A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ...

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Mathematical Models and Methods in Applied Sciences Vol. 15, No. 7 (2005) 1119–1139 c World Scientific Publishing Company

A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

RODOLFO ARAYA∗ and ABNER H. POZA† Departamento de Ingenier´ıa Matem´ atica, Universidad de Concepci´ on Casilla 160-C, Concepci´ on, Chile ∗[email protected] †[email protected] ERNST P. STEPHAN Institut f¨ ur Angewandte Mathematik, Universit¨ at Hannover Welfengarten 1, D-30167 Hannover, Germany [email protected]

Received 13 September 2004 Revised 14 December 2004 Communicated by F. Brezzi In this work we introduce a new a posteriori error estimate of hierarchical type for the advection-diffusion-reaction equation. We prove the equivalence between the energy norm of the error and our error estimate using an auxiliary linear problem for the residual and an easy way to prove inf–sup condition. Keywords: Advection problem; boundary layers; a posteriori error estimate; bubble functions. AMS Subject Classification: 65N15, 65N30, 65J15

1. Introduction In this work we deal with the advection-diffusion-reaction equation. This kind of problems arises in many applications, for instance when linearizing the Navier– Stokes problem, or in the transport of a pollutant in a river, etc. Especially interesting is the case when the convective term is dominant, in this case the solution of the equation frequently has exponential or parabolic boundary layers (for details, see Roos10 ). The standard Galerkin approximation usually fails in this situation because this method introduces nonphysical oscillations. To overcome this problem there exists the standard technique of adding some numerical diffusion terms ∗Corresponding

author 1119

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R. Araya, A. H. Poza & E. P. Stephan

to the variational formulation to stabilize the finite element solution. Some examples of this approach are the streamline upwind Petrov–Galerkin method (SUPG) (Brooks3 ) and the unusual stabilized finite element method (USFEM) (Franca6 ). Unfortunately, this is not enough to stop the oscillations because normally the computed solutions exhibit overshooting and undershooting phenomena. Our goal is to construct an a posteriori error estimate of hierarchical type with some suitable bubble functions, to capture accurately the different layers of the solution. The design of an a posteriori error estimate for this kind of problem is not an easy task as the standard error estimates involve an equivalence constant which depend on negative powers of the diffusion parameter, which leads to poor results in the advective or reactive dominant case. A few references are: Verf¨ urth13 and 1 Araya for a residual error estimator combined with stabilized schemes, Sangalli11 to consider the relation between error estimates and the residual free bubbles method, and Knopp8 to study an error estimated related with the shock capturing techniques. A final reference is John’s7 work, where a numerical comparison between different a posteriori error estimates is carried out. In contrast with the standard hierarchical approach, in our analysis we do not need a saturation assumption to hold. Our proof of the a posteriori error estimate is based on a discrete Babuska–Brezzi condition with specific factors depending on the diffusion parameter ε and the mesh size. For a different approach in how to overcome orfler and and prove the saturation assumption see the papers by Nochetto9 and D¨ Nochetto.5 As far as we know, this is the first time that a hierarchical a posteriori error estimator, which does not use the saturation assumption, is applied to the advection-diffusion equation. The paper is organized as follows. In Sec. 2 we introduce the advection-diffusionreaction problem under consideration and list some auxiliary results. In Sec. 3 we introduce the standard residual equation and a new bilinear form used to obtain the main result. In Sec. 4 we perform the error analysis and derive the hierarchical error estimate. We introduce an (LBB) condition, which is used to prove the equivalence between the hierarchical error estimate and the energy norm of the true error. We also give an example for a space of bubble functions satisfying the discrete (LBB) condition. Finally, in Sec. 5 we report some numerical results. 2. The Boundary Value Problem We consider the advection-diffusion-reaction equation −ε ∆u + a · ∇u + b u = f u=0

in Ω, on ΓD ,

∂u = g on ΓN , ∂n ¯D ∪ where Ω ⊂ R2 is a bounded polygonal domain with Lipschitz boundary Γ = Γ 2 2 ¯ N and ΓD ∩ ΓN = ∅, f ∈ L (Ω), g ∈ L (ΓN ) and n is the outer normal vector Γ ε

