Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations with a large reaction coefficient

Analysis of a new stabilized finite element method for the reaction-convection-diffusion equations

with a large reaction coefficient

Huo-Yuan Duan¹, Po-Wen Hsieh², Roger C. E. Tan³, and Suh-Yuh Yang⁴ March 24, 2011; revised July 14, 2011; June 2, 2012; July 21, 2012

Abstract

In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction-convection-diffusion equations.

The equation under consideration involves a small diffusivityε and a large reaction coefficient σ, leading to high P´eclet number and high Damk¨ohler number. In addition to giving error estimates of the approximations inL² andH¹ norms, we explicitly establish the dependence of error bounds on the diffusivity, theL^∞ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh P´eclet number and a large mesh Damk¨ohler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method.

The results obtained are also compared with those of existing stabilization methods.

Keywords:reaction-convection-diffusion equation; boundary layer; interior layer; finite element method; stabilization method.

AMS subject classifications: 65N15; 65N30.

1. Introduction

Let Ω be an open bounded convex polygonal domain inR² with boundary ∂Ω. In this paper, we study the stabilized finite element approximations to the following Dirichlet boundary value

1School of Mathematical Sciences, Nankai University, Tianjin 300071, China. E-mail: hyduan@nankai.edu.cn.

2Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan.

E-mail: pwhsieh0209@gmail.com.

3Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543. E-mail:

scitance@nus.edu.sg.

4Corresponding author. Department of Mathematics, National Central University, Jhongli City, Taoyuan County 32001, Taiwan. E-mail: syyang@math.ncu.edu.tw. Tel.: +886-3-4227151 ext. 65130; fax: +886-3- 4257379.

problem for the reaction-convection-dominated equation:

½ −ε∆u+a· ∇u+σu = f in Ω,

u = 0 on∂Ω, (1)

whereuis the physical quantity of interest (e.g., temperature in heat conduction or concentration of some chemical substance); ε > 0 is a constant diffusion coefficient (called diffusivity); a ∈ (C¹(Ω))² is a given convection (velocity) field with∇ ·a= 0 in Ω; σ >0 is a constant reaction coefficient; and f ∈L²(Ω) is a given source-like function. Here, when we say that the reaction- convection-diffusion equation in problem (1) is reaction-convection-dominated, we mean that the diffusivity ε is relatively small compared with the module of the convection field a or the reaction coefficient σ. In this paper, we are particularly interested in the case of small ε and largeσ. In other words, we will have high P´eclet number P eand high Damk¨ohler number Da,

P e:= U L

ε and Da:= σL U ,

whereU is a characteristic velocity andLa characteristic length of the considered problem.

Typically, problem (1) may arise from the time-discretization of transient convection-diffusion problems, where the reaction coefficient σ is inversely proportional to the time step length. In- deed, the common finite element approach to transient convection-diffusion problems is based on semi-discrete formulations where only the spatial dependence is approximated by finite element methods, and the resulting stiff system is then discretized by applying finite differences in time domain to obtain a fully discrete problem [1]. For problems involving fast chemical reactions, a small time step length is needed in order to account for the stiffness due to the fast reactions [2, 3]. In an implicit time discretization with a small time step, the resulting fully discrete prob- lem is analogue to the finite element formulation of the reaction-convection-diffusion problem (1) with a large reaction coefficientσ (cf. [2]).

It is well known that the solution u of reaction-convection-dominated problem (1) may ex- hibit localized phenomena such as boundary and interior layers [1, 4, 5]. Boundary and interior layers are some narrow regions in the immediate vicinity of the domain boundary∂Ω or in the interior of the domain Ω where the solution or its gradient changes rapidly. The presence of boundary or interior layers is very common in many partial differential equations arising from physical science (say aerodynamics) and engineering applications (say chemical engineering).

The problem (1) we are considering with a small diffusivity and a large reaction coefficient is such a typical model. More specifically, when serving as an intermediate model for the study of

more advanced incompressible Navier-Stokes equations in the time-splitting method, the occur- rence of boundary and interior layers is because of high Reynolds number.

As for the numerical solution of the reaction-convection-dominated problem (1), conventional numerical methods usually produce low accuracy or suffer from instability in the presence of boundary or interior layers [4, 5, 6]. In other words, it is rather difficult to numerically resolve the solution within the neighborhood of the layer regions. For example, the standard Galerkin method using continuous piecewise linear (P1) or bilinear (Q1) elements performs very poorly since large spurious oscillations exhibit not only in the layers but also in other regions. To overcome this difficulty, a big class of so-called stabilized finite element methods (FEMs) has been developed and intensively studied for almost thirty years, see, e.g., [7, 8, 9, 10, 11, 12] or a recent review by Franca, Hauke and Masud [13]. The stabilized FEMs are formed by adding to the standard Galerkin method with some consistent variational terms, relating to the residuals of the partial differential equations. An example for the derivation of such a stabilized FEM is the bubble-enriched method. The method adopts a standard Galerkin formulation but enriches the continuous piecewise P1 (or Q1) elements with suitable bubble functions, e.g., residual-free bubbles, see the early tutorial paper [14] by Brezzi and Russo or [15, 16, 17] and many references cited therein. A static condensation approach leads to a stabilization method [18]. A feature for all stabilization methods is that some mesh-dependent stabilization parameter is involved with, which may be viewed as a discrete version of P´eclet number. It is now clear that the stabiliza- tion parameter plays a key role in the stabilization method. To a great degree, it accounts for why those additional stabilization terms not only can enhance the numerical stability but also can improve the accuracy in the finite element solutions. Nowadays, stabilized FEMs are very popular for the reaction-convection-dominated problem (1).

