Higher Order Mortar Finite Element Methods in 3D ...

Numerische Mathematik manuscript No. (will be inserted by the editor)

Higher Order Mortar Finite Element Methods in 3D with Dual Lagrange Multiplier Bases⋆ B. P. Lamichhane1 , R. P. Stevenson2 , B. I. Wohlmuth1 1

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Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany, e-mail: {lamichhane,wohlmuth}@mathematik.uni-stuttgart.de Department of Mathematics, Utrecht University, The Netherlands, e-mail: [email protected]

The date of receipt and acceptance will be inserted by the editor

Summary Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension. We provide a new theoretical framework within this relaxed setting, which opens a new and simpler way to construct dual Lagrange multiplier bases for higher order finite element spaces. As examples, we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements. Numerical results illustrate the performance of our approach. Key words. Mortar finite elements, Lagrange multipliers, biorthogonal bases, nonmatching triangulations. AMS subject classification. 35N55, 65N30. 1 Introduction Over the recent past years, mortar methods with dual Lagrange multiplier bases have become an active area of research, see, e.g., [28, ⋆ This work was supported in part by the Deutsche Forschungsgemeinschaft, SFB 404, C12, the Netherlands Organization for Scientific Research and by the European Community’s Human Potential Programme under contract HPRN-CT2002-00286.

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B. P. Lamichhane et al.

31,18,29,19,20]. Originally introduced to couple spectral and finite element approximations [5–7], mortar methods have a wide range of applications. Being a powerful tool to couple different discretization schemes or nonmatching triangulations, these nonconforming techniques provide a more flexible approach than standard conforming approaches. Examples where such situations occur are when, in a parallel environment, mesh generators are used independently on different subdomains, or when different physical models or, within one model, strongly different parameter values naturally lead to the use of different discretization schemes on different subdomains. The central idea of mortar techniques is to replace the strong, pointwise continuity across the interface by a weak one, which requires that the jump of the solution across the interface is orthogonal to a suitable Lagrange multiplier space. Under this relaxed condition, optimal a priori estimates of the discretization error have been demonstrated, cf. [4,3,7,29,18], and various iterative methods have been introduced to solve the arising linear system, see [9,8,31,15,27]. Generally, the mortar approach has the disadvantage that even when the boundary value problem is elliptic, the arising linear system is of saddle point type, usually for which iterative methods are known to be less efficient than for symmetric positive definite systems. However, when working with dual Lagrange multiplier bases, the degrees of freedom associated to the multiplier can be locally eliminated leading to a sparse, positive definite system, on which, for example, efficient multigrid methods can be applied, see [31,29]. For the lowest order finite elements in 3D, dual Lagrange multiplier bases were constructed in [18,29]. However, the construction of such bases for higher order elements is more complicated, see [19,20,23]. In the present paper, we develop a general framework facilitating the construction of Lagrange multiplier bases, in particular for higher order elements. In a first step, we remove the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension, with which the introduction of additional degrees of freedom at the multiplier side only to satisfy this condition on the dimensions is avoided. As a result, we can work with a subspace of the trace space for which it is easier to construct a dual Lagrange multiplier basis. In a second step, we describe a general procedure for the construction of multiplier basis functions near the interface boundaries, where one has to deal with the complication that homogeneous boundary conditions are incorporated in the trace space of the approximation space at the slave side, but not in the Lagrange

Mortar finite elements with dual Lagrange multiplier bases

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multiplier space. Although the theory presented in this paper applies to general space dimensions, our concrete realizations of dual Lagrange multiplier bases will take place in the three dimensional case. This paper is organized as follows: In the rest of this section, we describe the problem setting, and fix some notations related to the mortar method. In Section 2, we prove some optimal a priori estimates without assuming that the trace space with zero boundary condition on the interface from the slave side has the same dimension as the multiplier space. As in [18,12], we allow locally refined finite element partitions and geometrically nonconforming subdivisions into subdomains. Furthermore, as in [20,12], we do not require that the partitions match on the boundaries of the interfaces. For the case of having quasi-uniform meshes on each subdomain, we will prove an error estimate showing that the possibly disadvantageous effect of having at the slave side a finer mesh than at the master side is much milder than that is indicated by existing error estimates, e.g., in [29,20]. A general framework for the construction of dual multiplier bases, in particular for the modifications needed along the interface boundaries, is given in Section 3. Examples of dual multiplier bases are given for quadratic and cubic tetrahedral elements, and for quadratic serendipity hexahedral elements. Finally, in Section 4, we present some numerical results concerning the discretization errors in the L2 -norm, the piecewise H 1 -norm, and in a weighted L2 -norm for the Lagrange multiplier. We consider the following elliptic second order boundary value problem in variational form: Find u ∈ H01 (Ω) such that Z (1.1) a∇u · ∇v + cuvdx = f (v), v ∈ H01 (Ω), Ω

