Nitsche-mortaring for singularly perturbed convection

Adv Comput Math (2012) 36:581–603 DOI 10.1007/s10444-011-9195-2

Nitsche-mortaring for singularly perturbed convection–diffusion problems Torsten Linß · Hans-Görg Roos · Martin Schopf

Received: 10 May 2010 / Accepted: 20 January 2011 / Published online: 4 October 2011 © Springer Science+Business Media, LLC 2011

Abstract In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented. Keywords Finite element method · Mortar method · Shishkin mesh · Convection–diffusion · Supercloseness · Singular perturbation · Uniform convergence Mathematics Subject Classifications (2010) 65N30 · 65N50

1 Introduction Consider the singularly perturbed boundary value problem Lu := −εu − b · ∇u + cu = f

in

,

(1a)

u=0

on

∂,

(1b)

Communicated by Martin Stynes. T. Linß (B) · H.-G. Roos · M. Schopf Institut für Numerische Mathematik, TU Dresden, 01062 Dresden, Germany e-mail: [email protected]

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¯ → R2 and c, f : ¯ → R is assumed to be smooth and where the data b : 0 < ε 1 is a small perturbation parameter. The solution u of (1) typically exhibits boundary layers, i.e. regions near the boundary where the solution and its derivatives change rapidly. Interior layers and corner singularities may also occur, but these will not be considered within this paper. For a problem posed in d dimensions, a numerical method is called uniformly convergent with respect to the perturbation parameter ε of order p > 0 in the norm · ∗ if the difference between the exact solution u and the numerical solution u M based on M degrees of freedom satisfies u − u M ∗ ≤ CM− p/d with a constant C that is independent of ε. Typically, for finite element methods for (1) convergence is studied in the energy norm v2ε := ε|v|21 + v20 . For both, problems with exponential layers and for problems with characteristic layers, there exist uniformly convergent finite element methods on layeradapted grids, see [9, 17]. Optimal FEM use, for instance, bilinear elements in the layer region 1 and linear or bilinear elements with or without stabilisation outside the layer region, i.e., on 2 := \ 1 . The good performance of these methods can be explained by certain supercloseness phenomena, see e.g., [8, 22]. For domains of more complicated shape, the construction of such hybrid admissible boundary layer meshes can be found in [12, Section 3.3]. However, the restriction to admissible meshes and continuous finite element approximations leads to certain difficulties, for instance, in the case of L-shaped domains; see [12, Fig. 2.6.13]. Therefore, it may be advantageous to use meshes that do not match at the interface of the two subdomains 1 and 2 . In this manner it is possible to combine highly structured meshes within the layer region with unstructured ones on the remaining domain. Different FE-discretisations may also be used on different subdomains. In general, the resulting approximation will be continuous within the subdomains but discontinuous at the interface. In this respect the proposed method gains the ability to handle non-matching discretisations by introducing less additional degrees of freedom compared to the discontinous Galerkin method. Within the framework of domain decomposition methods several approaches to handle non-matching meshes are well established: saddle point formulations, Lagrange multipliers and Nitsche mortaring [3, 4, 14]. The latter is based on Nitsche’s idea of weakly imposing Dirichlet boundary conditions [14]. Solutions on different subdomains are matched by using an adequately modified bilinear form. We prefer this approach, because the error analysis can be solely based on typical finite element ingredients. Moreover, it is well known that Nitsche’s technique for handling boundary conditions has advantages for singularly perturbed problems [18]. For singularly

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perturbed reaction–diffusion problems, Nitsche mortaring in combination with Bakhvalov-type meshes was studied in [7]. For the sake of simplicity we restrict ourselves in the present paper to the convection–diffusion problem (1) with exponential layers and use the Galerkin technique in both 1 and 2 . It is just a technical question to extend the analysis to the use of some stabilised finite element method in 2 . Throughout, we assume that ¯ b = (b 1 , b 2 ) > (β1 , β2 ) > (0, 0) on . In this case, u possesses exponential boundary layers of width O(ε ln 1/ε) at the outflow boundary of ; see, e.g., [17]. Furthermore, without loss of generality let 1 c + div b ≥ c0 > 0, (2) 2 with a constant c0 . Note, if ε is sufficiently small then (2) can be ensured by a simple change of variable v(x, y) = eκ(x+y) u(x, y) for some suitably chosen constant κ; see, e.g., [19].

2 Nitsche mortaring ¯ = ¯1∪ ¯2 Let be decomposed into two subdomains 1 and 2 , such that ¯ ¯ and 1 ∩ 2 = ∅. We shall refer to = 1 ∩ 2 as the interface of 1 and 2 . Note, the restriction to two subdomains is not significant but simplifies the presentation, cf. [7]. We denote the restriction of a function v onto i by vi := v|i (i = 1, 2). Moreover, we shall not symbolically distinguish between a function v on and the vector v = (v1 , v2 ). The trace and the generalised normal derivative of a Sobolev function v ∈ H s () will be denoted by v|∂ (s > 1/2) and ∂v (s > 3/2), respectively. For the sake of simplicity the symbol ·|∂ will ∂n ∂ often be omitted. Throughout, C denotes a generic positive constant which is independent of the diffusion parameter ε and the mesh but can take different values at different occurrences. Now consider the problem −εui − b · ∇ui + cui = fi

in

i , i = 1, 2,

(3a)

ui = 0

on

∂ ∩ ∂i i = 1, 2,

(3b)

u1 = u2

on

,

(3c)

on

,

(3d)

∂u2 ∂u1 + =0 ∂n1 ∂n2

where ni denotes the unit outward normal of i (i = 1, 2). If u ∈ H s (), s > 3/2, then the variational formulation of (1) and (3) are equivalent. Suppose ¯ Then this equivalence is valid in the classical sense too. that u ∈ C2 (). Throughout we assume u ∈ H 2 ().