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to Γ. For this problem we make the following assumptions: (H1) a ∈ W 1,∞ (Ω)2 and b ∈ L∞ (Ω). (H2) − 21 ∇ · a + b ≥ 1. (H3) Γ− := {x ∈ Γ : a(x) · n(x) < 0 } ⊂ ΓD . Let H := {v ∈ H 1 (Ω): v = 0 on ΓD }. The standard variational formulation of problem (P) is: Find u ∈ H such that B(u, v) = (f, v)L2 (Ω) + (g, v)L2 (ΓN ) ,

∀ v ∈ H,

(2.1)

where B(u, v) := ε(∇u, ∇v)L2 (Ω) + (a · ∇u, v)L2 (Ω) + (bu, v)L2 (Ω) . We define a symmetric bilinear form C : H × H → R by ε ∇u · ∇v + u v, C(u, v) := Ω

(2.2)

(2.3)

Ω

which we use to define the energy norm |||v|||Ω := C(v, v)1/2

∀ v ∈ H.

(2.4)

Lemma 2.1. The variational problem (2.1) has a unique solution and B(v, v) ≥ |||v|||2Ω

∀ v ∈ H.

(2.5)

Proof. The result is a simple consequence of the assumptions (H1) to (H3) and integration by parts. Let {Th }h>0 be a family of shape-regular partitions of Ω into triangles. For k ∈ N, let Hh := {ϕ ∈ H : ϕ|T ∈ Pk , ∀ T ∈ Th }, where Pk denote the space of polynomials of degree at most k. We consider a stabilized formulation of (2.1) given by: Find uh ∈ Hh such that Bδ (uh , vh ) = Fδ (vh ) ∀ vh ∈ Hh .

(2.6)

The definitions of Bδ and Fδ depend on the choice of the stabilized scheme. See Brezzi,2 Brooks,3 Franca6 and Knopp8 for different examples of stabilization algorithms. In this work we will use the following notation a b ⇔ a ≤ c b, a b⇔a b

and b a,

where the positive constant c is independent of the mesh size h and the diffusion parameter ε.

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Finally, we denote by Ih : L2 (Ω) → Hh the interpolation operator of Clément.4 This operator satisfies the following local error estimates ωT ) we have that Lemma 2.2. For all T ∈ Th , F ⊂ ∂T and v ∈ H 1 (˜ v − Ih v0,T min{hT ε−1/2 , 1} |||v|||ω˜ T v − Ih v0,F ε−1/4 min{hT ε−1/2 , 1}1/2 |||v|||ω˜ T where ω ˜ T :=

|||Ih v|||T |||v|||ω˜ T ,

T ∩T =φ

T .

Proof. See Lemma 3.2 in Verf¨ urth.13 3. The Residual Equation Let Eh denote the set of all edges in Th . We can decompose Eh in the following way Eh = Eh,Ω ∪ Eh,D ∪ Eh,N , where Eh,Ω , Eh,D and Eh,N are the sets of internal, Dirichlet and Neumann edges, respectively. It is clear that the error eh ∈ H, defined by eh := u − uh ,

(3.1)

B(eh , v) = (f, v)L2 (Ω) + (g, v)L2 (ΓN ) − B(uh , v) v ∈ H.

(3.2)

satisfies the residual equation:

Instead of working with this residual equation to obtain some knowledge about eh , we will use the following elliptic equation: Find e˜ ∈ H such that C(˜ e, v) = B(eh , v)

∀ v ∈ H.

(3.3)

Remark 3.1. Equation (3.3) has a unique solution e˜ because the symmetric bilinear form C is elliptic. Also it is clear that (3.3) is equivalent to C(˜ e, v) = Rh (v)

∀ v ∈ H.

(3.4)

Here the residual functional Rh : H → R is given by Rh (v) := (f, v)L2 (Ω) + (g, v)L2 (ΓN ) − B(uh , v) or, equivalently, Rh (v) :=

∀ v ∈ H,

(RT , v)L2 (T ) + (RF , v)L2 (F ) , T ∈Th

(3.5)

(3.6)

F ∈Eh

where RT := (f + ε∆uh − a · ∇uh − buh )|T ,

(3.7)