where Ω ⊂ Rd is a bounded polytope, a, c ∈ L∞ (Ω), a > a0 > 0 and c ≥ 0. The domain Ω is decomposed into K non-overlapping polytopes Ωk , i.e., Ω=

K [

Ωk

with

k=1

Ωk ∩ Ωℓ = ∅

for k 6= ℓ.

Defining, for each 1 ≤ k ≤ K, H∗1 (Ωk ) := {v ∈ H 1 (Ωk ), v|∂Ω∩∂Ω = 0}, k

QK

we assume that f ∈ k=1 (H∗1 (Ωk ))′ . We assume that a and c are piecewise smooth with respect to the subdivision into subdomains, but

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allow that they have jumps over the interfaces. However, in this paper, no attempt is made to obtain results that hold uniformly in the sizes of such jumps, cf. [17]. For any {k, ℓ} with 1 ≤ k 6= ℓ ≤ K, for which the closures of Ωk and Ωℓ have a (d − 1)-dimensional intersection, we set γ i = Ω k ∩ Ω ℓ , where 1 ≤ i ≤ N is a number uniquely associated to this {k, ℓ}, and thus with N being the total number of sets {k, ℓ} having this property. For each 1 ≤ i ≤ N , one of the subdomains Ωk and Ωℓ on both sides of the interface is referred to be at the slave or non-mortar side, and its index is denoted as s(i), where the other is referred to be at the master or mortar side, with its index denoted as m(i). For each δ from some index set I, and each 1 ≤ k ≤ K, we assume a finite element space Xkδ ⊂ H∗1 (Ωk ) with respect to a partition of Ωk into “elements”, e.g., d-simplices or d-rectangles. We assume that for any δ ∈ I, γi is the union of complete (d − 1)-dimensional faces of (A.1) elements at the slave side. Note that (A.1) does allow the partition into subdomains to be geometrically nonconforming; γi needs not to be a full face of both Ωm(i) and Ωs(i) , or even not to be a full face of either Ωm(i) or Ωs(i) , see Figure 1.1. In case the decomposition into subdomains is geometrically

Fig. 1.1: An interface that is not a full face of either Ωm(i) or Ωs(i) ; it satisfies (A.1) when the square partitioned into triangles is a face of Ωs(i) , but not when this is a face of Ωm(i) . conforming, (A.1) is automatically satisfied. For 1 ≤ i ≤ N and δ ∈ I, we define the trace space δ δ W0,i = {v δ |γi : v δ ∈ Xs(i) , v δ |∂Ωs(i) \γi = 0}. δ As a consequence of Xs(i) ⊂ H∗1 (Ωs(i) ) being a finite element space, δ ⊂ H 1 (γ ). We will refer to the partition of γ we can assume that W0,i i 0 i into the faces of elements as referred to in (A.1) as the finite element δ . For enforcing a weak coupling of the finite partition underlying W0,i element functions over the interfaces, for each 1 ≤ i ≤ N and δ ∈ I


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we introduce a Lagrange multiplier space Miδ ⊂ L2 (γi ). Thinking of Miδ as being a finite element space, the partition underlying Miδ will δ . be equal to that of W0,i Q Q δ Setting the product spaces X δ := K Xkδ , M δ := N i=1 Mi , and k=1 R PK defining the bilinear forms a(v, w) = k=1 Ωk a∇v · ∇w + cvwdx PN R and b(v, µ) = i=1 γi [v]γi µdσ, where [v]γi denotes the jump of v on γi from the master to the slave side, the mortar finite element discretization now consists in finding (uδ , λδ ) ∈ X δ × M δ such that a(uδ , v δ )+b(v δ , λδ ) = f (v δ ), b(uδ , µδ ) = 0,

vδ ∈ X δ , µδ ∈ M δ ,

(1.2)

or, equivalently, considering the first component only, in finding uδ ∈ V δ := {v δ ∈ X δ : b(v δ , µδ ) = 0, µδ ∈ M δ } such that a(uδ , v δ ) = f (v δ ),

vδ ∈ V δ .