In this paper, we will continuously focus on developing efficient stabilized FEMs for solving the reaction-convection-dominated problem (1). In [18], Franca and Farhat proposed a so-called unusual stabilized FEM using continuous piecewiseP1 elements for problem (1). For the case of a=0, they proved that the error estimate is optimal inH¹-seminorm independent of the values of εand σ. In addition, forε≤σh²_T for all elements T’s, optimal order inL² norm can also be obtained without using the duality argument. This property is particularly attractive for those fast chemical reaction problems for which the time step length (inversely proportional to σ) is required to be very small. They also considered the problem (1) including the convection term a· ∇u and suggested a stabilization parameter to deal with all the three effects from reaction,

convection and diffusion simultaneously, but no analysis is given therein. Later, Franca and Valentin [19] constructed a new stabilization parameter, for general finite elements, which im- proves the accuracy of the unusual stabilized finite element solutions to problem (1). Moreover, the improvement is also justified therein from an error analysis. Some further results have also been achieved by Duan [20]. More recently, Hauke, Sangalli and Doweidar [21] proposed a new stabilized FEM for solving the reaction-convection-diffusion problems. Their method combines two types of stabilization integrals, namely an adjoint stabilization and a gradient adjoint stabi- lization, and two stabilization parameters are involved therein. These two parameters are chosen based on imposing one-dimensional nodal exactness.

Motivated by the works of Franca-Farhat [18] and Franca-Valentin [19], in the present paper, we devise a new stabilized FEM for problem (1), with emphasis on small diffusivityεand large reaction coefficient σ. As usual, we employ the continuous piecewiseP1 (or Q1) elements and we use the residual of the differential equation in problem (1) to produce the stabilization term.

However, a novel stabilization parameter is designed. A difference from the stabilization param- eters in [18] and [19] is that our stabilization parameter is deterministic and explicit, without the comparisons among the three effect-terms: reaction, convection, and diffusion. Another difference is that our stabilization parameter is always the same no matter if the convection a is present or not in problem (1). When dealing with nonlinear problems, say the incompressible Navier-Stokes equations, where the convection terma depends on the solution, or dealing with those problems where the convection may be nonhomogeneous, the property of the indepen- dence of the convection in our stabilization parameter is very useful. The computational cost can be therefore reduced in evaluating the stabilization parameter. The third difference is that the test function involved in the stabilization term is taken as the form −ε∆v+σv, instead of the adjoint-operator form −ε∆v−a· ∇v+σv in [18] and [19]. Our error analysis is also different from [18] and [19]. We use the finite element solution ue_h of the problem (1) without the convection terma· ∇uas the finite element interpolating function to the exact solution, not the usual nodal finite element interpolantI_hu, see Theorem 4 in Section 3.

Our analysis then gives the error estimates of the finite element solution u_h inL² and H¹ norms. We prove that with respect to the mesh sizeh, the new stabilized FEM shows an optimal error bound of O(h¹) inH¹ norm and a non-optimal error bound inL² norm. However, under additional suitable assumptions, the optimal error estimate of O(h²) in L² norm can also be obtained if we employ the standard duality argument [22, 23]. More importantly, the explicit de-

pendence ofε,kak_∞,σandhare established in all error bounds. From the error analysis, we can see that our stabilization method is particularly suitable for the reaction-convection-dominated problem (1) with small ε and large σ. In fact, the H¹-norm error bound (40) is independent of ε and is inversely proportional to the mesh Damk¨ohler number. It mathematically justified that the proposed stabilization method is valid for all diffusivity and large mesh Damk¨ohler number (see Remark 3 in Section 3). To the best of the authors’ knowledge, in the presence of convection, we are not aware of similar results in the literature.

We present several numerical examples involving boundary or interior layers to illustrate the high performance of the newly proposed stabilized FEM. The numerical results obtained are also compared with those of the unusual stabilized FEMs of Franca-Farhat [18] and Franca-Valentin [19] and the combined stabilized FEM of Hauke-Sangalli-Doweidar [21] for the same problems with the same settings of reaction, convection, diffusion and meshes. Strong numerical evidence indicate that our stabilization method is more stable than that of [18], [19] and [21]. Especially in the case of large reaction coefficient and small diffusivity, the methods of [18] and [19] exhibit badly oscillatory behavior. In contrast, our method still performs very well, and the boundary and interior layers are clearly visible.

The remainder of this paper is organized as follows. In Section 2, we introduce the new stabi- lized FEM for problem (1) and give an error analysis for the case ofa=0. For later use, we also present a brief review of existing stabilized FEMs of Franca-Farhat [18], Franca-Valentin [19]

and Hauke-Sangalli-Doweidar [21]. In Section 3, with the help of the analysis given in Section 2, error estimates of the new stabilized FEM for the case of nonzero convection field (a6=0) are derived. In Section 4, several numerical examples are presented to illustrate the effectiveness of the newly proposed stabilized FEM. Finally, in Section 5 summary and conclusions are given.