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and

 ∂uh   − ε   ∂nF F   ∂uh RF := g−ε   ∂n  F    0

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if F ∈ Eh,Ω , if F ∈ Eh,N ,

(3.8)

if F ∈ Eh,D ,

here [·]F stands for the jump across F in the direction of nF (see Verf¨ urth13 ). The next result gives us an equivalence result between the energy norm of the error eh and the solution e˜ of (3.3). Lemma 3.1. Let eh be the error defined in (3.1) and e˜ the solution of (3.3). Then e||| ≤ |||eh ||| ≤ |||˜ e|||. (1 + b∞,Ω + ε−1/2 a∞,Ω )−1 |||˜ Proof. Using (2.5) and (3.3) we have that |||eh |||2 ≤ B(eh , eh ) = C(˜ e, eh ) ≤ C(˜ e, e˜)1/2 C(eh , eh )1/2 = |||˜ e||||||eh |||, which give us the upper bound. For the lower bound we proceed as follows: e, e˜) = B(eh , e˜) |||˜ e|||2 = C(˜ = (ε∇eh , ∇˜ e)L2 (Ω) + (a · ∇eh , e˜)L2 (Ω) + (beh , e˜)L2 (Ω) ≤ ε1/2 ∇eh 0,Ω ε1/2 ∇˜ e0,Ω + (a · ∇eh , e˜)L2 (Ω) + b∞,Ω eh 0,Ω ˜ e0,Ω ≤ |||˜ e||||||eh |||(1 + b∞,Ω ) + ε−1/2 a∞,Ω ε1/2 ∇eh 0,Ω ˜ e0,Ω ≤ |||˜ e||||||eh |||(1 + b∞,Ω ) + ε−1/2 a∞,Ω |||eh ||||||˜ e|||. The result follows dividing by |||˜ e|||. 4. Hierarchical Error Estimate Consider a finer finite element space Wh such that Hh ⊂ Wh ⊂ H. We assume that there exist subspaces H1 , . . . , Hm of Wh such that Wh = H0 +

m

Hi ,

(4.1)

i=1

with H0 = Hh . Associated with each subspace Hi there is a projection operator Pi : H → Hi given by the solution of the local problem C(Pi v, wi ) = C(v, wi ),

∀ wi ∈ Hi , Pi v ∈ Hi .

(4.2)

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With the above definitions our hierarchical a posteriori error estimator ηH is defined by the local additive decomposition

m 1/2 C(Pi e˜, Pi e˜) . (4.3) ηH := i=1

Notice that Pi e˜ ∈ Hi solves the local problem: C(Pi e˜, vi ) = Rh (vi ),

∀ vi ∈ Hi .

(4.4)

If the subspaces Hi are sufficiently local, the computation of Pi e˜ will be inexpensive to carry out. In the following development, we will define one subspace HTb for each element T ∈ Th and also one HFb for each edge F ∈ Eh,Ω ∪ Eh,N . In this way, (4.3) can be written as

1/2 ηH = C(PT e˜, PT e˜) + C(PF e˜, PF e˜) . (4.5) T ∈Th

F ∈Eh,Ω ∪Eh,N

At this stage we assume that the subspaces HTb and HFb , which we called bubble subspaces, satisfy HTb ⊂ H01 (T ) and HFb ⊂ H01 (ωF ), where ωF := T. F ⊂∂T

Finally, we define the following mesh-dependent constants θT := min{hT ε−1/2 , 1} θF := ε

−1/4

min{hF ε

if T ∈ Th ,

−1/2

, 1}

1/2

if F ∈ Eh,N ∪ Eh,Ω .

Our main assumption is the next (LBB) type condition: (LBB) There is a positive constant β, independent of h and ε, such that sup b bT ∈HT

sup b bF ∈HF

(bT , RT )L2 (T ) ≥ β θT RT 0,T , CT (bT , bT )1/2 (bF , RF )L2 (F ) ≥ β θF RF 0,F , CωF (bF , bF )1/2

∀ T ∈ Th , ∀ F ∈ Eh,N ∪ Eh,Ω .

Remark 4.1. In Sec. 4.1 we will give an example of bubble subspaces satisfying this (LBB) condition. Lemma 4.1. If the (LBB) condition holds, then C(PT e˜, PT e˜)1/2 θT−1 v0,T Rh (v)

T ∈Th

+

F ∈Eh,Ω ∪Eh,N

for all v in H.

