a new edge stabilization method for the convection-dominated diﬀusion-convection ... · 2018. 7....

A new edge stabilization method for theconvection-dominated diffusion-convection equations⋆

Huoyuan Duan1 and Yu Wei2

1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China,[email protected],

2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China,[email protected]

Abstract. We study a new edge stabilization method for the finite element discretization ofthe convection-dominated diffusion-convection equations. In addition to the stabilization ofthe jump of the normal derivatives of the solution across the inter-element-faces, we addition-ally introduce a SUPG/GaLS-like stabilization term but on the domain boundary other thanin the interior of the domain. New stabilization parameters are also designed. Stability anderror bounds are obtained. Numerical results are presented. Theoretically and numerically,the new method is much better than other edge stabilization methods and is comparable tothe SUPG method, and generally, the new method is more stable than the SUPG method.

Keywords: diffusion-convection equation · stabilized finite element method · stability · errorestimates.

1 Introduction

When discretizing the diffusion-convection equations by the finite element method, the standardGalerkin variational formulation very often produces oscillatory approximations in the convection-dominated case(cf. [14], [3]). As is well-known, this is due to the fact that there lacks controllingthe dominating convection in the stability of the method. For obtaining some stability in the di-rection of the convection, over more than thirty years, numerous stabilized methods have beenavailable. Basically, all the stabilization methods share the common feature: from some residualsrelating to the original problem to get the stability in the streamline direction. The stabilizationmethod is highly relevant to the variational multiscale approach [6]: solving the original prob-lem locally(e.g., on element level) to find the unresolved component of the exact solution in thestandard Galerkin method. Some extensively used stabilization methods are: SUPG(StreamlineUpwind/Petrov-Galerkin) method or SD(Streamline Diffusion) method(cf. [12], [7]), residual-freebubble method(cf. [15]), GaLS method(cf. [22], [4], [9], [11], [16], [10]), local projection method(cf.[2], [3]), edge stabilization method(cf. [17], [20]), least-squares method(cf. [8], [5], [23]), etc. Allthese stabilization methods can generally perform well for the convection-dominated problem, i.e.,the finite element solution is far more stable and accurate than that of the standard method. Theedge stabilization method is such method, which uses the jump residual of the normal derivativesof the exact solution, [∇u · n] = 0 across any inter-element-face F . This method is also known

⋆ Supported by NSFC under grants 11571266, 11661161017, 91430106, 11171168, 11071132, the WuhanUniversity start-up fund, the Collaborative Innovation Centre of Mathematics, and the ComputationalScience Hubei Key Laboratory (Wuhan University).

ICCS Camera Ready Version 2018To cite this paper please use the final published version:

DOI: 10.1007/978-3-319-93713-7_4

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as CIP(continuous interior penalty) method [1] for second-order elliptic and parabolic problems.In [17], the edge stabilization method is studied, suitable for the convection-dominated problem. Ithas been as well proven to be very useful elsewhere(e.g., cf. [18], [19], [20], [21], etc).

In this paper, we study a new edge stabilization method, motivated by the one in [17]. Precisely,letting F int

h be the set of the interior element faces, and hF the diameter of element face F , andF∂

h the set of the element faces on ∂Ω, we define the new edge stabilization as follows:

Jh(u, v) =∑

F∈Finth

βτint,F

∫F

[∇u · n][∇v · n] +∑

F∈F∂h

ατ∂,F

∫F

(−ε∆u+ b · ∇u)(b · ∇v). (1.1)

Here α, β are positive constants, and τint,F , τ∂,F are mesh-dependent parameters, which will bedefined later. The role of τint,F , τ∂,F is approximately the same as h2

F . The new method is consistentin the usual sense(cf. [13], [14]), and it allows higher-order elements to give higher-order convergentapproximations, whenever the exact solution is smooth enough. The first stabilization term on theright of (1.1) is essentially the same as [17]. However, the additional second term on the right of(1.1) is crucial. It ensures that the new method can wholly control the term b ·∇u on every elementand can give the same stability as the SUPG method. Differently, the stabilization in [17] cannothave the same stability. See further explanations later. We analyze the new method, and give thestability and error estimates. Numerical experiments are provided to illustrate the new method, alsoto compare it with the method in [17] and the SUPG method. As will be seen from the numericalresults, in the presence of boundary and interior layers, the new edge stabilization method is muchbetter than the method in [17] and is comparable to the SUPG method. In general, the new edgestabilization is more stable than the SUPG method.