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2.1 Weak formulation In order to formulate (3) weakly we define the broken Sobolev space V = v ∈ L2 () : vi ∈ H 1 (i ) for i = 1, 2, v|∂ = 0 . i ∈ L2 (). Therefore, equality of Note that vi ∈ H 1 (i ) does not imply ∂v ∂n directional derivatives on has to be understood in a different space and depends on geometrical properties of the interface . If the interface is described by the entire boundary of a single subdomain, i.e. = ∂i , then one can choose H −1/2 (), the dual space of H 1/2 (). If ∂ ∩ ∂i = ∅ for i = 1, 2 then a suitable space is the dual space of 1/2 H00 (). The latter can be interpreted as an interpolation space between the space L2 () and the Sobolev space H01 () := {v ∈ H 1 () : v|∂ = 0}, see [11]. Denoting the trivial extension by zero of a function v ∈ H 1/2 () onto ∂i by v, ˜ we have 1/2 H00 () = v ∈ H 1/2 () : v˜ ∈ H 1/2 (∂i ) ; see [5]. 1/2

Note, H00 () is a proper subset of and continuously embedded in H 1/2 (). −1/2 1/2 We shall use ·, · to denote the duality pairing on H00 () × H00 (). When dividing the domain we shall choose 1 to cover the boundary layer regions and 2 the remainder of the domain. The outer normal to ∂ and ∂2 is denoted by n. Hence, n := n2 on the interface . We shall assume that n·b≤0

on .

This means that there is no convective flow from the subdomain 1 into the subdomain 2 . This is a reasonable assumption since 1 covers the layer (Fig. 1).

Fig. 1 Dissection of

n1 Ω2 Ω1 Γ n2


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Along the interface we define the jump of a function v ∈ V by [[v]] := v1 | − v2 | . Then a weak formulation of (3) is: Find u ∈ V such that aσM (u, v) = ( f, v)

∀v = (v1 , v2 ) ∈ V,

(4a)

where aσM (u, v) := aG (u, v) + aσI (u, v), aG (u, v) := ε

2

(∇ui , ∇vi )i −

2

i=1

and aσI (u, v)

b · ∇ui − cui , vi

(4b) i

,

(4c)

i=1

∂u1 ∂u2 ∂v1 ∂v2 := −ε α1 − α2 , [[v]] + σ ε [[u]] , α1 − α2 ∂n1 ∂n2 ∂n1 ∂n2 − b · n[[u]], v1 + (γ I [[u]], [[v]]) .

(4d)

Here αi ≥ 0 are positive constants with α1 + α2 = 1, σ is a real constant and γ I : → R>0 is a positive function on . The first term in (4d) arises from the application of Green’s formula. All other terms which contain the jump [[u]] of u are introduced artificially. They vanish if u ∈ H 2 (). Therefore, σ can attain arbitrary values. However, we shall only consider σ ∈ {−1, 0, 1}. Adapting the terminology of the discontinuous Galerkin method, we call the mortaring symmetric, incomplete and nonsymmetric (or more precisely, asymmetric) if σ = −1, σ = 0 and σ = 1, respectively. The introduction of the second summand of the right hand side of (4d) is motivated by symmetrisation (σ = −1) or coercivity (σ = 1). In the latter respect the third summand is needed to handle terms that arise due to an application of an integral theorem for the convective term. The term (γ I [[u]], [[v]]) penalises jumps of the solution along the interface thus weakly enforcing continuity. 2.2 Discretisation Let us consider (1) on the unit square = (0, 1)2 . Exponential boundary layers will form at the edges x = 0 and y = 0 of the domain. We shall resolve these layers by means of a Shishkin mesh [13], but other layer-adapted meshes may also be used, see [9]. Define the transition point λ by

1 2ε ln N , λ = (λ1 , λ2 ) with λi = min , 2 βi where N, the number of mesh intervals in each coordinate direction, is an even positive integer. We assume λi =

2ε 1 ln N ≤ , βi 2

i = 1, 2

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as otherwise N > eC/ε and therefore the method can be analysed in a conventional manner. The domain is divided into subdomains 2 = (λ1 , 1) × (λ2 , 1) ¯ 2 ; see Fig. 2. and 1 = \ ¯ On 1 a piecewise uniform mesh will be used. This is constructed as follows: Each of the intervals [0, λ1 ], [λ1 , 1], [0, λ2 ] and [λ2 , 1] is uniformly dissected ¯ 1 , i, j = 0, . . . , N, with into N/2 subintervals giving the grid points (xi , y j) ∈ for i = 0, . . . , N/2, 2iλ1 /N xi = 1 − 2(N − i)(1 − λ1 )/N for i = N/2 + 1, . . . , N, and

yj =

2 jλ2 /N 1 − 2(N − j)(1 − λ2 )/N

for j = 0, . . . , N/2, for j = N/2 + 1, . . . , N.

Drawing lines through these mesh points parallel to the x-axis and y-axis, we ¯ 1 which consists of rectangles only. obtain our triangulation T1 of the domain We denote the lengths of the edges of these rectangles T ∈ T1 by Hi =

2(1 − λi ) 2(1 − 2ε ln N/βi ) = , N N

hi =

2λi 4ε ln N = , N βi N

i = 1, 2.

Clearly, N −1 ≤ H1 , H2 ≤ 2N −1 and h1 , h2 = O(εN −1 ln N). ¯ 2 into shape regular triangles (or Let T2 be an admissible triangulation of rectangles or a combination of both). Let denote the mesh parameter of this triangulation, i.e. = maxT∈T2 diam T. Note, the combined triangulation T := T1 ∪ T2 is not required to be admissible; see Fig. 3. For any triangular (or rectangular) element T ∈ T of the triangulation and any of its edges F ⊂ ∂ T, we denote by h⊥F the height of the element T perpendicular to the edge F.

Fig. 2 Dissection of based on a Shishkin mesh

n

Ω2 Γ λ Ω1


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1 0.9 0.8 0.7 0.6 0.5 0.4

H2

−

h

0.3 0.2 0.1

h2

0 0

0.2

(a)

0.4

2

0.6

0.8

1

h1

H1 (b) mesh parameters

Delaunay triangulation

Fig. 3 Example triangulations T

Restricting the triangulations T1 of 1 and T2 of 2 on , one obtains triangulations E1 and E2 of the interface : Ei = {E : ∃T ∈ Ti with E = T ∩ and |E| > 0} ,

where |E| denotes the length of the side E. There are many triangulations E of the interface to choose from. In [15] the author recommends the setting E = E2 and α2 = 1 but gives criteria for more general triangulations, too. A natural choice is given by what we will refer to as induced triangulation E∩ : E∩ = {E : ∃E1 ∈ E1 and ∃E2 ∈ E2 with E = E1 ∩ E2 and |E| > 0} ;

see Fig. 4. In this article we shall choose E = E∩ . Fig. 4 Induced triangulation E∩ of the interface

1

ε1 ε ε2 2

Ω1

Γ Γ Γ

Ω2

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We introduce the FE space V h := {v ∈ V : v|T ∈ P1 (T)

∀T ∈ T1 , v|T ∈ Q1 (T) ∀T ∈ T2 } .