1/2

C(PF e˜, PF e˜)

+

1/2

C(PT e˜, PT e˜)

T ⊂ωF

θF−1 v0,F ,

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Proof. Using the Cauchy–Schwarz inequality, the (LBB) condition and the definition of the projection PT e˜, we obtain θT RT 0,T ≤

(bT , RT )L2 (T ) 1 sup β bT ∈HTb CT (bT , bT )1/2

=

Rh (bT ) 1 sup β bT ∈H b CT (bT , bT )1/2 T

=

C(PT e˜, bT ) 1 sup β bT ∈H b C(bT , bT )1/2 T

C(PT e˜, PT e˜)1/2 . Also, for F ∈ Eh,Ω ∪ Eh,N we have (bF , RF )L2 (F ) 1 sup β bF ∈H b C(bF , bF )1/2 F Rh (bF ) − T ⊂ωF (RT , bF )L2 (T ) 1 sup = β bF ∈H b C(bF , bF )1/2 F

θF RF 0,F ≤

≤

1 C(PF e˜, bF ) sup β bF ∈H b C(bF , bF )1/2 F

RT 0,T bF 0,T 1 sup β bF ∈HFb C(bF , bF )1/2 T ⊂ωF

C(PF e˜, PF e˜)1/2 + θT RT 0,T , +

(4.6)

T ⊂ωF

since for each T ⊂ ωF we have

bF 20,T bF · bF = T CT (bF , bF ) CT (bF , bF )

ε h−2 T

· bF b · bF + T bF · bF T F

b T F

1 εh−2 T

+1

1 1 = −2 −1 h2 , 1}−1 min{ε max{εhT , 1} T

θT2 . At this point, we need to assume that the solution uh of the stabilized scheme (2.6) satisfies

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(STAB) For all v ∈ H we have that

B(u − uh , Ih v)

θT RT 0,T |||v|||ω˜ T .

T ∈Th

This assumption is satisfied by several stabilized schemes (see Sec. 5 for an example). Using (STAB) assumption we can prove the following result Lemma 4.2. If (LBB) and (STAB) hold, then C(˜ e, e˜)

C(PT e˜, PT e˜) + T ∈Th

C(PF e˜, PF e˜).

(4.7)

F ∈Eh,Ω ∪Eh,N

Proof. From Lemma 4.1, we have C(PT e˜, PT e˜)1/2 θT−1 v0,T Rh (v)

T ∈Th

+

1/2

C(PF e˜, PF e˜)

+

F ∈Eh,Ω ∪Eh,N

θF−1 v0,F .

1/2

C(PT e˜, PT e˜)

T ⊂ωF

(4.8) Let Ih : H → Hh be the interpolation operator defined in Sec. 2. Using inequality (4.8) with v = e˜ − Ih e˜, (STAB) assumption and the Cauchy–Schwarz inequality, we have e) C(˜ e, e˜) = Rh (˜ = Rh (˜ e − Ih e˜) + Rh (Ih e˜)

C(PT e˜, PT e˜)1/2 θT−1 ˜ e − Ih e˜0,T

T ∈Th

T ⊂ω

+

T ∈Th

·

θT RT 0,T |||˜ e|||ω˜ T

C(PT e˜, PT e˜) +

1/2 C(PF e˜, PF e˜)

F ∈Eh,Ω ∪Eh,N

θT−2

˜ e−

Ih e˜20,T

+

1/2 θF−2

˜ e−

Ih e˜20,F

F ∈Eh,N ∪Eh,Ω

C(PT e˜, PT e˜)1/2 |||˜ e|||ω˜ T .

T ∈Th

F ∈Eh,Ω ∪Eh,N

F

T ∈Th

+

C(PF e˜, PF e˜)1/2

C(PT e˜, PT e˜)1/2 θF−1 ˜ e − Ih e˜0,F

T ∈Th

+

+

(4.9)

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Then using the properties of the operator Ih given in Lemma 2.2 and the regularity of the mesh, we obtain θT−2 ˜ e − Ih e˜20,T C(˜ e, e˜) T ∈Th

θF−2 ˜ e − Ih e˜20,F C(˜ e, e˜).

F ∈Eh,N ∪Eh,Ω

In fact,

θT−2 ˜ e − Ih e˜20,T

T ∈Th

2

θT−2 min{h2T ε−1 , 1}|||˜ e|||ω˜ T C(˜ e, e˜).