2. The new stabilized finite element method

Throughout this paper, we will use the standard notation and definitions for the Sobolev spaces H^m(Ω) for nonnegative integers m (cf. [22, 23, 24, 25]). The associated inner prod- uct and norm are denoted by (·,·)_m and k · k_m, respectively. As usual, L²(Ω) = H⁰(Ω) and H₀¹(Ω) ={v∈H¹(Ω) andv|_∂Ω= 0}. Let{T_h}_0<h≤1 be a family of triangulations of Ω. A trian- gulationT_hof Ω into elementsTconsisting of triangles or quadrilaterals is performed in the usual way; the intersection of any two elements is a vertex, or an edge or empty, and Ω =∪_T∈ThT.

For each triangulation the subscript h ∈(0,1] refers to the level of refinement of the triangu- lation. In particular, the mesh size h is defined as h = max{h_T : T ∈ T_h}, where h_T denotes the diameter of element T. We always assume that the family {T_h}_0<h≤1 of triangulations is shape regular [22, 23, 24]. As usual, (·,·)_m,T and k · k_m,T denote the associated inner product and norm in H^m(T), respectively, whereT is a given element in T_h.

Let V₁ ⊆ H₀¹(Ω) be the continuous piecewise linear (or bilinear) finite element space over the triangulationT_h. The standard interpolation theory [22, 23] ensures that ifu∈H²(Ω) then there exists an interpolationI_hu∈ V₁ such that

ku− I_huk_0,T +h_Tku− I_huk_1,T +h²_Tku− I_huk_2,T ≤Ch²_Tkuk_2,T ∀ T ∈ T_h, (2) whereC is a positive constant independent of T and h_T. We also remark that in this paper we useC to denote a generic positive constant, possibly different at different occurrences, which is always independent of h and other parameters introduced.

The new stabilized FEM for solving problem (1) that we wish to consider can be written as:

F ind u_h ∈ V₁ such that B(u_h, v_h) =L(v_h) ∀v_h ∈ V₁, (3) where the bilinear formB(·,·) and linear form L(·) are, respectively, defined as

B(u, v) := ε(∇u,∇v)₀+ (a· ∇u, v)₀+σ(u, v)₀

− X

T∈Th

τ

³

−ε∆u+a· ∇u+σu,−ε∆v+σv

´

0,T, (4)

L(v) := (f, v)₀− X

T∈Th

τ

³

f,−ε∆v+σv

´

0,T, (5)

and the stabilization parameterτ >0 is given by τ = h²

σh²+ 6ε. (6)

The choice ofτ is inspired by the stabilization parameter designed in [19] (see Remark 1 at the end of this section). It is also interesting to note that the stabilization parameterτ here is inde- pendent of the convection field a as well as the elements T’s. In other words, our stabilization parameterτ is not elementwise defined. It is defined globally.

From (4), we immediately have the following numerical stability estimate of the new stabi- lized FEM (3):

Lemma 1. For everyv_h ∈ V₁ ⊆H₀¹(Ω), we have

B(v_h, v_h) =εk∇v_hk²₀+ 6εσ

σh²+ 6εkv_hk²₀. (7) Proof. Using Green’s formula [25] with the fact that ∇ ·a= 0 in Ω, we obtain (a· ∇v_h, v_h) = 0 for all v_h∈ V₁. This yields the assertion (7) by a direct computation.

Note that the stability estimate (7) ensures the unique solvability of the new stabilized FEM (3). We also remark that the stabilized FEM (3) is a consistent formulation, since the equation in (3) is satisfied when the finite element solutionu_his replaced by the exact solutionu∈H²(Ω).

As a consequence, we have the following orthogonality property:

B(u−u_h, v_h) = 0 ∀ v_h∈ V₁. (8) We now focus on the error estimates of the new stabilized FEM (3) for the problem (1) with a =0. The estimates obtained will be employed to deal with the case of nonzero convection field,a6=0, which is discussed in Section 3. For the clarity of the presentation, when a=0 we use eu and ue_h instead of u and u_h, respectively, to denote the exact solution and its associated stabilized finite element solution for a given source functionfe. In other words, we assume that e

u solves the following problem:

½ −ε∆eu+σeu = fe in Ω, e

u = 0 on∂Ω, (9)

and eu_h ∈ V₁ satisfies

B(eu_h, v_h) =L(v_h) ∀ v_h∈ V₁. (10) Following the ideas of the proof of Theorem 3.1 in [18], we may show the convergence result.

Theorem 2.Assume thata=0. Let eu∈H₀¹(Ω)∩H²(Ω)be the solution of problem (9). Then the stabilized finite element solutionue_h defined by (10) converges toueas follows:

k∇(eu−ue_h)k²₀+ 6σ

σh²+ 6εkeu−eu_hk²₀ ≤Ch²keuk²₂. (11) Proof. LetI_hue∈ V₁ be the interpolant ofueinV₁. Defineη:=eu− I_heuande_h:=I_hue−ue_h∈ V₁. Then the exact errore:=ue−eu_h satisfies e=η+e_h. From (7) and (8), we have

εk∇e_hk²₀+ 6εσ

σh²+ 6εke_hk²₀ =B(e_h, e_h) =B(e_h−e, e_h) =−B(η, e_h)