Note that elements of V h are continuous on each of the subdomains 1 and 2 and that this discretisation is conforming (V h ⊂ V). We discretise (4) by means of a hybrid Galerkin-FEM with Nitsche mortaring: Find uh ∈ V h such that aσM (uh , v h ) = ( f, v h )

∀(v1h , v2h ) = v h ∈ V h .

(5)

∂v h

For all v h ∈ V h , we have ∂nii ∈ L2 () and vih ∈ L2 () (i = 1, 2). Thus, for functions from V h we can identify the duality pairing with the L2 scalar product. Suppose γ I is piecewise constant on each E ∈ E : γ I | E = γ E > 0. Then we can rewrite the bilinear form in (5) as aσM (uh , v h )

=ε

2 i=1

∇uih , ∇vih i

−

2

b · ∇uih − cuih , vih

i

i=1

∂u1h ∂u2h h −ε − α2 , [[v ]] α1 ∂n1 ∂n2 E E∈E

∂v1h ∂v2h h α1 +σ ε − α2 , [[u ]] ∂n1 ∂n2 E E∈E − γ E ([[uh ]], [[v h ]]) E . b · n[[uh ]], v1h E +

E∈E

E∈E

Lemma 1 (Galerkin orthogonality) The weak solution u = (u1 , u2 ) ∈ H 2 () of (3) satisf ies aσM (u, v h ) = ( f, v h ) ∀(v1h , v2h ) = v h ∈ V h . Therefore, the mortar method is consistent with aσM (u − uh , v h ) = 0 ∀v h ∈ V h . When studying coercivity of the bilinear form aσM we shall use the following norm in V: |||v|||2 := ε

2 2 1 |vi |21,i + c0 vi 20,i + (γ I [[v]],[[v]]) − b · n [[v]], [[v]] . 2 i=1 i=1

Note, the mapping ||| · ||| : V → R≥0 is a norm because γ I ≥ γ0 > 0 and n · b ≤ 0 on . We will refer to ||| · ||| as broken energy norm. Lemma 2 (Coercivity, asymmetric mortaring) We have 2 a+1 M (v, v) ≥ |||v|||

∀v ∈ V.


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Proof The argument uses standard techniques, in particular 1 1 2 − b · ∇v v dx = − b · ni v ds + div b v 2 dx 2 ∂i 2 i i

for i = 1, 2.

For σ = 0 and σ = −1, i.e. incomplete and symmetric mortaring, we shall prove coercivity in V h of the related bilinear forms. In order to do so, we need the well-known trace inequalities in finite dimensional spaces. Lemma 3 Let p be an arbitrary polynomial p ∈ Pt (T) (or p ∈ Qt (T)), t ∈ N. Then there exists a constant C0 = C0 (t) such that −1 p20,F ≤ C0 h⊥F p20,T for all elements T ∈ T of the triangulation and all of its edges F ⊂ ∂ T. Remark 1 The constant C0 can be specified explicitly. For example, C0 = 6 for linear elements on triangles [21] and C0 = 36 for bilinear elements on rectangles [6]. ⊥ Lemma 4 For any E ∈ E let hi,E denote the height of the element T ∈ Ti perpendicular to the side F = T ∩ for which F ∩ E = ∅. Furthermore, for any E ∈ E def ine h⊥E and α E by ⎧ ⊥ ⎪ if α1 = 0, ⎨h2,E −1 ⊥ ⊥ α E = h E := h1,E (6) if α2 = 0, ⎪ ⎩ ⊥ ⊥ min{h1,E ,h2,E } otherwise.

Then ∂v h 2 ∂v2h C0 2 h 2 h 1 α1 − α2 , [[v ]] ≤ α1 v1 1,1 + α22 v2h 1,2 ∂n1 ∂n2 μ E E∈E 2 α E [[v h ]]0,E +μ E∈E

for all μ0 > 0 and v ∈ V , where C0 is the constant from Lemma 3. h

h

Proof The proof of this lemma is very technical. Therefore, we defer it to Appendix. Lemma 5 (Coercivity in V h ) Let E = E∩ be that triangulation of the interface induced by T . Let σ ∈ {−1, 0, 1}. Suppose −1 γ E ≥ τ εC0 (σ − 1)2 max α12 , α22 h⊥E with some constant τ > 1. Then there exists a constant C > 0 such that aσM (v h , v h ) ≥ C|||v h |||2

∀v h ∈ V h .

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Remark 2 The coercivity of aσM in V h guarantees the uniqueness of a solution uh of (5). Remark 3 In the case of symmetric or incomplete mortaring we have for the choice of γ E : ε−1 if α1 = 0 and (7) γE ≥ C N(ln N)−1 if α1 ∈ (0, 1]. Proof of Lemma 5 Let v h ∈ V h be arbitrary. Then aσM (v h , v h )

=

h h a+1 M (v , v )

+ ε(σ − 1)

E∈E

∂v h ∂v h α1 1 − α2 2 , [[v h ]] ∂n1 ∂n2

. E

Since v h ∈ V, Lemma 2 applies and yields aσM (v h , v h ) ≥

2

ε|vi |21,i + c0

i=1

−

2 vi 20,i + (γ I [[v h ]], [[v h ]]) i=1

∂v h ∂v h 1 b · n [[v h ]], [[v h ]] + (σ − 1)ε α1 1 − α2 2 , [[v h ]] . 2 ∂n1 ∂n2 E E∈E

The inequality a ≥ − |a| ∀a ∈ R and Lemma 4 yield for the last term aσM (v h , v h )

≥ε

2 i=1

+

E∈E

C0 αi2 1 − (1 − σ ) μ

|vi |21,i

2 + c0 vi 20,i i=1

2 1 b · n [[v h ]], [[v h ]] . (γ E − εμ(1 − σ )α E ) [[v h ]]0,E − 2

Fix μ > C0 (σ − 1) max{α12 , α22 }. Then C > 0, which is there exists a constant independent of ε, N and , such that 1 − (σ − 1)C0 αi2 μ−1 ≥ C. Moreover, the choice of γ E implies the existence of a constant C > 0 with −1 γ E − εμ(1 − σ )α E = γ E − τ εC0 (σ − 1)2 max α12 , α22 h⊥E ≥ Cγ E .

