²+12ν(σwh−ν∆wh, σvh−ν∆vh)0,K

=(g,vh)0−X

K∈T_h

h²

σh²+12ν(g, σvh−ν∆vh)0,K

=(σu−ν∆u,v_h)0− X

K∈T_h

h²

σh²+12ν(σu−ν∆u, σvh−ν∆v_h)0,K

=σ(u,v_h)0+ν(∇u,∇v_h)0− X

K∈Th

h²

σh²+12ν(σu−ν∆u, σv_h−ν∆v_h)0,K. (34) To give error estimates of the the newly proposed stabilized FEM (5), we also need the following result about the pressurep. A more general form of the result can be found in [9] (see Theorem A.2 on pp. 101-102).

Lemma 4.Assume that p ∈H¹(Ω)∩L²₀(Ω). Letep_h ∈ Q_h be the H¹-seminorm projection of p on the finite element spaceQ_h, that is,

(∇eph,∇qh)0=(∇p,∇qh)0 ∀qh∈ Q_h. (35) Then there exists a constant C>0independent of h such that

kp−

ep_hk₀≤Chkpk₁. (36)

Moreover, if p∈H²(Ω)∩L²₀(Ω)then we have hk∇(p−

eph)k₀+kp−

ephk₀≤Ch²kpk₂. (37)

Now, we are in the position to derive the error estimates of the newly proposed stabilized FEM (5).

Theorem 5.Let(u,p)∈(H¹₀(Ω)∩H²(Ω))²×(H¹(Ω)∩L²₀(Ω))be the solution of problem (1) and(uh,ph)∈ V_h× Q_h⊆ (H₀¹(Ω))²×(H¹(Ω)∩L²₀(Ω))the corresponding stabilized C⁰ piecewise linear (or bilinear) finite element solution given by (5). Then there exists a constant C>0independent ofν,σand h such that

ku−u_hk₀ ≤ C

h r

h²+ ν σ +

rν σ

r h²+ ν

σ+h rν

σ+ h

√σ

kuk₂+ h

√σkpk₁

!

, (38)

ku−uhk₁ ≤ C

h+ rν

σ+

√νh

√σh²+12ν+ h

√σh²+12ν

kuk2+ h

√σh²+12νkpk1

!

, (39)

kp−phk₀ ≤ C ν

r 1+ ν

σh² +ν+√ ν

kuk₂+(h+√ ν)kpk₁

!

, (40)

k∇(p−p_h)k₀ ≤ C k∇(p−

ep_h)k₀+ ν

r 1+ ν

σh² +ν+√ ν

kuk₂+√ νkpk₁

!

, (41)

whereeph∈ Q_his the H¹-seminorm projection of p onQ_hgiven by (35). Moreover, if p∈H²(Ω)∩L²₀(Ω)then we have kp−phk₀ ≤ C

ν r

1+ ν

σh² +ν+√ ν

kuk₂+(h²+√ ν)kpk₂

!

, (42)

k∇(p−ph)k₀ ≤ C ν

r 1+ ν

σh² +ν+√ ν

kuk₂+(h+√ ν)kpk₂

!

. (43)

Proof.Letw_h∈ V_handep_h∈ Q_hbe the functions stated in Lemma 3 and Lemma 4, respectively. Utilizing the inf-sup estimate (9) and the orthogonality property (11), we have

k(uh−wh,ph−eph)kh ≤ sup

0,(vh,q_h)∈V_h×Q_h

B((uh−wh,ph−eph),(vh,qh))

k(v_h,q_h)k_h (44)

= sup

0,(vh,qh)∈Vh×Qh

B((u−w_h,p−

eph),(v_h,qh)) k(vh,qh)kh

.

Since∆wh|_K=0 for allK ∈ T_hand∆vh|_K =0 for allvh ∈ V_handK ∈ T_h, we obtain from Green’s formula [9], (34) and (35) that

B((u−wh,p−eph),(vh,qh))