=

¯¯

¯¯ε(∇η,∇e_h)₀+σ(η, e_h)₀− X

T∈Th

h²

σh²+ 6ε(−ε∆η+ση, σe_h)_0,T

¯¯

¯¯

=

¯¯

¯¯ε(∇η,∇e_h)₀+ X

T∈Th

εσ

σh²+ 6ε(h²∆η, e_h)_0,T + X

T∈Th

6εσ

σh²+ 6ε(η, e_h)_0,T

¯¯

¯¯,

where we have used the fact that ∆e_h = 0 onT for allT ∈ T_h, sincee_h ∈ V₁. Now using H¨older’s inequality and Young’s inequality, CD≤ δ

2C²+ 1

2δD² for all C, D∈R andδ >0, we obtain εk∇e_hk²₀+ 6εσ

σh²+ 6εke_hk²₀ ≤ εk∇ηk₀k∇e_hk₀+ X

T∈Th

εσ

σh²+ 6εkh²∆ηk_0,Tke_hk_0,T

+ X

T∈Th

6εσ

σh²+ 6εkηk_0,Tke_hk_0,T

≤ ε

2k∇ηk²₀+ ε

2k∇e_hk²₀+ X

T∈Th

εσ σh²+ 6ε

³1

6h⁴k∆ηk²_0,T +6

4ke_hk²_0,T

´

+ X

T∈Th

6εσ σh²+ 6ε

³

kηk²_0,T +1

4ke_hk²_0,T

´

≤ ε

2k∇ηk²₀+ ε

2k∇e_hk²₀+ X

T∈Th

6εσ σh²+ 6ε

³

h⁴k∆ηk²_0,T +kηk²_0,T

´

+1 2

6εσ

σh²+ 6εke_hk²₀. (12)

Dividing both left-hand and right-hand sides of (12) by ε and rearranging the equation, it immediately follows from the interpolation property (2) that

1 2

³

k∇e_hk²₀+ 6σ

σh²+ 6εke_hk²₀

´

≤ 1

2k∇ηk²₀+ X

T∈Th

6σ σh²+ 6ε

³

h⁴k∆ηk²_0,T +kηk²_0,T

´

≤ 1

2k∇ηk²₀+ X

T∈Th

6 h²

³

h⁴k∆ηk²_0,T +kηk²_0,T

´

≤ Ch²keuk²₂. (13)

Hence, we have by the triangle inequality that 1

4

³

k∇ek²₀+ 6σ

σh²+ 6εkek²₀

´

≤ 1 4

³

2(k∇ηk²₀+k∇e_hk²₀) + 2 6σ

σh²+ 6ε(kηk²₀+ke_hk²₀)

´

≤ Ch²keuk²₂. This completes the proof.

In the rest of this section, we give a brief review of three well-known stabilized FEMs for problem (1) that are respectively proposed by Franca-Farhat [18], Franca-Valentin [19], and Hauke-Sangalli-Doweidar [21].

In [18], Franca and Farhat introduced the following unusual stabilized FEM:

F ind u_h∈ V₁ such that B_{F F}(u_h, v_h) =L_{F F}(v_h) ∀ v_h∈ V₁, (14)

where the bilinear formB_{F F}(·,·) and linear form L_{F F}(·) are respectively defined as follows:

B_{F F}(u, v) := ε(∇u,∇v)₀+ (a· ∇u, v)₀+σ(u, v)₀

− X

T∈Th

³

−ε∆u+a· ∇u+σu, τ_T(−ε∆v−a· ∇v+σv)

´

0,T, (15) L_{F F}(v) := (f, v)₀− X

T∈Th

³

f, τ_T(−ε∆v−a· ∇v+σv)

´

0,T, (16)

and τ_T is an element-dependent stabilization parameter. This formulation is suggested by using the static condensation procedure for the Galerkin method enriched with bubble func- tions [18]. The “unusual” feature of this stabilization method is the subtraction of a term P

T∈Th(σu, τ_Tσv)_0,T from σ(u, v)₀ of the standard Galerkin FEM.

The second stabilized FEM discussed in Franca-Valentin [19] is almost the same as the method of Franca-Farhat [18]. The only difference between them is the choice of the stabiliza- tion parameter τ_T in (15) and (16) that will be specified later.

More recently, Hauke, Sangalli and Doweidar [21] proposed another new stabilized FEM that combines the adjoint stabilization with a gradient adjoint stabilization:

F ind u_h∈ V₁ such that B_HSD(u_h, v_h) =L_HSD(v_h) ∀ v_h∈ V₁, (17) where the bilinear formB_HSD(·,·) and linear form L_HSD(·) are respectively defined as follows:

B_HSD(u, v) := ε(∇u,∇v)₀+ (a· ∇u, v)₀+σ(u, v)₀

− X

T∈Th

³

−ε∆u+a· ∇u+σu, τ_1T(−ε∆v−a· ∇v+σv)

´

0,T,

− X

T∈Th

³

∇(−ε∆u+a· ∇u+σu), τ_2T∇(−ε∆v−a· ∇v+σv)

´

0,T, (18) L_HSD(v) := (f, v)₀− X

T∈Th

³

f, τ_1T(−ε∆v−a· ∇v+σv)

´

0,T,

− X

T∈Th

³

∇f, τ_2T∇(−ε∆v−a· ∇v+σv)

´

0,T, (19)

and τ_1T and τ_2T are two element-dependent stabilization parameters involved here.