(4.10)

T ∈Th

In the same way, we have θF−2 ˜ e − Ih e˜20,F

F ∈Eh,N ∪Eh,Ω

2

θF−2 ε−1/2 min{hT ε−1/2 , 1}|||˜ e|||ω˜ T

F ∈Eh,N ∪Eh,Ω

C(˜ e, e˜). The main result of this section is the following theorem: Theorem 4.1. Let e˜ be the solution of the variational problem (3.3). If (LBB) and (STAB) conditions hold, then |||˜ e|||Ω ηH , where ηH is given by (4.5) and where the equivalence constants are independent of h and ε. Proof. For simplicity we write C(PT e˜, PT e˜) + T ∈Th

C(PF e˜, PF e˜) =

m

C(Pi e˜, Pi e˜),

(4.11)

i=1

F ∈Eh,N ∪Eh,Ω

for some positive integer m. By construction of Pi e˜ and the Cauchy–Schwarz inequality, we have that 2 m 2 m 2

m C(Pi e˜, Pi e˜) = C(˜ e, Pi e˜) = C e˜, Pi e˜ i=1

i=1

≤ C(˜ e, e˜) C

i=1

m i=1

Pi e˜,

m

Pi e˜ .

(4.12)

i=1

Using Cauchy–Schwarz’s inequality again, we obtain m m m C Pi e˜, Pi e˜ = C(Pi e˜, Pj e˜) i=1

i=1

i=1 j∈Ni

m 1 1 C(Pi e˜, Pi e˜) + C(Pj e˜, Pj e˜) ≤ 2 2 i=1 j∈Ni m

≤ Kmax

i=1

C(Pi e˜, Pi e˜),

(4.13)

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where Ni denotes the set of indices of the spaces Hj which are neighbors of the space Hi , i.e. Ni := { j : ∃ vj ∈ Hj and vi ∈ Hi such that C(vi , vj ) = 0},

(4.14)

and where Kmax denotes the maximal number of neighbors, i.e. Kmax := max{card(Nl ) : 1 ≤ l ≤ m}.

(4.15)

Then, from (4.12) and (4.13) we finally obtain m

C(Pi e˜, Pi e˜) ≤ Kmax C(˜ e, e˜).

(4.16)

i=1

The other inequality is a direct consequence of Lemma 4.2. Remark 4.2. Using the fact that HTb ⊂ H01 (T ) and HFb ⊂ H01 (ωF ), and the regularity of the mesh, we have that Kmax is uniformly bounded. If we use the equivalence given by Lemma 3.1 and Theorem 4.1, we obtain the following result: Theorem 4.2. If (LBB) and (STAB) conditions holds, then 2 2 2 ηH,T

|||u − uh |||Ω

ηH,T , (1 + b∞,Ω + ε−1/2 a∞,Ω )−1 T ∈Th

where

ηH,T :=

C(PT e˜, PT e˜) +

+

(4.17)

T ∈Th

1 2

C(PF e˜, PF e˜)

F ∈∂T ∩Eh,Ω

1/2 C(PF e˜, PF e˜)

,

(4.18)

F ∈∂T ∩Eh,N

with equivalence constants which are independent of h and ε. 4.1. An example of bubble functions For each element T ∈ Th we define the element bubble function bT by λx , bT := 27

(4.19)

x∈N (T )

where N (T ) is the set of all the vertices of the element T and λx denotes the continuous, piecewise linear, nodal function which is equal to 1 at x and vanishes at all other vertices. Following Verf¨ urth,13 let Tˆ be the standard reference element, of vertices (1,0), (0,1) and (0,0). Given any number α ∈ (0, 1] denote by Φα : R2 → R2 the transformation which maps (x, y) onto (x, αy). Let Tˆα := Φα (Tˆ ),

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ˆ1,α , λ ˆ 2,α and λ ˆ 3,α its barycentric coordinates (see Fig. 1). and denote by λ Set

ˆ 1,α on Tˆα , ˆ 3,α λ 4λ bFˆ ,α := 0 on Tˆ \Tˆα , where Fˆ := {(t, 0) ∈ R2 : 0 ≤ t ≤ 1}. Let F ∈ Eh,Ω and denote by T1 , T2 two triangles which have F in common. Denote by GF,i , i = 1, 2, the orientation preserving affine transformation which maps Tˆ onto Ti and Fˆ onto F (see Fig. 2). Set

bFˆ ,α ◦ G−1 F,i on Ti , i = 1, 2, bF,α := (4.20) 0 on Ω\ωF . If F ∈ Eh,N the function bF,α is defined in the same way with the obvious modifications. ˆ the ˆ the hyperplane defined by Π ˆ := {(x, 0) : x ∈ R} and let Q ˆ : R2 → Π Let Π 2 ˆ ˆ orthogonal projection from R to Π. We introduce the lifting operator PFˆ : Pk (Fˆ ) → Pk (Tˆ ) by ˆ PˆFˆ (ˆ σ) = σ ˆ ◦ Q.