=σ(u−w_h,v_h)0+ν(∇(u−w_h),∇v_h)0−(p−ep_h,∇ ·v_h)0−(∇ ·(u−w_h),q_h)0

−X

K∈T_h

h²

σh²+12ν(σ(u−w_h)−ν∆u+∇(p−

eph), σv_h+∇qh)0,K

+X

K∈T_h

12ν

σh²+12ν(∇ ·(u−wh),∇ ·vh)0,K

=σ(u−w_h,v_h)₀+ν(∇(u−w_h),∇v_h)₀+(∇(p−

ep_h),v_h)₀+(u−w_h,∇q_h)₀

−X

K∈T_h

h²

σh²+12ν(σ(u−w_h)−ν∆u+∇(p−eph), σvh+∇qh)0,K

+X

K∈T_h

12ν

σh²+12ν(∇ ·(u−wh),∇ ·vh)_0,K

= 12ν

σh²+12ν(∇(p−eph),vh)0+ 12ν

σh²+12ν(u−wh,∇qh)0− h²

σh²+12ν(−ν∆u,∇qh)0

+ 12ν

σh²+12ν(∇ ·(u−wh),∇ ·vh)0

= 12ν

σh²+12ν(eph−p,∇ ·vh)0+ 12ν

σh²+12ν(u−wh,∇qh)0− h²

σh²+12ν(−ν∆u,∇qh)0

+ 12ν

σh²+12ν(∇ ·(u−w_h),∇ ·v_h)0. Applying H¨older inequality onB((u−w_h,p−

ep_h),(v_h,q_h)) with (31), (32) and (36), we have B((u−w_h,p−

ep_h),(v_h,q_h))≤C









r 12ν σh²+12νkp−

ep_hk₀+ 12ν h

√σh²+12νku−w_hk₀ + νh

√σh²+12νk∆uk₀+

r 12ν

σh²+12νk∇ ·(u−w_h)k₀









k(v_h,q_h)k_h

≤C ν

√σ+ νh

√σh²+12ν +

√νh

√σh²+12ν

kuk₂+

√νh

√σh²+12νkpk1

!

k(v_h,qh)kh.

Therefore, form (44) and the above estimate, we get k(u_h−w_h,p_h−

ep_h)k_h≤C ν

√σ+ νh

√σh²+12ν+

√νh

√σh²+12ν

kuk₂+

√νh

√σh²+12νkpk₁

! ,

which combined with the definition of energy norm (8) implies k∇(u_h−w_h)k₀ ≤ C

rν σ+

√νh

√σh²+12ν+ h

√σh²+12ν

kuk₂+ h

√σh²+12νkpk₁

!

, (45)

k(u_h−w_h)k0 ≤ C rν

σ r

h²+ ν σ+h

rν σ+ h

√σ

kuk2+ h

√σkpk1

!

, (46)

k∇(ph−eph)k0 ≤ C ν

r 1+ ν

σh² +ν+√ ν

kuk₂+√ νkpk1

!

. (47)

Now the Poincar´e-Friedrichs inequality [7, 9] ensures that kuh−whk₁ ≤ C

rν σ+

√νh

√σh²+12ν+ h

√σh²+12ν

kuk2+ h

√σh²+12νkpk1

!

, (48)

kph−

ephk₀ ≤ C ν

r 1+ ν

σh² +ν+√ ν

kuk₂+√ νkpk₁

!

. (49)

Finally, combining the triangle inequality with (31), (32), (36), (37) and (45)-(49) yields the conclusion. 2

Note that if the family{T_h}of triangulations is quasi-uniform [4, 38, 39], then the error estimates (38)-(40) can be further improved as follows:

Theorem 6.Let(u,p)∈(H¹₀(Ω)∩H²(Ω))²×(H¹(Ω)∩L²₀(Ω))be the solution of problem (1) and(uh,ph)∈ V_h× Q_h⊆ (H₀¹(Ω))²×(H¹(Ω)∩L²₀(Ω))the corresponding stabilized C⁰ piecewise linear (or bilinear) finite element solution given by (5). Assume that the family{T_h}of triangulations is quasi-uniform. Then there exists a constant C > 0 independent ofν,σand h such that

ku−u_hk₀ ≤ Ch

h+ rν

σ+ 1

√σ

kuk₂+ 1

√σkpk₁

!

, (50)

ku−u_hk₁ ≤ Ch

1+ 1

√σh²+12ν

kuk2+ 1

√σh²+12νkpk1

!

, (51)

kp−phk₀ ≤ C (ν+√

ν)kuk2+(h+√ ν)kpk1

. (52)

Proof. The proof is very similar to that of Theorem 5, except we replace the finite element solution-projection w_h∈ V_hstated in Lemma 3 with the usualL²projection ofuontoV_h. That is, letw_hbe the solution of

(u−w_h,v_h)₀=0 ∀v_h∈ V_h. Since the family{T_h}of triangulations is quasi-uniform, we have (cf. [4, 38, 39])

ku−whk₀≤Ch²kuk2 and k∇(u−wh)k0≤Chkuk2.