The stabilization parameters τ_T, τ_1T and τ_2T appearing in the above-mentioned stabilized FEMs (14) and (17) are respectively given as follows:

• The Franca-Farhat stabilized FEM [18]: the element stabilization parameter τ_T in

(15) and (16) is chosen as τ_T := h_T

2|a|_pξ(P_T), P_T := 2|a|_ph_T

σh²_T +ε and ξ(P_T) :=

½ P_T, if 0≤P_T <1,

1, if P_T ≥1, (20) where | · |_p denotes the `^p-norm. In practice, we take p = 2, i.e., |a(x, y)|₂ = (a²₁(x, y) + a²₂(x, y))^1/2. It is also understood that if a=0 then the parameterτ_T will be reduced to

τ_T = h²_T

σh²_T +ε. (21)

• The Franca-Valentin stabilized FEM[19]: the element stabilization parameterτ_T in (15) and (16) is chosen as

τ_T := h²_T

σh²_Tξ(P_1T) + (2ε/m₁)ξ(P_2T), P_1T := 2ε m₁σh²_T, P_2T := m₁|a|_ph_T

ε , ξ(z) :=

½ 1, if 0≤z <1,

z, if z≥1. (22)

C₁ X

T∈Th

h²_Tk∆vk²_0,T ≤ k∇vk²₀ ∀ v∈ V₁ and m₁:= min

½1 3, C₁

¾ . In computations, we choosep= 2 and takem₁ = 1/3 since k∆vk_0,T = 0 forv∈ V₁.

• The Hauke-Sangalli-Doweidar stabilized FEM [21]: the element stabilization pa- rameters τ_1T and τ_2T in (18) and (19) are chosen based on imposing one-dimensional nodal exactness. First, we define

θ_1T :=

µ

2D_T + D_T² sinh(P_T)

−cosh(P_T) + cosh(γ_T)−D_T sinh(P_T)

¶₋₁ ,

θ_2T := 3 +D_T² +3D_T P_T +D_T

³

−3D_Tcosh(γ_T) + (−3 +D_T²) sinh(P_T)

´

−2 cosh(P_T) + 2 cosh(γ_T)−D_T sinh(P_T) , where

P_T := kak_∞,Th_T

2ε and D_T := σh_T

kak_∞,T, (23)

kak_∞,T is the L^∞ norm of convection arestricted on the element T, and γ_T is defined as γ_T :=p

P_T(2D_T +P_T).

Then τ_1T andτ_2T in (18) and (19) are respectively given by τ_1T := h_T

kak_∞,Tθ_1T, (24)

τ_2T := kak²_∞,T

6σ³ θ_2T. (25)

The parameters τ_1T and τ_2T must be computed with some care, see [21] for details. In particular, when σ→ ∞, then the parameters tend to

τ_1T = 1

2σ and τ_2T =−h²_T

12σ. (26)

Remark 1. The choice of the stabilization parameter τ given in (6) for the newly proposed stabilized FEM (3) is inspired by the element stabilization parameterτ_T described in (22). One can observe that if a =0, 0< ε ¿1 and σ À 1, then the element stabilization parameterτ_T can be expressed as

τ_T = h²_T σh²_T + 6ε, which is exactly equal to τ given in (6) as we replace h_T by h.

Remark 2. Notice that all the element stabilization parameters τ_T given in (20) and (22) and τ_1T and τ_2T given in (24) and (25) are depending on the module of convection field a and the elements T’s. On the contrary, the stabilization parameterτ given in (6) for our new stabilized FEM (3) is independent of the convection fieldaas well as the elementsT’s. Thus, whenais a spatially varying convection field, i.e.,a=a(x, y), it is easier to implement the newly proposed stabilized FEM (3) than the other three stabilized FEMs. We also remark that although our stabilization parameter τ is defined globally, we have never assumed that the meshes used in the proposed stabilized FEM (3) are structured. Indeed, in Section 4 (see Example 2), we will provide a numerical example of varying convection fielda on an unstructured mesh to illustrate the high performance of the proposed stabilization method.

3. Error estimates of the new stabilized finite element method

We now proceed to estimate the error of the finite element solution u_h of the new stabilized FEM (3). Let u ∈ H₀¹(Ω)∩H²(Ω) be the solution of problem (1) with the convection term a· ∇u. Define a source function by

fe:=−ε∆u+σu. (27)

Then thisuobviously solves problem (9) with the given source function f. As a consequence ofe Theorem 2, we obtain the following results:

Lemma 3. Let u ∈ H₀¹(Ω)∩H²(Ω) be the solution of problem (1) with the convection term a· ∇u. Letue_h∈ V₁ be the unique solution of problem (10) associated with the source function fegiven in (27), that is, eu_h satisfies the following equation:

ε(∇eu_h,∇v_h)₀+σ(eu_h, v_h)₀− X

T∈Th

h²

σh²+ 6ε(−ε∆eu_h+σue_h,−ε∆v_h+σv_h)_0,T

= (−ε∆u+σu, v_h)₀− X

T∈Th

h²

σh²+ 6ε(−ε∆u+σu,−ε∆v_h+σv_h)_0,T, ∀ v_h ∈ V₁.(28)

Then there exists a constantC >0 independent ofε,σ and h such that

ku−eu_hk₁ ≤Chkuk₂, (29)

and

ku−ue_hk₀≤Ch rh²

6 + ε

σ kuk₂. (30)

Proof. The estimate (29) follows estimate (11) combining with the usual Poincar´e-Friedrichs inequality [25], kvk₀ ≤C_pfk∇vk₀ for all v∈H₀¹(Ω). The estimate (30) is a direct consequence from (11).