The proposition of the lemma follows.

3 Solution decomposition and interpolation For our analysis, we assume that the solution u of (1) can be decomposed as u = S + E1 + E2 + E12 ,

(8)


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¯ and 0 ≤ i + j ≤ m with m = 3 the parts of the solution where for all (x, y) ∈ satisfy i+ j ∂ (9a) ∂ xi ∂ y j S(x, y) ≤ C, i+ j ∂ −i −β1 x/ε , (9b) ∂ xi ∂ y j E1 (x, y) ≤ Cε e i+ j ∂ − j −β2 y/ε , (9c) ∂ xi ∂ y j E2 (x, y) ≤ Cε e i+ j ∂ −(i+ j) −(β1 x+β2 y)/ε e . (9d) ∂ xi ∂ y j E12 (x, y) ≤ Cε Remark 4 In [10] a technique is introduced to derive compatibility conditions on the data of the problem that guarantee the existence of such a decomposition. By u I let us denote the piecewise bilinear or linear nodal interpolant of u on T . Using the assumptions (9) and the anisotropic interpolation estimates of [1, 2] for general meshes, one can prove the following interpolation-error estimates; cf. [17, 19]. Theorem 1 Suppose ε1/2 ≤ (ln N)−2 . Then the interpolation error on our hybrid Shishkin mesh satisf ies u − u I ≤ C(N −2 ln2 N), (10a) ∞,1 u − u I ≤ C(N −2 + 2 ), (10b) ∞,2 ε ∇ u − u I ∞,1 ≤ C(N −1 ln N), (10c) ε ∇ u − u I ∞,2 ≤ C(N −1 + N −2 (ln N)−4 −1 + (ln N)−4 ), (10d) u − u I ≤ CN −2 , (10e) 0,1 u − u I ≤ C 2 , (10f) 0,2 ε1/2 ∇ u − u I 0,1 ≤ CN −1 ln N, (10g) ε1/2 ∇ u − u I 0,2 ≤ C(N −2 + (ln N)−2 + N −2 (ln N)−2 −1 ). (10h) Invoking (10a) and (10b), we readily get an estimate for the jump of the interpolation error along the interface : [[u − u I ]] ≤ u1 − u1I ∞, + u2 − u2I ∞, ∞, ≤ u1 − u1I ∞,1 + u2 − u2I ∞,2 ,

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where we have used the continuity of u on and uiI on i . Thus, [[u − u I ]]

0,

≤ 2[[u − u I ]]∞, ≤ C(N −2 ln2 N + 2 ).

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4 Error analysis In this section we derive error estimates and show that the finite element solution enjoys a super closeness property. Subsequently, we assume γ I to be constant (for fixed N) on the interface , i.e. γ I ≡ γ N ∈ R>0 . Let uh ∈ V h be the finite element solution of (5). A triangle inequality yields for the error of the method |||u − uh ||| ≤ |||u − u I ||| + |||u I − uh |||. For the interpolation error η := u − u I , the estimates of Section 3 imply

1/2 2 2 1 2 2 |||η||| = ε |ηi |1,i + ηi 0,i + (γ I [[η]],[[η]]) − b · n [[η]], [[η]] 2 i=1 i=1 ≤ C N −2 ln2 N + N −4 + (ln N)−4 2 + N −4 (ln N)−4 −2 + N −4 + 4 1/2 + γ N N −4 ln4 N + 4 + b∞ N −4 ln4 N + 4 . We obtain the following bound for the interpolation error. Lemma 6 Suppose γ N ≤ CN (ln N)−1 and N −1 (ln N)−3 ≤ ≤ N −3/4 ln3/4 N. Then the interpolation error on a Shishkin mesh satisf ies |||η||| ≤ CN −1 ln N. Next, we estimate ξ := u I − uh , the difference between the nodal interpolant of the exact solution and the finite element solution. A standard argument using Lemmas 5 and 1 gives |||ξ |||2 ≤ CaσM (η, ξ ).

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Recalling the definition of the bilinear form aσM , we have aσM (η, ξ ) = ε

2 i=1

(∇ηi , ∇ξi )i −

2

b · ∇ηi − cηi , ξi

i

(13a)

i=1

∂η1 ∂η2 − ε α1 − α2 , [[ξ ]] ∂n1 ∂n2

∂ξ1 ∂ξ2 + σ ε α1 − α2 , [[η]] ∂n1 ∂n2 − b · n [[η]], ξ1 + (γ I [[η]], [[ξ ]]) .

(13b) (13c) (13d) (13e)

i Note, that ∂u ∈ L2 (), because u is smooth. Therefore, we can identify the ∂ni dual pairing with the L2 scalar product in the term (13b). The same applies to the term (13c), since ξ ∈ V h . The terms on the right-hand side of (13) will be bounded separately.