2 Diffusion-Convection Equations

We study the following diffusion-convection problem: Find u such that

−ε∆u+ b · ∇u = f in Ω, u = 0 on ∂Ω. (2.1)

Here ε > 0 denotes the diffusive constant, b the convection/velocity field, and f the source function.The convection-dominated case means that ε ≪ ||b||L∞(Ω); or, the dimensionless quantity Pecletnumber: Pe = V L/ε is very large. Here V and L are the characteristic velocity and the length scalesof the problem. In this paper, we shall use the standard Sobolev spaces [13]. The standard Galerkinvariational problem is to find u ∈ H1

0 (Ω) such that

A(u, v) := ε(∇u,∇v)L2(Ω) + (b · ∇u, v)L2(Ω) = (f, v)L2(Ω) ∀v ∈ H10 (Ω). (2.2)

From (2.2), the finite element method reads as follows: find uh ∈ Uh ⊂ H10 (Ω) such that

A(uh, vh) = (f, vh) ∀vh ∈ Uh. (2.3)

It has been widely recognized whether (2.3) performs well or not depends on whether the followingdiscrete Peclet number is large or not:

Peh = ||b||L∞(Ω)h/ε discrete Peclet number, (2.4)


DOI: 10.1007/978-3-319-93713-7_4

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A new edge stabilization method for the convection-dominated diffusion-convection equations 3

where h is the mesh size of the triangulation Th of Ω. We assume that Ω is partitioned into afamily of triangles, denoted by Th for h > 0 and h → 0, such that Ω = ∪T∈Th

T . The mesh sizeh := maxT∈Th

hT , where hT denotes the diameter of the triangle element T ∈ Th. Concretely, lettingPℓ denote the space of polynomials of degree not greater than the integer ℓ ≥ 1.

Uh = vh ∈ H10 (Ω) : vh|T ∈ Pℓ(T ), ∀T ∈ Th, vh|∂Ω = 0. (2.5)

3 Edge stabilization

In this paper, we shall consider a stabilized Ah(·, ·) by the residual of the normal derivatives ofthe exact solution, i.e.,

[∇u · n] = 0 ∀F ∈ F inth , (3.1)

and the residual of the partial differential equation (2.1), i.e.,

−ε∆u+ b · ∇u− f = 0 ∀T ∈ Th. (3.2)

Corresponding to the new edge stabilization (1.1), we define the right-hand side as follows:

Lh(v) =∑

F∈F∂h

ατ∂,F

∫F

f(b · ∇v), (3.3)

where, denoting by hF the diameter of F ,

τint,F :=h3F ∥b∥2L∞(F )

∥b∥L∞(F )hF + ε, τ∂,F :=

h3F

∥b∥L∞(F )hF + ε. (3.4)

The stabilizing parameters τ∂,F and τint,F are motivated by [9], [10] and [16].Now, the new edge stabilized finite element method is to find uh ∈ Uh such that

Ah(uh, vh) := A(uh, vh) + Jh(uh, vh) = Rh(vh) := (f, vh)L2(Ω) + Lh(vh) ∀vh ∈ Uh. (3.5)

This method is consistent, i.e., letting u be the exact solution of (2.1), we have

Ah(u, vh) = Rh(vh) ∀vh ∈ Uh. (3.6)

4 Stability and Error Estimates

Without loss of generality, we assume that divb = 0.Define

|||uh|||2h := ε||∇uh||2L2(Ω) +∑T∈Th

τT ||b · ∇uh||2L2(T )

+∑

F∈Finth

βτint,F

∫F

|[∇uh · n]|2 +∑

F∈F∂h

ατ∂,F

∫F

|b · ∇uh|2,

where

τT =h2T

||b||L∞(Ω)hT + ε.

Now we can prove the following Inf-Sup condition.


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Theorem 1. ( [27]) The Inf-Sup condition

supvh∈Uh

Ah(uh, vh)

|||vh|||h≥ C|||uh|||h ∀uh ∈ Uh

holds, where the constant C is independent of h, ε,b, uh, only depending on Ω and ℓ.

The above result crucially relies on the following Lemma 1.Denote by Wh = wh ∈ L2(Ω) : wh|T ∈ Pℓ−1(T ), ∀T ∈ Th, and let W c

h = Wh ∩H10 (Ω) ⊂ Uh.

Introduce the jump of v across F ∈ Fh. If F ∈ F inth , which is the common side of two elements

T+ and T−, denoting the both sides of F by F+ and F−, we define the jump [v] = (v|T+)|F+ −(v|T−)|F− . If F ∈ F∂

h , letting T be such that F ⊂ ∂T , we define the jump [v] = (v|T )|F .

Lemma 1. For any wh ∈ Wh, there exists a wch ∈ W c

h, which can be constructed by the averagingapproach from wh, such that, for all T ∈ Th,

||wh − wch||L2(T ) ≤ Ch

1/2T

∑F⊂∂T

||[wh]||L2(F ).

For a linear element, the argument for constructing the finite element function wch ∈ W c

h from thediscontinuous wh ∈ Wh through a nodal averaging approach can be found in [25]. For higher-orderelements, we refer to [24](see Theorem 2.2 on page 2378) for a general nodal averaging approach.An earlier reference is [26], where a similar nodal averaging operator can be found. In [17], a proofis also given to prove a similar result for any wh|T := hTb · ∇uh for all T ∈ Th, but there is a fault.In fact, the authors therein made the mistake in those elements whose sides locate on ∂Ω, e.g., forT ∈ Th with three sides F1, F2, F3, letting F1 ∈ F∂

h and F2, F3 ∈ F inth ,

||wch − wh||L2(T ) ≤ Ch

1/2T

∑F⊂∂T

F∈Finth

||[wh]||L2(F ) = Ch1/2T

∑F2,F3

||[wh]||L2(F ).