With a suitable adjustment in the argument for proving Theorem 5, we obtain k(u_h−w_h,p_h−

ep_h)k_h≤C √

νh+ νh

√σh²+12ν+

√νh

√σh²+12ν

kuk₂+

√νh

√σh²+12νkpk₁

!

. (53)

Similar to that in Theorem 5, estimates (50) and (51) can be immediately derived from (53). On the other hand, combining the Poincar´e-Friedrichs inequality with (8), (53) and (47), we get that

kph−ephk₀ ≤ Ck∇(ph−eph)k0

≤ C







(ν+ √

ν)kuk2+min









 ν

r 1+ ν

σh², ν r

1+σh² ν











kuk2+√ νkpk1









≤ C (ν+√

ν)kuk2+√ νkpk1

. (54)

Now, estimate (52) follows from (54) with the triangle inequality and (36). 2

From the error estimates (38)-(43) and (50)-(52), we can find that the proposed stabilized FEM (5) usingC⁰piece- wiseP₁-P₁ (orQ₁-Q₁) elements is particularly suitable for the generalized Stokes system (1) with a small viscosity νand a large reaction coefficientσ. To the best of our knowledge, they have never been achieved before in the error analysis of other stabilization methods in the literature.

Remark 3.In practical computations, there usually holdsν≤1≤σh²for a certain range ofhor the time stepδt=h² (i.e.,σ=δt⁻¹ =h⁻²) for allh. With this assumption, (38) and (39) can be rewritten as

ku−u_hk₀ ≤ Ch²(kuk₂+kpk1), (55)

ku−uhk₁ ≤ Ch(kuk2+kpk1), (56)

whereC >0 is independent ofν,σandh. Note that, in (55), it does not require the usual assumption of a convex domain. Also, note that no quasi-uniform meshes are required in (55) and (56).

Remark 4.From the continuous inf-sup condition [7], sup

0,v∈(H¹₀(Ω))²

(∇ ·v,q)₀ kvk1

≥Ckqk₀ ∀q∈L²₀(Ω),

and (50), (51) and (54), we can further prove that

kph−ephk₀ ≤ C hkuk2+hkpk1+h(ν+ √

ν)kuk2+h√ νkpk1

+h 1

√σh²+12ν+ √

σh²+12ν+√

σh+σh+√ σν

kuk2

+h 1

√σh²+12ν+ √

σh²+12ν+√ σh

kpk₁

! ,

which combined with the triangle inequality and (36) leads to kp−p_hk₀ ≤ Ch kuk₂+kpk₁+(ν+√

ν)kuk₂+√ νkpk₁ + 1

√σh²+12ν+√

σh²+12ν+√

σh+σh+ √ σν

kuk₂ + 1

√σh²+12ν+√

σh²+12ν+√ σh

kpk₁

!

. (57)

Note that, in (57), the reaction coefficientσ is controlled by a factorh or a factorν. Also, note thatkp−phk₀ is bounded by the minimum of the right-hand sides of (52) and (57).

Remark 5.We remark that ifσh² νand 0 < ν1 then it is reasonable to neglect the div-div stabilization term in the stabilized FEM (5) and in such case, method (5) will be reduced to the Barrenechea-Valentin stabilized FEM (16), provided the mesh sizehis replaced by the element-dependent parameterhK. However, without the divergence- free stabilization effect, we are only able to derive less sharp error estimates than those obtained in Theorem 5. For example, with a suitable adjustment in the argument for proving Theorem 5, we can only obtain

ku−uhk₁≤C h+ 1 σh +

rν σ

!

(kuk2+kpk1), (58)

where the term 1/σhwill dominate in some cases. That is why we need to keep the div-div stabilization term in method (5) even if the associated stabilization parameter 12ν/(σh²+12ν) is very small whenσh²νand 0< ν1.

Finally, we like to emphasize that, with the div-div stabilization term in (5), the error estimates given in Theorem 5 are valid for the entire range of the viscosity 0< ν≤1 as well as the entire range of the reaction coefficientσ≥1.

Note thatσis inversely proportional to the time stepδt.