Now, the error estimates of the newly proposed stabilized FEM (3) can be stated as follows:

Theorem 4. Letu∈H₀¹(Ω)∩H²(Ω) be the solution of problem (1) with the convection term a· ∇u. Let u_h ∈ V₁ ⊆H₀¹(Ω) be the corresponding stabilized finite element solution given in (3). Then there exists a constantC >0independent of ε,a,σ and h such that

ku−u_hk₁ ≤ C

³

h+kak_∞ σ

´

kuk₂, (31)

ku−u_hk₁ ≤ Ch

³

1 +kak_∞

√σε

´

kuk₂, (32)

and

ku−u_hk₀≤Ch

³rh² 6 + ε

σ + kak_∞ σ

´

kuk₂. (33)

Proof. As stated in Lemma 3, letue_h be the unique solution of problem (10) associated with the source functionfegiven in (27). Utilizing the coercivity estimate (7), orthogonality property (8), equation (28) and estimate (29), we have

εk∇(u_h−eu_h)k²₀+ 6εσ

σh²+ 6εku_h−ue_hk²₀ = B(u_h−eu_h, u_h−eu_h)

= B(u_h−u+u−ue_h, u_h−eu_h)

= B(u−eu_h, u_h−eu_h)

= 6ε

σh²+ 6ε(a· ∇(u−ue_h), u_h−ue_h)₀

≤ 6ε

σh²+ 6εCkak_∞hkuk₂ku_h−ue_hk₀, which implies

ku_h−ue_hk₀ ≤Ckak_∞h

σ kuk₂ (34)

and then

εk∇(u_h−eu_h)k²₀ ≤ 6ε

σh²+ 6εCkak_∞hkuk₂ku_h−ue_hk₀

≤ 6ε

σh²+ 6εCkak_∞hkuk₂Ckak_∞h

σ kuk₂. (35)

It follows from (35) that

k∇(u_h−eu_h)k²₀≤C 6

σ(σh²+ 6ε)kak²_∞h²kuk²₂. Thus, we have

k∇(u_h−ue_h)k²₀ ≤C 6

σ²h²kak²_∞h²kuk₂, and

k∇(u_h−ue_h)k²₀ ≤C 6

6σεkak²_∞h²kuk₂.

Now the usual Poincar´e-Friedrichs inequality used in the proof of Lemma 3 ensures that ku_h−eu_hk₁ ≤ Ckak_∞

σ kuk₂, (36)

ku_h−eu_hk₁ ≤ Ckak_∞h

√σε kuk₂. (37)

Finally, combining the triangle inequality with (29), (30), (34), (36) and (37) yields the conclu- sion. This completes the proof.

Remark 3. Mathematically, estimates (32) and (33) ensure the convergence of the stabilized finite element solution u_h in H¹ and L² norms as the mesh size h approaches to zero. More specifically, with respect to the mesh sizeh, the convergence order is optimal inH¹ norm, while it is not optimal in L² norm. Moreover, if we introduce the mesh P´eclet number P e_h and the mesh Damk¨ohler numberDa_h by

P e_h := kak_∞h

2ε , (38)

Da_h := σh

kak_∞ providedkak_∞6= 0, (39)

then according to theH¹-norm error bound (31) and the L²-norm error bound (33), we have ku−u_hk₁ ≤ Ch

³ 1 + 1

Da_h

´

kuk₂, (40)

ku−u_hk₀ ≤ Ch²

³r1

6 + 1

2P e_hDa_h + 1 Da_h

´

kuk₂. (41)

It now becomes clear that if the mesh P´eclet numberP e_h≥1 and the mesh Damk¨ohler number Da_h≥1, then our stabilization method will perform well with respect to bothH¹andL²norms.

As is well known, it is due to large mesh P´eclet number and large mesh Damk¨ohler number that the standard Galerkin method and some stabilization methods fail in capturing boundary or interior layers. Moreover, note that the multiplicative constant in the H¹-norm error bound (40) is independent of ε,a,σ and h. So, the applicability of our method is now clear. Namely, the proposed stabilization method is valid for the entire range of diffusivity ε and the entire

range of large mesh Damk¨ohler numberDa_h≥1. These theoretical results will be confirmed by the numerical experiments given in Section 4.

Remark 4. The non-optimal L²-norm error bound in (33) is obtained without a duality argu- ment. However, if we employ the duality argument [22, 23] with additional assumptions, we can have optimalL²-norm error bound in terms of the mesh size h, i.e., O(h²). That is, on an open bounded convex polygonal domain or an open bounded smooth domain on which the solution of the adjoint problem to problem (1) can be inH²(Ω), from the standard duality argument we can establish the following L²-norm error bound for u−u_h:

ku−u_hk₀ ≤Ch²Λkuk₂, (42) where Λ is given by

Λ = ε

³

1 +kak_∞

√σε

´

+hkak_∞

³

1 +kak_∞

√σε

´ +hσ

³rh² 6 + ε

σ +kak_∞ σ

´

+ 1

σh²+ 6ε n

εσ+hσ¡

kak_∞+σ¢³r h²

6 + ε

σ +kak_∞ σ

´o

. (43)

Clearly, regardless of the values of the physical parameters, as the mesh is refined, or ashtends to zero, we have a quadratic, or optimal,L²-norm convergence order. This quadratic or optimal L²-norm convergence order is obviously compatible with the the linear convergence order in H¹-norm in (32).