This result can hold only for wh|∂Ω = 0. In general, it is not necessarily true that wh = 0 on ∂Ω. Ofcourse, for some problems, say, nonlinear Navier-Stokes equations of the no-slip Dirichlet velocityboundary condition, the convection field b is the velocity itself of the flow, trivially b|∂Ω = 0, andconsequently, the result of [17] will be correct. Now, it is clear the reason why we introduce thesecond stabilization on ∂Ω on the right of (1.1). With this stabilization, we can obtain the resultin Lemma 1 to correct the one of [17] and ensure that the new method is still consistent as usual.If the method is consistent, for a higher-oder element(applicable when the exact solution is smoothenough), a higher-order convergence can be obtained.

Theorem 2. ( [27]) Let u and uh be the exact solution and finite element solution of (2.1) and(3.8), respectively. Then,

|||u− uh|||h ≤ C((

ε1/2hℓ + (||b||L∞(Ω)h+ ε)1/2hℓ)|u|Hℓ+1(Ω)

+(||b||L∞(Ω)h+ ε)−1/2||b||L∞(Ω)hℓ+1|u|Hℓ+1(Ω) + ε1/2hℓ+1|∆u|Hℓ(Ω)

).

In the case of convection-dominated case, i.e., Peh ≫ 1, or ε ≪ ||b||L∞(Ω)h, we find that

|||u− uh|||h ≤ C(ε1/2hℓ + ||b||1/2L∞(Ω)hℓ+1/2)|u|Hℓ+1(Ω) + Cε1/2hℓ+1|∆u|Hℓ(Ω).


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Denote by

||v||2SUPG := ε||∇v||2L2(Ω) +∑T∈Th

τT ||b · ∇v||2L2(T )

the norm which is often used in the SUPG method or other methods such as the residual-free bubblemethod(or which is equivalent to the norms used in the literature for the SUPG method and othermethods, at least, in the convection-dominated case of Peh ≫ 1). Using this norm, we restate theabove error bounds as follows:

||u− uh||SUPG ≤ (ε1/2hℓ + ||b||1/2L∞(Ω)hℓ+1/2)|u|Hℓ+1(Ω) + Cε1/2hℓ+1|∆u|Hℓ(Ω).

In comparison with the SUPG method, here the error bounds are essentially the same [15], only upto a higher-order error bound Cε1/2hℓ+1|∆u|Hℓ(Ω). Therefore, the new edge stabilization methodin this paper is theoretically comparable to the SUPG method. The numerical results will furthershow that the new edge stabilization method is comparable to the SUPG method. Moreover, in thenew edge stabilization method, we have more stability than the SUPG method, i.e., the stability ismeasured in the norm ||| · |||h, where the jump of the normal derivatives of the solution(includingthe normal derivatives of the solution on ∂Ω) are controlled. Numerically, for some meshes, the newedge stabilization method is indeed more stable than the SUPG method.

In comparison with the edge stabilization method [17], we have already observed the advantagesof the new method in this paper. Theoretical results have confirmed the observations. Numericalresults will further give the supports.

5 Numerical experiments

In this section, we give some numerical results for illustrating the performance of the new edgestabilization method, the SUPG method and the edge stabilization method [17] for solving theconvection-dominated diffusion-convection equations with boundary and inner layers.

We study two types of meshes as shown in Fig 1. In the first case(denoted mesh-1) the squareelements are cut into two triangles approximately along the direction of the convection; in the secondcase (mesh-2) they are cut almost perpendicular to the direction of the convection. We choosedomain Ω := (0, 1)2, the convection field |b| = 1 which is constant, f = 0, and nonhomogeneousboundary condition u|∂Ω = U . The geometry, the boundary conditions and the orientation of bare shown in Fig 2. At h = 1/64 and ε = 10−5 and ε = 10−8, using the linear element, we havecomputed the finite element solutions using three methods: SUPG method, BH method [17], Newmethod in this paper. For mesh-1, the elevations and contours are given by Figs 3-4. For mesh-2,the elevations and contours are given by Figs 5-6. For mesh-1, from Figs 3-4, we clearly see thatthe New method is comparable to the SUPG method and is much better than the BH method. Formesh-2, from Figs 5-6, we clearly see that the New method is better than the SUPG method andis still much better than the BH method.


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(a) mesh-1 (b) mesh-2

Fig. 1: Meshes

Fig. 2: Boundary conditions and flow orientation: U = 1 thick edge and U = 0 thin edge.


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Fig. 3: The elevation and contour of the finite element solution, mesh-1, ε = 10−5, h = 1/64.


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DOI: 10.1007/978-3-319-93713-7_4

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DOI: 10.1007/978-3-319-93713-7_4

https://dx.doi.org/10.1007/978-3-319-93713-7_4

a new edge stabilization method for the convection-dominated diﬀusion-convection ... · 2018. 7....

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