Remark 6.Note that the stabilization parameters of the proposed stabilized FEM (5) is only designed for theC⁰ piecewiseP1-P1(orQ1-Q1) elements. If the higher-order finite elements are used, the stabilization parameters in the method need to be redesigned. Indeed, quite similar to (19), we may design the two stabilization parameters as

h²

σh²+(4ν/C_k) and (4ν/Ck)

σh²+(4ν/C_k), (59)

where constantCktakes value according to the local inverse inequality,

Ckh²_Kk∆vhk²_0,K ≤ k∇vhk²_0,K ∀vh∈Vk,k≥2,

andVk⊆H¹₀(Ω) denotes theC⁰piecewisePkfinite element space. For example ofk=2, from [15] for various orders of finite elements, one can find thatC₂ =1/48. Thus, ifσh²νand 0< ν1 then our stabilization parameters in method (5) should be replaced by

h²

σh²+192ν and 192ν σh²+192ν.

In the Example 1 of Section 4 below, we will provide some numerical results of the proposed stabilized FEM (5) with the above new stabilization parameters and usingC⁰piecewiseP2-P2elements to verify its validness.

Remark 7.The stabilization parameters in our stabilized FEM (5) are fixed and element-independent. This somehow prevents the use of adaptive meshes. Based on our recent numerical experiences, for an adaptive mesh, it should be better to use element-dependent stabilization parameters, i.e., replacinghbyh_K in the parameters. However, as is often seen in the literature of stabilized FEMs, adaptive mesh computation seems to be generally unnecessary for an effective stabilized FEM because accurate results can be already obtained even for boundary or interior layer problems; see also an example depicted in Figure 11 below.

4. Numerical experiments

This section is devoted to numerical experiments to illustrate the super performance of the proposed stabilized FEM (5) for the generalized Stokes system (1) with a small viscosity and a large reaction coefficient. The performance of the proposed stabilized FEM (5) will be also evaluated against the results from the Barrenechea-Blasco (BB) method (12), the Barrenechea-Valentin (BV) method (16), and the Bochev-Gunzburger-Lehoucq (BGL) method (20). We will consider four test problems that are used in [1, 17, 30]. The relative errors of numerical solutions of the first example will be calculated and the convergence rates will be estimated. By comparing the accuracy and stability, we may conclude that the present stabilized FEM (5) achieves better results than the Barrenechea-Blasco method and the Bochev-Gunzburger-Lehoucq method while comparable with the Barrenechea-Valentin method whenσh² νand 0< ν1.

Example 1.(P1-P1 and P2-P2 finite elements) This example is quoted from [17]. We will first study the detailed convergence behavior of various stabilized FEMs withP1-P1 finite elements on the unit square domainΩ =(0,1)× (0,1). We assume that the smooth exact solution (u,p) of problem (1) is given by

u1(x,y) = −256x²(x−1)²y(y−1)(2y−1), u2(x,y) = −u1(y,x),

p(x,y) = 150(x−1/2)(y−1/2).

Substituting the solution (u,p) into problem (1), we obtain the source-like function f. Notice thatu=0on∂Ωand R

Ωp = 0. We concentrate on the uniform triangular meshes of Ω. Here, a uniform triangular mesh is formed by dividing each square, with side-lengthh^∗in a uniform square mesh, into two triangles by drawing a diagonal line from the left-down corner to the right-up corner. Thus, we havehK=h= √

2h^∗for allK∈ T_h.

Numerical results forν=10<sup>−`</sup>,` =2,3,4, andσ=10^m,m=2,3,4,5, are reported in Table 1 - Table 8, where the convergence orders are estimated. From the numerical results, we may observe the following:

• The newly proposed stabilized FEM (5) usingP₁-P₁ finite elements displays optimal orders of convergence in the L² norm andH¹ norm for both velocity field and pressure. Moreover, the convergence behavior are independent of the small viscosityνand the large reaction coefficientσ. These observations are consistent with the theoretical analysis given in Theorem 5 and Theorem 6. More importantly, the proposed method (5) also

presents a robust behavior in the sense that no pressure instabilities appear, even if we use a rather coarse mesh withh^∗=1/20; see Figure 1 for the contours of pressure approximations.

• The Barrenechea-Valentin method (16) can produce robust results. Their convergence behavior are very similar to those of the newly proposed stabilized FEM (5) whenσh²νand 0< ν1.

• On the contrast, the convergence behavior of the Barrenechea-Blasco method (12) and the Bochev-Gunzburger- Lehoucq method (20) are obviously dependent on the reaction coefficientσ. The accuracy and stability of both methods are deteriorated when the reaction coefficientσis getting larger. Meanwhile, the pressure oscillations appear; see Figure 1.