Fig. 1.

Fig. 2.

Triangles Tˆ and Tˆα .

Affine transformation GF,i , i = 1, 2.

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Let F ∈ Eh,Ω an interior edge such that ωF = T1 ∪ T2 . Let Ti ∈ ωF and let GF,i the affine transformation defined in Fig. 2. We define the lifting operator PF,Ti : Pk (F ) → Pk (Ti ) by PF,Ti (σ) = PˆFˆ (σ ◦ GF,i ) ◦ G−1 F,i . Finally, we can define a lifting operator ∀ σ ∈ Pk (F ),

PF (σ) :=

PF,T1 (σ) PF,T2 (σ)

in T1 in T2 .

Lemma 4.3. Let k be an arbitrary integer, then v20,T (v, bT v)L2 (T ) |||vbT |||T min{hT ε−1/2 , 1}−1 v0,T σ20,F (σ, bF,αF σ)L2 (F ) |||bF,αF PF (σ)|||ωF ε1/4 min{hF ε−1/2 , 1}−1/2 σ0,F , holds for all T ∈ Th , all F ∈ Eh , and all polynomials v, σ of degree at most k defined on T ∈ Th and F ∈ Eh , respectively, with constants which only depend on k and the regularity of the mesh. Here, αF : = min{ε1/2 h−1 F , 1}. Proof. See Verf¨ urth,13 Lemma 3.3. We need the following assumption to show an example of bubble subspaces which satisfies the (LBB) condition: (F) f and g are piecewise polynomials. We can define now our bubble subspaces by HTb := span{bT RT }

∀ T ∈ Th

HFb

∀ F ∈ Eh,Ω ∪ Eh,N ,

:= span{bF,αF PF (RF )}

where bT and bF,αF are the bubble functions defined in (4.19) and (4.20), respectively. From the definition of HTb we have that bT RT ∈ HTb . Hence sup

b ˜ bT ∈HT

(RT , bT RT )L2 (T ) (RT , ˜bT )L2 (T ) θT RT 0,T C(bT RT , bT RT )1/2 θT RT 0,T C(˜bT , ˜bT )1/2

RT 20,T

θT RT 0,T min{hT ε−1/2 , 1}−1 RT 0,T ≥ β.

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The same analysis is also valid for an edge F in Eh,Ω ∪ Eh,N . In fact, we have sup

b ˜ bF ∈HF

(RF , bF,αF RF )L2 (F ) (RF , ˜bF )L2 (F ) 1/2 ˜ ˜ θF RF 0,F C(bF,αF PF (RF ), bF,αF PF (RF ))1/2 θF RF 0,F C(bF , bF )

RF 20,F θF RF 0,F ε1/4 min{hF ε−1/2 , 1}−1/2 RF 0,F

≥ β. Remark 4.3. More general bubble subspaces can be used. Indeed, it is only necessary to choose HTb and HFb such that bT RT ∈ HTb

∀ T ∈ Th

and bF,αF PF (RF ) ∈ HFb

∀ F ∈ Eh,N ∪ Eh,Ω .

5. Numerical Examples In this section we report three numerical experiments obtained using the hierarchical a posteriori error estimate introduced in Sec. 4. In all the experiments we have used piecewise linear finite elements, the bubble functions defined in Sec. 4.1 and the following USFEM stabilized scheme (see Franca6 for details): For vh , wh ∈ Hh , we define Bδ (vh , wh ) := B(vh , wh ) (δT (−ε∆vh + a · ∇vh + bvh ), −ε∆wh − a · ∇wh + bwh )L2 (T ) − T ∈Th

and Fδ (vh ) := (f, vh )L2 (Ω) + (g, vh )L2 (ΓN ) −

(δT f, −ε∆vh − a · ∇vh + bvh )L2 (T ) .