Remark 5. In the absence of the convection term, a = 0, the error estimates (31), (32) and (33) become

ku−u_hk₁ ≤ Chkuk₂, (44)

ku−u_hk₀ ≤ Ch rh²

6 + ε

σ kuk₂. (45)

This establishes optimal order of convergence of the approximationu_h inH¹ norm independent of the values of ε and σ. Furthermore, the optimal order of convergence in L² norm can be also ensured provided ε≤ σh². These results are consistent with that obtained in [18] for the problems without involving convection effect.

Remark 6.In this paper, we focus on the case of small diffusivity ε, large reaction coefficient σ and nonzero convection fielda 6=0. In the case where the reaction coefficientσ is small but not zero, however, our new stabilized FEM (3) is still convergent, see (32) and (33). It is also

interesting to note that if σ = 0 particularly then the stabilization term in our new stabilized FEM (3) will be canceled out since v_h ∈ V₁. In this case, our method becomes the standard Galerkin method, and it is well known that the finite element solution would only be stable when the mesh P´eclet number P e_h := (kak_∞h)/(2ε) is not too large.

Remark 7. Regarding spatially varying diffusivityε=ε(x, y) or spatially varying reaction co- efficient σ =σ(x, y), we have not considered such cases in this paper, as most of the literature did. If we have to deal with spatially varying smooth diffusivity and reaction coefficient, we may use piecewise constant finite element interpolations to approximate them, and then use the present stabilization method to seek the finite element solution. This will be dealt with elsewhere.

4. Numerical experiments

This section is devoted to numerical experiments. We will consider several typical test problems that are frequently used in the literature. The performance of the newly proposed stabilized FEM (3) will be evaluated against the results from the Franca-Farhat stabilized FEM (14) with (20), the Franca-Valentin stabilized FEM (14) with (22), and the Hauke-Sangalli-Doweidar sta- bilized FEM (17) with (24) and (25). By comparing the accuracy and stability, we conclude that when the reaction-convection-diffusion problem (1) possesses a small diffusivity ε and a large reaction coefficientσ, the present stabilized FEM can always provide accurate and stable results while the approximations generated by the other stabilized FEMs may exhibit spurious oscillations near the boundary or interior layer regions.

Example 1. In this example, we will study the convergence behavior of the newly proposed stabilized FEM (3) by constructing a problem with an exact solution. Taking the domain Ω := (0,1)×(0,1) and the convection fielda= (a₁, a₂)^> := (1/2,√

3/2)^>, we assume that the exact solutionu of problem (1) is given by

u(x, y) = µ x²

2a₁+εx a²₁+

³ 1 2a₁+ ε

a²₁

´e^−aε1 −e^{−aε1(1−x)} 1−e^−aε1

¶µ y² 2a₂+εy

a²₂+

³ 1 2a₂+ ε

a²₂

´e^−aε2 −e^{−aε2(1−y)} 1−e^−aε2

¶ . Then u = 0 on ∂Ω. Substituting the solution u into problem (1), we can obtain the inhomo- geneous source-like function f. Notice that the solution is dependent on the diffusivityεwhile it is independent of the reaction coefficient σ. When the diffusivityεis small enough, a strong boundary layer appears near the right up corner (cf. Figures 2-5).

We will concentrate on various stabilized FEMs on uniform square meshes and uniform triangular meshes. Here, a uniform triangular mesh is formed by dividing each square, with side-length h^∗ in a uniform square mesh, into two triangles by drawing a diagonal line from the left-down corner to the right-up corner. In the computations of all stabilized FEMs, we use the continuous piecewise linear (P1) elements for uniform triangular meshes while the continuous piecewise bilinear (Q1) elements for uniform square meshes.

We consider the values of ε= 0.01, σ = 10<sup>`</sup> for `= 2,3,4 and compare the error behavior of the solutions generated by the new stabilized FEM (3) and the other three stabilized FEMs for h^∗ = 2<sup>−`</sup>, ` = 5,6,7,8. Notice that the mesh parameter is given by h = √

2h^∗, which is the diameter of each triangular (or square) element. The numerical raw data for P1 elements are reported in Table 1, where the asymptotic convergence orders as h^∗ → 0⁺ are estimated using the results of h^∗ = 1/128 and h^∗ = 1/256. The corresponding mesh P´eclet numbers, P e_h := (kak_∞h)/(2ε), and mesh Damk¨ohler numbers,Da_h:= (σh)/kak_∞, are also collected in Table 2. From Table 1, for a not too small diffusivityε= 0.01, we may observe that

• For the present stabilized FEM (3), the asymptotic convergence order in H¹ norm is optimal, and it is near optimal inL² norm when the reaction coefficientσ is not too large.

These observations are consistent with the error estimates (32) and (42).

• The accuracy of the numerical results of the Hauke-Sangalli-Doweidar stabilized FEM is higher than that of the other three stabilized FEMs, and the Franca-Farhat stabilized FEM shows a lower accuracy.

• The error behavior of the results generated by the Franca-Valentin stabilized FEM and the newly proposed stabilized FEM are very similar, especially for large reaction coefficients.