(i) We start with the term (13b). The Cauchy–Schwarz inequality yields ∂η1 ∂η2 Sb := ε α1 − α2 , [[ξ ]] ∂n1 ∂n2 ∂η1 ∂η2 ε 1/2 = 1/2 α1 − α2 γ N [[ξ ]] ds ∂n1 ∂n2 γN ∂η2 ε ∂η1 1/2 ≤ 1/2 α1 ∂n − α2 ∂n (γ N [[ξ ]],[[ξ ]]) 1 2 0, γN ∂η1 ∂η2 ε |||ξ ||| ≤ C 1/2 α1 + α2 ∂n ∂n 1 ∞, 2 ∞, γN ≤C

ε 1/2

γN

α1 ∇η1 ∞, + α2 ∇η2 ∞,

∂ηi ≤ |∇ηi | . |||ξ |||, because ∂ni

Based on the continuity of ∇ηi within elements E ∈ Ei of the triangulation of the two subdomains i , we can invoke (10c) and (10d) to obtain ε α1 ∇η1 ∞,1 + α2 ∇η2 ∞,2 |||ξ ||| −1/2 ≤ Cγ N α1 N −1 ln N + α2 N −1 + N −2 (ln N)−4 −1 + (ln N)−4 |||ξ |||. −1/2

Sb ≤ Cγ N

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(ii) The term (13c) will be bounded against ε1/2 |ξ1 |21,1 + |ξ2 |21,2 which is contained in |||ξ |||. The triangle inequality and the Cauchy–Schwarz inequality yield ∂ξ1 ∂ξ2 Sc := ε |σ | α1 − α2 , [[η]] ∂n1 ∂n2 ∂ξ1 ∂ξ2 ≤ ε |σ | [[η]]0, α1 . ∂n + α2 ∂n 1 0, 2 0, Using the arithmetic-quadratic means inequality, we have 2 2 1/2 2 ∂ξ1 2 ∂ξ2 + α2 Sc ≤ Cε|σ | [[η]]0, α1 ∂n1 0, ∂n2 0, ⎛ ⎞1/2 ∂ξ1 2 ∂ξ2 2 2 ⎠ . ≤ C|σ |ε [[η]]0, ⎝α12 ∂n + α2 ∂n 1 0,E 2 0,E E∈E E∈E 1

2

∂ξi Note, that the normal derivatives ∂n are polynomials on each T ∈ T . i T

Therefore, we can apply Lemma 3 to ∂ξi /∂ni 20,E , i = 1, 2. We get Sc ≤ Cε|σ | [[η]]0, ⎛ 1 × ⎝α12 h⊥ 1,E E∈E 1

1 ∂ξ1 2 2 + α 2 ∂n h⊥ 1 0,T1 (E) 2,E E∈E 2

⎞1/2 ∂ξ2 2 ⎠ ∂n 2 0,T2 (E)

with h⊥ 1,E defined in Lemma 4. The shape regularity of the triangulation T2 implies h⊥ 2,E ≥ C, while ⊥ −1 h⊥ ≥ min{h , h }, i.e., h ≥ CεN ln N. We set 1 2 1,E 1,E if α1 = 0, ⊥ h = −1 εN ln N otherwise. Then Sc ≤ Cε|σ | [[η]]0, ⎛ ∂ξ1 2 1 × ⎝α12 + α22 −1 −1 εN ln N E∈E ∂n1 0,T1 (E) E∈E

≤ Cε|σ | [[η]]0,

1

2

⎞1/2 ∂ξ2 2 ⎠ ∂n 2 0,T2 (E)

1/2 1 2 2 ∇ξ ∇ξ + , 1 2 0,T 0,T h⊥ T∈T1

∂ξi ≤ |∇ξi | . because ∂ni

T∈T2


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Next, inequality (11) yields ε −2 2 Sc ≤ C|σ | N ln N + 2 |||ξ ||| ⊥ h if α1 = 0, Cε1/2 N −2 ln2 N −1/2 + 3/2 |||ξ ||| −3/2 3/2 ≤ ln N + N 1/2 (ln N)−1/2 2 |||ξ ||| otherwise. C N

(15)

(iii) The triangle inequality yields for the term (13d) b · n [[η]],ξ1 = b · n [[η]],[[ξ ]] + b · n [[η]],ξ2 ≤ b · n [[η]],[[ξ ]] + b · n [[η]],ξ2 . By means of the Hölder inequality, we get b · n [[η]],ξ1 ≤ b · n [[η]]0, [[ξ ]]0, + ξ2 0, ∞, −1/2 1/2 ≤ C [[η]]0, γ N (γ N [[ξ ]],[[ξ ]]) + ξ2 0, . Lemma 3 implies ξ2 20,E ≤ C ξ2 20, = E∈E2

1

h⊥ E∈E2 2,E

ξ2 20,T2 (E) ≤ C−1 ξ2 20,2 ,

because of the shape regularity of the triangulation T2 . Therefore, b · n [[η]],ξ1 ≤ C [[η]]0, γ −1/2 + −1/2 |||ξ |||. N Combining this with (11), we get b · n [[η]],ξ1 ≤ C N −2 ln2 N + 2 γ −1/2 + −1/2 |||ξ |||. N

(16)

(iv) The last term (13e) is estimated using the Cauchy-Schwarz inequality and (11) |(γ N [[η]], [[ξ ]]) | ≤ γ N [[η]]0, [[ξ ]]0, ≤ γ N1/2 [[η]]0, (γ N [[ξ ]], [[ξ ]])1/2 1/2 −2 2 ≤ Cγ N N ln N + 2 |||ξ |||. (17) Finally, collecting (12)–(17), we obtain the following estimate:

|aG (η, ξ )| |||ξ ||| ≤ C |||ξ ||| −1/2 + α1 γ N N −1 ln N + α1 N −3/2 ln3/2 N + N 1/2 (ln N)−1/2 2 −1/2 −1 + α2 γ N N + N −2 (ln N)−4 −1 + (ln N)−4 + ε1/2 N −2 ln2 N −1/2 + ε1/2 3/2 −1/2 1/2 + N −2 ln2 N + 2 γ N + −1/2 + γ N N −2 ln2 N + 2 . (18)

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In accordance with Lemma 5 we set γ N = CN (ln N)−1 ,

(19)

which implies (7) and ensures the coercivity of the bilinear form aσM . This choice also balances the error components in (18) if in addition one has ∼ N −1 ln N

Then

(N → ∞).

(20)

|aG (η, ξ )| −3/2 3/2 +N ln N . |||ξ ||| ≤ C |||ξ |||

Theorem 2 (Supercloseness) Assume the solution u of (1) can be decomposed according to (8) and (9). Suppose ε1/2 ≤ (ln N)−2 . Let u I ∈ V h be the nodal interpolant of u and let uh ∈ V h be the FE approximation with Nitsche mortaring, i.e., the solution of (5). Suppose the mortaring parameter γ I ≡ γ N is chosen according to (19). Let T2 be a shape regular triangulation of 2 with mesh parameter satisfying (20). Furthermore, let the Galerkin bilinear form satisfy the inequality aG (u − u I , v) ≤ CN −3/2 ln3/2 N|||v||| ∀v ∈ V h . (21) Then |||u I − uh ||| ≤ CN −3/2 ln3/2 N.