The results depicted in Figure 1 are using a uniform triangulation withh^∗ = 1/20. For further testing the per- formance of various stabilized FEMs, we next consider theP1-P₁finite elements on an unstructured triangular mesh that is depicted in Figure 2. This mesh is constructed by dividing each side of the squareΩinto equal segments with lengthh^∗ =1/20 and then using theFreeFem++(see [42]) to generate an unstructured quasi-uniform mesh. In the numerical simulation of the present stabilized FEM, the mesh parameterhis taken ash =h^∗ =1/20. The contour plots of pressure approximations produced by various stabilized FEMs are displayed in Figure 3. Again, one can find that the present stabilized FEM (5) and the Barrenechea-Valentin method (16) generate more accurate results with a better stability than the numerical results produced by the other two stabilized FEMs when 0< ν1 andσ1.

We now consider the proposed stabilized FEM (5) with the new stabilization parameters described in Remark 6 and usingC⁰piecewiseP2-P2 finite elements on the uniform triangular meshes. The numerical results are reported in Table 9 and Table 10. In this case, our stabilization method still shows superior accuracy and stability. We remark that the Barrenechea-Valentin stabilized method usingP2-P₂finite elements exhibits the similar behavior.

Table 1.L²relative errors ofu_handphproduced by the proposed stabilized FEM (5) usingP1-P₁finite elements for Example 1 with various viscosityνand reaction coefficientσ.

L²error ν σ h^∗=1/20 1/40 1/60 1/80 1/100 order

10² 2.8661e-2 7.3342e-3 3.2841e-3 1.8549e-3 1.1904e-3 1.98 uh 10⁻² 10³ 2.6070e-2 6.6626e-3 2.9812e-3 1.6828e-3 1.0794e-3 1.98 10⁴ 2.5901e-2 6.6352e-3 2.9711e-3 1.6772e-3 1.0757e-3 1.98 10⁵ 2.5889e-2 6.6367e-3 2.9734e-3 1.6791e-3 1.0771e-3 1.98 10² 2.8738e-2 7.3559e-3 3.2918e-3 1.8578e-3 1.1916e-3 1.98 u_h 10⁻³ 10³ 2.6073e-2 6.6626e-3 2.9803e-3 1.6816e-3 1.0782e-3 1.98 10⁴ 2.5902e-2 6.6353e-3 2.9711e-3 1.6772e-3 1.0756e-3 1.98 10⁵ 2.5889e-2 6.6367e-3 2.9734e-3 1.6791e-3 1.0771e-3 1.98 10² 2.8746e-2 7.3587e-3 3.2931e-3 1.8584e-3 1.1916e-3 1.98 u_h 10⁻⁴ 10³ 2.6074e-2 6.6626e-3 2.9802e-3 1.6815e-3 1.0780e-3 1.98 10⁴ 2.5902e-2 6.6353e-3 2.9711e-3 1.6772e-3 1.0756e-3 1.98 10⁵ 2.5889e-2 6.6367e-3 2.9734e-3 1.6791e-3 1.0771e-3 1.98 10² 4.1816e-3 8.3636e-4 3.3515e-4 1.9508e-4 1.3661e-4 2.01 ph 10⁻² 10³ 4.6320e-3 1.1110e-3 4.6547e-4 2.4804e-4 1.5393e-4 2.13 10⁴ 4.6973e-3 1.1755e-3 5.2090e-4 2.9182e-4 1.8604e-4 2.01 10⁵ 4.7042e-3 1.1829e-3 5.2810e-4 2.9870e-4 1.9248e-4 1.98 10² 4.6380e-3 1.1155e-3 4.6547e-4 2.4273e-4 1.4367e-4 2.21 ph 10⁻³ 10³ 4.6931e-3 1.1710e-3 5.1619e-4 2.8689e-4 1.8085e-4 2.03 10⁴ 4.6998e-3 1.1784e-3 5.2365e-4 2.9426e-4 1.8805e-4 2.00 10⁵ 4.7005e-3 1.1792e-3 5.2444e-4 2.9506e-4 1.8884e-4 2.00 10² 4.6942e-3 1.1723e-3 5.1751e-4 2.8816e-4 1.8204e-4 2.03 p_h 10⁻⁴ 10³ 4.6998e-3 1.1784e-3 5.2362e-4 2.9423e-4 1.8802e-4 2.00 10⁴ 4.7005e-3 1.1791e-3 5.2439e-4 2.9501e-4 1.8880e-4 2.00 10⁵ 4.7005e-3 1.1792e-3 5.2447e-4 2.9509e-4 1.8888e-4 2.00