It has been pointed out in [19] that, under the presence of convection (a6=0), the element parameterhthat yields the best numerical results is computed using the largest streamline dis- tance in the element. In other words, the mesh parameterhis defined as the largest diameter of element in the direction ofa. We illustrate in Figure 1 how this diameter is measured. We then perform the same numerical test as that reported in Table 1 but using the new mesh parameter, h=p

4/3h^∗. The results are presented in Table 3 while the corresponding mesh P´eclet numbers and mesh Damk¨ohler numbers are collected in Table 4. By comparing the results reported in Table 1 and Table 3, we can find that with this new mesh parameter, the Franca-Farhat, the Franca-Valentin and the present stabilized FEMs show a better accuracy; however, the accuracy

of the Hauke-Sangalli-Doweidar stabilized FEM seems no significant improvement.

We also perform the numerical simulations of all stabilized FEMs using Q₁ elements. The results are presented in Table 5 and Table 6. Not surprisely, we reach the same conclusions with the stabilized FEMs using P₁ elements.

a

h

h

h

*

h h *

* h

Figure 1. The largest diameterhof element in the direction ofa.

Next, we consider the influence of large σ on the approximations. When the diffusivityεis getting small enough, sayε= 10⁻⁴, a strong boundary layer appears near the upper right corner.

Numerical results of various stabilization methods using Q₁ elements with σ = 10²,10³,10⁴, h^∗ = 1/32 and h = p

4/3h^∗ are displayed in Figures 2-5, where the mesh P´eclet number and the mesh Damk¨ohler number are also given. From the numerical results, we may observe the following:

• The Franca-Farhat stabilized FEM totally loses the accuracy of solution for all reaction coefficients σ= 10²,10³,10⁴ (see Figure 2).

• Whenσ is not too large, σ= 10², a little bit of oscillation occurs near the boundary layer region in approximate solutions of the Franca-Valentin and the present stabilized FEMs.

However, the solution oscillation will not disappear in the Franca-Valentin stabilized FEM even if the σ is rather large, say σ= 10⁴ (see Figure 3). In contrast, increasing the value of σ enhances the stability of the newly proposed stabilized FEM (see Figure 5), and no oscillations are seen near the upper right corner. Moreover, in this example, the Hauke- Sangalli-Doweidar stabilized FEM always shows accurate and stable results for all reaction coefficients σ= 10²,10³,10⁴ (see Figure 4).

Table 1. Relative errors of the numerical solutionu_hproduced by various stabilized FEMs using P₁ elements of Example 1 with ε= 0.01, where the mesh parameter h is defined as the diameter of each element, i.e., h=√

2h^∗.

Stabilized FEM Norm σ h^∗= 1/32 h^∗= 1/64 h^∗= 1/128 h^∗= 1/256 order≈ 10² 0.542664 0.229839 0.084465 0.025609 1.72 Franca-Farhat L² 10³ 0.495957 0.206057 0.079441 0.026796 1.57 10⁴ 0.490113 0.202888 0.078810 0.027331 1.53 10² 0.162923 0.060560 0.012592 0.003052 2.04 Franca-Valentin L² 10³ 0.157827 0.066537 0.020575 0.004090 2.33 10⁴ 0.157181 0.066382 0.020998 0.006097 1.78 10² 0.098339 0.036609 0.011132 0.002959 1.91 Hauke-Sangalli- L² 10³ 0.086737 0.026260 0.008758 0.002889 1.60

Doweidar 10⁴ 0.085896 0.024157 0.005822 0.001697 1.78

10² 0.136393 0.047246 0.013455 0.003497 1.94

Present L² 10³ 0.151014 0.058636 0.019052 0.005386 1.82

10⁴ 0.152707 0.060281 0.020250 0.006007 1.75 10² 0.750730 0.537620 0.319686 0.144556 1.15 Franca-Farhat H¹ 10³ 0.720791 0.501352 0.310167 0.161819 0.94 10⁴ 0.717177 0.496635 0.309019 0.167606 0.88 10² 0.590442 0.357501 0.178308 0.089704 0.99 Franca-Valentin H¹ 10³ 0.586519 0.366109 0.192290 0.091127 1.08 10⁴ 0.586033 0.366125 0.194467 0.096778 1.01 10² 0.559820 0.334683 0.177227 0.089663 0.98 Hauke-Sangalli- H¹ 10³ 0.564546 0.334462 0.176277 0.089782 0.97

Doweidar 10⁴ 0.565540 0.337248 0.177856 0.089571 0.99

10² 0.575448 0.343122 0.178963 0.089911 0.99

Present H¹ 10³ 0.586561 0.360107 0.190556 0.093545 1.03

10⁴ 0.587967 0.363196 0.194915 0.096980 1.01

Table 2. The mesh P´eclet numbers and mesh Damk¨ohler numbers corresponding to Table 1 and Table 5, whereε= 0.01,a= (1/2,√

3/2)^>, and h=√ 2h^∗.

σ h^∗= 1/32 h^∗= 1/64 h^∗= 1/128 h^∗= 1/256

10² 1.9137 0.9568 0.4784 0.2392

mesh P´eclet number P e_h 10³ 1.9137 0.9568 0.4784 0.2392

10⁴ 1.9137 0.9568 0.4784 0.2392

10² 5.1031 2.5516 1.2758 0.6379

mesh Damk¨ohler numberDa_h 10³ 51.0310 25.5155 12.7578 6.3789 10⁴ 510.3104 255.1552 127.5776 63.7888