(22)

Remark 5 The estimate (22) constitutes a supercloseness result, because in general, for the actual error one has only |||u − uh ||| ≤ |||u − u I ||| + |||u I − uh ||| ≤ CN −1 ln N, and this result is sharp. Recovery techniques [20, 23] can be used to design a recovery operator R with |||u − Ruh ||| ≤ CN −3/2 ln3/2 N. Remark 6 Assuming (21) for the error contribution from the Galerkin part of the discretisation is reasonable. For example, if the triangulation T2 is uniform and consists of rectangles, then aG (u − u I , v) ≤ CN −2 ln5/2 N|||v|||, see [8, 22]. In [16] Roos validates (21) for a triangulation generated by three families of parallel lines with = CN −1 and constant b , following a technique of Q. Lin. Remark 7 In order to compare the FEM with Nitsche mortaring with other methods on Shishkin meshes one also wants to consider the choice = CN −1 . Then the asymptotic behaviour of both triangulations T1 and T2 is identical. In particular this captures the case of matching triangulations. Then the method possesses the supercloseness property |||u I − uh ||| ≤ CN −3/2 ln2 N.


597

Remark 8 From (18) one can deduce maximum values for γ I and that still give first order accuracy, i.e., |||u − uh ||| ≤ CN −1 ln N. This choices are γ I = γ N ∼ N ln N −3/4

and ∼ N −3/4 ln3/4 N.

(23)

5 Numerical results For our numerical experiments we consider the following test problem −ε u − 2ux − 3u y + u = f in = (0, 1)2 u = 0 on ∂,

(24)

where we choose the right hand side f in such a way that u(x, y) = 2 sin(1 − x) 1 − e−2x/ε (1 − y)2 1 − e−3y/ε is the exact solution of (24). It exhibits typical boundary and corner layer behaviour. Unless defined otherwise we take ε = 10−8 . First, we examine the accuracy of the results of our method on tensor product grids (without hanging nodes along the interface ), as these standard Shishkin meshes are frequently used to solve singularly perturbed problems. Therefore, we have = CN −1 . Figure 5 contains plots of the mesh. Next, we verify our theoretical findings. Table 1a and b illustrate the uniformity of the method with respect to ε for both the actual error u − uh and the supercloseness error u I − uh , respectively. The results are in agreement with our theoretical findings in Remarks 5 and 7. The method is robust with respect to the perturbation parameter ε. For the actual error we observe rates which are smaller than one, but slowly converging towards one. For the distance between the interpolant of u and the numerical solution uh the rates

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0

0.2

(a)

0.4

0.6

0.8

Grid of regular type ( N = 8 )

1

0

0.2

(b)

0.4

0.6

0.8

Rectangular grid ( N = 8 )

Fig. 5 Tensor product meshes with subtriangulations matching at the interface

1

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Table 1 Errors and convergence rates in the energy norm on hybrid tensor product grids of regular type and symmetric mortaring (σ = −1, α1 = 1/2 and γ N = 0.5N(ln N)−1 ) (a) |||u − uh ||| N ε = 10−4

ε = 10−8

|||u − uh ||| 8 16 32 64 128 256 512

4.3357e-1 2.9260e-1 1.8450e-1 1.1113e-1 6.4914e-2 3.7111e-2 2.0878e-2

Rate

|||u − uh |||

0.5673 0.6652 0.7314 0.7755 0.8066 0.8298

4.3357e-1 2.9260e-1 1.8450e-1 1.1113e-1 6.4914e-2 3.7113e-2 2.0884e-2

(b) |||u I − uh ||| N ε = 10−4

|||u − uh |||

Rate

0.5673 0.6653 0.7314 0.7756 0.8066 0.8296

4.3357e-1 2.9260e-1 1.8450e-1 1.1113e-1 6.4914e-2 3.7113e-2 2.0884e-2

0.5673 0.6653 0.7314 0.7756 0.8066 0.8295

ε = 10−8

|||u I − uh ||| 8 16 32 64 128 256 512

ε = 10−12 Rate

8.9255e-2 3.7107e-2 1.4966e-2 5.6008e-3 1.9582e-3 6.4716e-4 2.0563e-4

Rate

|||u I − uh |||

1.2661 1.3099 1.4178 1.5160 1.5972 1.6539

8.9222e-2 3.7072e-2 1.4923e-2 5.5630e-3 1.9345e-3 6.3934e-4 2.0370e-4

ε = 10−12 Rate

|||u I − uh |||

Rate

1.2671 1.3128 1.4236 1.5239 1.5973 1.6501

8.9222e-2 3.7072e-2 1.4923e-2 5.5630e-3 1.9345e-3 6.3933e-4 2.0370e-4

1.2671 1.3128 1.4236 1.5239 1.5973 1.6501

of convergence are almost twice as big and greater than 3/2, possibly behaving like O N −2 ln2 N . Additional numerical experiments were carried out on sequences of regular grids covering various combinations of the parameter settings σ ∈ {−1, 0, 1} and α1 ∈ {0, 1/2}. For the error |||u − uh ||| a variation of at most 6% was observed and for the finest meshes same values were calculated. The error component |||u I − uh ||| varied less than 7% for N ≥ 16. For N = 256 a variation of less then 0.3% indicates that the effect of the parameters σ and α1 on the error of the method is asymptotically negligible. Table 2 shows that the application of bilinear elements outside the layer region (c.f. Fig. 5b) does not improve the results in comparison to linear ones. Finally, let us consider the case of non-matching subtriangulations with different asymptotic behaviour of the mesh parameters N −1 and (Fig. 6). Table 2 Errors and convergence rates on matching tensor product grids of rectangular type, symmetric mortaring (σ = −1, α1 = 1/2 and γ N = 0.5N(ln N)−1 ) 2