T ∈Th

In the expressions above we use a stabilization parameter δT defined as follows: δT (x) :=

b(x)h2T

max{1, P eR T (x)}

h2T , + (2ε/mk ) max{1, P eA T (x)}

A where P eR T (x) and P eT (x) are the local Peclet numbers defined by

P eR T (x) :=

2ε , mk b(x)h2T

P eA T (x) :=

mk |a(x)|hT ; ε

| · | denotes the standard Euclidean norm on R2 and mk := min{1/3, ck },

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Fig. 3.

Element parameter hT .

with ck being a positive constant satisfying h2T ∆vh 20,T ≤ ∇vh 20,Ω ck

∀ vh ∈ Hh ,

T ∈Th

which only depends on the polynomial degree k and the shape-regularity of the mesh. Finally hT is a measure of the element size. If a = 0, then it is reported in Franca6 that an element parameter hT which yields best numerical results is the largest streamline distance in the element, as shown in Fig. 3. If aT = 0, we take hT equal to the diameter of T . Lemma 5.1. Given T ∈ Th , let δT be defined as above. Then the following bounds hold ∀ x ∈ T : 1 εδT (x) ≤ h2T , (5.1) 6 1 |a(x)| δT (x) ≤ hT , (5.2) 2 (5.3) b(x)δT (x) θT . Proof. See Araya,1 Lemma 2. Lemma 5.2. The following local estimates hold for all vh ∈ Hh : ∇vh 0,T h−1 T θT |||vh |||T , ∆vh 0,T

h−2 T θT

|||vh |||T .

(5.4) (5.5)

Proof. See Araya,1 Lemma 3. The next result shows the validity of the (STAB) assumption for our stabilized scheme. Theorem 5.1. The USFEM stabilized scheme satisfies the (STAB) assumption, i.e. if uh is the discrete solution of the USFEM scheme, then θ: T RT 0,T |||v|||ω˜ T ∀ v ∈ H. B(u − uh , Ih v)

T ∈Th

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Proof. Using the definition of the bilinear forms B and Bδ we have that δT RT (−ε∆vh − a · ∇vh + bvh ), B(u − uh , vh ) = −

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(5.6)

T ∈Th T

for all vh ∈ Hh . Now, from (5.1) and (5.5) we have δT RT ε∆vh ≤ |RT | εδT |∆vh | T

T

1 ≤ hT 2 RT 0,T ∆vh 0,T 6

θT RT 0,T |||vh |||T . Next, from (5.2) and (5.4), δT RT a · ∇vh ≤ |RT | |a| δT |∇vh | T

T

1 ≤ hT RT 0,T ∇vh |0,T 2

θT RT 0,T |||vh |||T . Similarly, using (5.3) and the definition of ||| · |||T we have δT RT bvh ≤ |RT | bδT |vh | T

T

θT RT 0,T |||vh |||T . Then, we estimate the right-hand side of (5.6) by means of the last three inequalities. Finally, we replace vh by Ih v and use Lemma 2.2 to obtain θT RT 0,T |||Ih v|||T B(u − uh , Ih v)

T ∈Th

θT RT 0,T |||v|||ωeT .

T ∈Th

Remark 5.1. The (STAB) assumption is also true for the standard SUPG stabilized scheme (see Verf¨ urth,13 Eq. (4.8)). The adaptive procedure is given by: (i) Solve (2.6) in an initial mesh T0 and compute ηH,T ∀ T ∈ T0 , and η maxT ∈T0 ηH,T . (ii) If T ∈ T0 satisfies ηH,T ≥ 0.5 η, then T is subdivided. (iii) This process is repeated until a prescribed tolerance is attained.

:=

The meshes are generated using Triangle, an adaptive mesh generator developed by J. Shewchuk.12

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Fig. 4.

Boundary conditions for the reaction-diffusion problem.

Fig. 5. Reaction-diffusion problem: ε = 10−5 . Meshes and horizontal cuts of the approximate solutions (top left: d.o.f. = 25; top center: d.o.f. = 269; top right: d.o.f. = 1680).

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5.1. A reaction-diffusion problem The first test consists of solving a purely reaction-diffusion problem. We have chosen the following data: a = 0, b = 1 and f = 1. The boundary conditions are shown in Fig. 4. The exact solution of this problem is u(x, y) = 1 −

sinh(ε−1/2 x) . sinh(ε−1/2 )

In Fig. 5 and 6 we show the results obtained for different values of the diffusion parameter ε. We can see that the boundary layer present in the solution is well resolved without any significant oscillation. We include in Fig. 7 a plot of the estimated and exact error curves.