N

|||u − uh |||

Rate

|||u − uh ||| lnNN

|||u I − uh |||

Rate

|||u I − uh ||| (lnNN)2

8 16 32 64 128 256 512

4.3338e-1 2.9277e-1 1.8454e-1 1.1113e-1 6.4915e-2 3.7113e-2 2.0882e-2

0.5659 0.6658 0.7317 0.7756 0.8066 0.8296

1.6673 1.6895 1.7039 1.7102 1.7125 1.7134 1.7139

8.9600e-2 3.8237e-2 1.5356e-2 5.6754e-3 1.9613e-3 6.4547e-4 2.0506e-4

1.2285 1.3162 1.4360 1.5329 1.6034 1.6543

1.3262 1.2734 1.3091 1.3440 1.3650 1.3757 1.3813

Nitsche-mortaring for convection–diffusion problems 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

599

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 6 Meshes of regular type with non-matching subtriangulations

Table 3 Errors and rates in the energy norm on non-matching grids of regular type, symmetric mortaring (σ = −1, α1 = 1/2), (a, b) γ N = 0.5N(ln N)−1 and (c) γ N = 0.5N(ln N)−3/4 (a) ∼ N −1 ln N N

|||u − uh |||

8 16 32 64 128 256 512 1,024

4.3357e-1 2.9337e-1 1.8465e-1 1.1114e-1 6.4918e-2 3.7113e-2 2.0884e-2

Rate 0.5635 0.6679 0.7324 0.7757 0.8067 0.8295

|||u I − uh |||

Rate

8.9222e-2 4.0137e-2 1.6216e-2 6.4850e-3 2.3949e-3 8.9699e-4 3.4133e-4 1.3409e-4

1.1525 1.3075 1.3223 1.4372 1.4168 1.3939 1.3479

(b) Components of |||u I − uh ||| for ∼ N −1 ln N N

u I − uh ε

8 16 32 64 128 256 512 1,024

8.9085e-2 3.7676e-2 1.5288e-2 5.7543e-3 1.9373e-3 6.3732e-4 2.0245e-4 6.4194e-5

1/2

Rate

γ N [[u I − uh ]]0,

1.2416 1.3012 1.4097 1.5706 1.6039 1.6545 1.6570

3.7344e-3 1.1716e-2 4.8467e-3 2.8018e-3 1.3548e-3 6.1734e-4 2.7138e-4 1.1690e-4

Rate

|||u I − uh |||

Rate

0.5619 0.6683 0.7320 0.7759 0.8070 0.8299

8.8079e-2 4.3328e-2 2.0209e-2 1.0592e-2 6.2007e-3 3.5018e-3 1.9293e-3

1.0235 1.1003 0.9320 0.7725 0.8243 0.8600

Rate 1.2734 0.7906 1.0482 1.1340 1.1858 1.2150

(c) ∼ N −3/4 (ln N)3/4 N

|||u − uh |||

8 16 32 64 128 256 512

4.3344e-1 2.9361e-1 1.8476e-1 1.1123e-1 6.4964e-2 3.7132e-2 2.0889e-2

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Table 3 supports our analysis of the mortar method. The actual errors converge as predicted, i.e., |||u − uh ||| = O N −1 ln N . Moreover, we observe a reduction of the degree of supercloseness if is chosen according to (20) or (23). In the former case the distance between the interpolant of u and the numerical solution uh converges with a rate smaller than 3/2. Table 3b shows that in this case the norm component containing the constant γ N is crucial for sufficiently large N and that this component converges with a slowly growing rate, possibly behaving as predicted (O N −3/2 ln3/2 N ). Furthermore, Table 3c confirms Remark 8: in accordance with our analysis the supercloseness error u I − uh as well as the error u − uh converge with similar rates smaller than one. Note that the supercloseness property is not lost but transformed: this method attains same errors on asymptotically coarser meshes.

Appendix: Proof of Lemma 4 For E ∈ Ei we denote by E (E) := {F ∈ E : |F ∩ E| > 0} the set of elements of the induced triangulation that contain more than one point of the segment E. Note that E (E) isnever empty as E , E1 and E2 are triangulations of the interface and that E = F∈E (E) F ∀E ∈ Ei , i = 1, 2. Furthermore for E ∈ Ei we define Ti (E) to be that rectangle (i = 1) or triangle (i = 2), that contains the complete side E, i.e. Ti (E) ∩ = E. We ⊥ denote by hi,E the height of the element Ti (E) perpendicular to the side E. We ⊥ ⊥ := hi,E for all F ∈ E (E), see extend the last definition to elements of E by hi,F Fig. 7 for some illustration. For example, E (Er ) = {Ea , Eb , Ec }, E (Es ) = {Eb }, T1 (Er ) = Tr , T2 (Es ) = Ts and h2,Et = h2,Ec = h2,Ed = |Ts ∩ Tt |. Let ∂v h ∂v h I1 := α1 1 − α2 2 , [[v h ]] . ∂n1 ∂n2 E E∈E Fig. 7 Definitions along the interface

ε1

1

ε ε2

Ea Tr

Er

2

Eb Es Ec

Ts Tt

Ed Et Ω1

Γ Γ

Γ

Ω2


601

We start by using a triangle inequality and the Cauchy–Schwarz inequality ∂v h h ∂v ∂v1h ∂v2h h h 1 2 |I1 | = − α2 , [[v ]] = α1 − α2 , [[v ]] α1 ∂n1 ∂n2 ∂n1 ∂n2 E E∈E

∂v1h ∂v h h ≤ α1 , [[v ]] + α2 2 ∂n1 ∂n2 h α1 ∂v1 , [[v h ]] + ≤ ∂n1 E E∈E E∈E 1

2

, [[v ]] h α2 ∂v2 , [[v h ]] ∂n2 E h

h h ∂v1 h ∂v2 h [[v ]] + [[v ]] . α1 α ≤ 2 ∂n 0,E 0,E ∂n2 0,E 1 0,E E∈E1

E∈E2

Young’s inequality yields h 2 h 2 α12 ∂v μ E 1 [[v ]] + |I1 | ≤ 0,E 2μ E ∂n1 0,E 2 E∈E1 α2 h 2 νE h 2 2 ∂v2 + [[v ]] 0,E + 2ν E ∂n2 0,E 2 E∈E

2

for arbitrary constants μ E > 0 and ν E > 0. Since ni | E is constant (i = 1, 2, E ∈ E ) and vh |T is a polynomial, the normal ∂v h derivative ∂nii is a polynomial too. Therefore, Lemma 3 yields T

|I1 | ≤ C0

E∈E1

h 2 ∂v1 α12 α22 + C 0 2μ E h⊥ 2ν E h⊥ 1,E ∂n1 0,T1 (E) 2,E E∈E 2

h 2 ∂v2 ∂n 2 0,T2 (E)

μE νE [[v h ]]2 + [[v h ]]2 . + 0,E 0,E 2 2 E∈E E∈E 1

(25)

2

Note, for example, if α1 = 0 then the third term vanishes, because the inner product that led to this term becomes zero. Defining the coefficient μ¯ F := μ E with E ∈ E1 such that F ⊂ E, we can switch to the induced triangulation. Therefore, we get for the third term in (25): μE μE [[v h ]]2 = [[v h ]]2 ds 0,E 2 2

E∈E1

E∈E1

F∈E (E) F

μ¯ F μ¯ E [[v h ]]2 . = [[v h ]]2 ds = 0,E 2 F 2 E∈E1 F∈E (E)

E∈E

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Treating the last term in (25) in the same way, we obtain h 2 ∂v1 α12 α22 |I1 | ≤ C0 + C 0 2μ E h⊥ 2ν E h⊥ 1,E ∂n1 0,T1 (E) 2,E E∈E E∈E +

1

2

h 2 ∂v2 ∂n 2

0,T2 (E)

2 max{μ¯ E , ν¯ E } [[v h ]]0,E .

E∈E

2 Note, if α1 = 0 then the last term becomes E∈E ν¯ E /2 [[v h ]]0,E . Next, let μ > 0 be an arbitrary constant. Set μ E = μ/ h⊥ 1,E for E ∈ E1 , and ! ⊥ for E ∈ E . Extending the definition of h to E ∈ E , we have the ν E = μ h⊥ 2 2,E 1,E identity μ E = μ¯ F for all F ∈ E (E). Thus, α 2 ∂v h 2 α 2 ∂v h 2 2 1 2 1 2 |I1 | ≤ C0 + C + α E [[v h ]]0,E 0 2μ ∂n1 0,T1 (E) 2μ ∂n2 0,T2 (E) E∈E1

E∈E2

E∈E

with α E given by (6). h ∂v Finally, the inequality ∂nii = ∇vih · ni ≤ |∇vi | |ni | = |∇vi | implies |I1 | ≤ C0

α2 α2 2 2 2 1 2 ∇v1h 0,T1 (E) + C0 ∇v2h 0,T2 (E) + μ α E [[v h ]]0,E 2μ 2μ E∈E E∈E E∈E 1

≤ C0

2 2 α12 ∇v h 2 + C0 α2 ∇v h 2 + μ α E [[v h ]]0,E 1 0,T 2 0,T μ μ T∈T1

≤

2

T∈T2

E∈E

2 C0 α12 h 2 C0 α22 h 2 v1 1,1 + v2 1,2 + μ α E [[v h ]]0,E . μ μ E∈E

This completes the proof of Lemma 4.

References 1. Apel, T.: Anisotropic finite elements: Local estimates and applications. In: Advances in Numerical Mathematics. Teubner, Leipzig (1999) 2. Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47(3–4), 277–293 (1992) 3. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982) 4. Becker, R., Hansbo, P., Stenberg, R.: A finite element method for domain decomposition with non-matching grids. Math. Model. Numer. Anal. 37(2), 209–225 (2003) 5. Braess, D., Dahmen, W., Wieners, C.: A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37(1), 48–69 (1999). doi:10.1137/S0036142998335431 6. Burman, E., Ern, A.: Continuous interior penalty hp-finite element methods for transport operators. In: de Castro, A.B., et al. (eds.) Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2005, The 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, 18–22 July 2005, pp. 504–511. Springer, Berlin (2006) 7. Heinrich, B., Pönitz, K.: Nitsche type mortaring for singularly perturbed reaction–diffusion problems. Computing 75(4), 257–279 (2005). doi:10.1007/s00607-005-0123-5


603

8. Linß, T.: Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer. Methods Partial Differ. Equ. 16(5), 426–440 (2000) 9. Linß, T.: Layer-adapted meshes for reaction-convection–diffusion problems. In: Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010) 10. Linß, T., Stynes, M.: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J. Math. Anal. Appl. 261, 604–632 (2001) 11. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin (1972) 12. Melenk, J.M.: hp-finite element methods for singular perturbations. In: Lecture Notes in Mathematics, vol. 1796. Springer, Berlin (2002) 13. Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. In: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996) 14. Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971). In German 15. Pönitz, K.: Finite-Elemente-Mortaring nach einer Methode von J. A. Nitsche für elliptische Randwertaufgaben. Ph.D. thesis, Technische Universität Chemnitz (2006). In German 16. Roos, H.G.: Superconvergence on a hybrid mesh for singularly perturbed problems with exponential layers. ZAMM, Z. Angew. Math. Mech. 86(8), 649–655 (2006). doi:10.1002/zamm. 200510264 17. Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. In: Convection–Diffusion–Reaction and Flow Problems, 2nd ed., vol. 24. Springer, Berlin (2008) 18. Schieweck, F.: On the role of boundary conditions for CIP stabilization of higher order finite elements. ETNA, Electron. Trans. Numer. Anal. 32, 1–16 (2008) 19. Stynes, M., O’Riordan, E.: A uniformly convergent Galerkin method on a Shishkin mesh for a convection–diffusion problem. J. Math. Anal. Appl. 214(1), 36–54 (1997) 20. Stynes, M., Tobiska, L.: The SDFEM for a convection–diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41(5), 1620– 1642 (2003) 21. Warburton, T., Hesthaven, J.S.: On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765–2773 (2003) 22. Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-d convection–diffusion problems. Math. Comput. 72(243), 1147–1177 (2003) 23. Zienkiewicz, O., Zhu, J.: Superconvergence and the superconvergent patch recovery. Finite Elem. Anal. Des. 19(1–2), 11–23 (1995)