1

1

1

0.6

0.6

0.6

0.2

0.2

0.2

-0.2

-0.2 0

0.2

0.4

0.6

0.8

1

-0.2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 6. Reaction-diffusion problem: ε = 10−10 . Meshes and horizontal cuts of the approximate solutions (top left: d.o.f. = 25; top center: d.o.f. = 269; top right: d.o.f. = 1167).

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Fig. 7.

Reaction-diffusion problem. Estimated and exact error curves (ε = 10−5 ).

5.2. An advection-diffusion-reaction problem. Analytical solution For this example, we have chosen the following data: a = (1, 0), b = 1 and f = 1 + u(x, y) where u(x, y) = x −

e−

1−x ε

1

− e− ε 1

1 − e− ε

,

the boundary conditions are shown in Fig. 8. The solution of this problem exhibits a boundary layer along the line x = 1 which is well captured by our adaptive scheme (see Fig. 9). For this problem, we also include a plot of the exact and estimate error (see Fig. 10).

Fig. 8.

Boundary conditions for the advection-diffusion problem.

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Fig. 9. Advection-reaction-diffusion problem: ε = 10−4 . Meshes and horizontal cuts of the approximate solutions (top left: d.o.f. = 25; top center: d.o.f. = 1459; top right: d.o.f. = 8775).

Fig. 10.

Advection-reaction-diffusion problem. Estimated and exact error curves (ε = 10−4 ).

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5.3. An advection-diffusion-reaction problem with an inner layer √ √ In this last example we have chosen the following data: a = ( 2/2, 2/2), b = 1 and f = 0. The boundary conditions are shown in Fig. 11. Note that in this case we do not know the exact solution, but we know that there is an inner layer parallel to the line y = x. We note in Fig. 12 that the inner layer is detected by our adapted scheme even if we use a small diffusion parameter (ε = 10−10 ).

Fig. 11.

Boundary conditions for the advection-diffusion-reaction problem.

Fig. 12. Advection-diffusion-reaction problem: ε = 10−10 . Meshes and level sets (left: d.o.f. = 175; center: d.o.f. = 1547; right: d.o.f. = 12209).

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Acknowledgments R. Araya was partially supported by FONDECYT-Chile through project 1040595 and by the DFG Graduiertenkolleg GRK 615. The work of E. P. Stephan was partially supported by BMBF grant 03STM1HV. References 1. R. Araya, E. Behrens and R. Rodr´ıguez, An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation, to appear in Appl. Numer. Math. 2. F. Brezzi, D. Marini and A. Russo, Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Comput. Meth. Appl. Mech. Engrg. 166 (1998) 51–63. 3. A. N. Brooks and T. J. R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. Meth. Appl. Mech. Engrg. 32 (1982) 199–259. 4. Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Num´ er. 2 (1975) 77–84. 5. W. D¨ orfler and R. H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math. 91 (2002) 1–12. 6. L. P. Franca and F. Valentin, On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation, Comput. Meth. Appl. Mech. Engrg. 190 (2000) 1785–1800. 7. V. John, A numerical study of a posteriori error estimators for convection-diffusion equations, Comput. Meth. Appl. Mech. Engrg. 190 (2000) 757–781. 8. T. Knopp, G. Lube and G. Rapin, Stabilized finite element methods with shock capturing for advection-diffusion problems, Comput. Meth. Appl. Mech. Engrg. 190 (2002) 1785–1800. 9. R. Nochetto, Removing the saturation assumption in a posteriori error analysis, Istit. Lombardo Accad. Sci. Lett. Rend. A127 (1993) 67–82. 10. H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations — Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics, Vol. 24 (Springer-Verlag, 1996). 11. G. Sangalli, A robust a posteriori estimator for the residual-free bubbles method applied to advection-diffusion problems, Numer. Math. 89 (2001) 379–399. 12. J. R. Shewchuk, Delaunay refinement algorithms for triangular mesh generation, Comp. Geom. Theor. Appl. 22 (2002) 21–74. 13. R. Verf¨ urth, A posteriori error estimators for convection–diffusion equations, Numer. Math. 80 (1998) 641–663.