(1.3)

Assuming that for any δ ∈ I and 1 ≤ i ≤ N , the constant function 1 ∈ Miδ ,

(A.2)

and that βiδ :=

inf

sup

µδi ∈Miδ \{0} w δ ∈W δ \{0} i 0,i

hµδi , wiδ iL2 (γi )

kµδi kL2 (γi ) kwiδ kL2 (γi )

> 0,

(1.4)

the problem (1.2) and thus (1.3) are known to have a unique solution, cf. [7,10]. In this paper, in order to avoid the repeated use of generic but unspecified constants, by C < ∼ D we mean that C can be bounded by a multiple of D, independently of parameters which C and D may depend on, where we, in particular, think of δ as well as the number < of subdomains K. Obviously, C > ∼ D is defined as D ∼ C, and C h D > as C < ∼ D and C ∼ D. 2 A priori estimates We start with establishing suitable mortar projectors without requirδ , as is usually done in the literature. ing that dimMiδ = dimW0,i Lemma 2.1 Let M, W be subspaces of some Hilbert space H with L := dimM = dimW < ∞.

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(a). It holds β :=

hµ, wi > 0, µ∈M \{0} w∈W \{0} kµkkwk inf

sup

(2.1)

if and only if there exists a projector Π onto W for which hΠv, µi = hv, µi

v ∈ H, µ ∈ M.

Moreover, kΠk = β −1 . (b). Let {m1 , . . . , mL } and {w1 , . . . , wL } be bases for M and W , respectively. Then β > 0 if and only if the matrix R := (hwℓ , mℓ′ i)1≤ℓ,ℓ′ ≤L is invertible. In particular, with λM , ΛM being the largest or smallest constant such that λM kck ≤ k

L X ℓ=1

cℓ mℓ k ≤ ΛM kck,

c = (cℓ )1≤ℓ≤L ∈ RL ,

and with λW , ΛW defined analogously for the basis for W , we have (ΛM ΛW )−1 ≤ βkR−1 k ≤ (λM λW )−1 . P P Furthermore, Πv = ℓ hv, mℓ i ℓ′ (R−1 )ℓ,ℓ′ wℓ′ .

Proof For the non-trivial part of (a), see e.g. [13, Theorem 2.1(a)], and a straightforward computation shows (b). Proposition 2.2 Thanks to (1.4), there exist a bounded projector δ with Πiδ : L2 (γi ) → L2 (γi ) onto W0,i hΠiδ v, µδi iL2 (γi ) = hv, µδi iL2 (γi )

v ∈ L2 (γi ), µδi ∈ Miδ .

Such a projector will be referred to as being √ a mortar projector. It can be selected such that kΠiδ kL2 (γi )→L2 (γi ) ≤ 2 (βiδ )−1 . δ denote the L2 (γ )-orthogonal projector Proof Let Qδi : L2 (γi ) → W0,i i δ , and let Q ˘ δ denote its restriction to M δ . For any µδ ∈ M δ , onto W0,i i i i i we have

˘ δ µδ kL2 (γ ) = kQ i i i =

sup δ \{0} wiδ ∈W0,i

sup δ \{0} wiδ ∈W0,i

˘ δ µδ , wδ iL2 (γ ) hQ i i i i kwiδ kL2 (γi )

hµδi , wiδ iL2 (γi ) kwiδ kL2 (γi )

≥ βiδ kµδi kL2 (γi ) .


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δ be defined by the orthogonal decomposition W δ = Let Uiδ ⊂ W0,i 0,i ⊥L2 (γ ) δ δ δ ˘ ˘ ˘ i ℑ Qi ⊕ Ui , where ℑQ denotes the range of Q. For any µi ∈ Miδ , δ δ ui ∈ Ui , we have

˘ δi µδi + uδi iL2 (γ ) = hQ ˘ δi µδi + uδi , Q ˘ δi µδi + uδi iL2 (γ ) , hµδi + uδi , Q i i

(2.2)

and so from q

δ 2 ˘ δ µδ k2 2 kQ i i L (γi ) + kui kL2 (γi ) q ≥ βiδ kµδi k2L2 (γi ) + kuδi k2L2 (γi ) √ ≥ βiδ 21 2 kµδi + uδi kL2 (γi ) ,

˘ δ µδ + uδ kL2 (γ ) = kQ i i i i

˜ δ := M δ + U δ ] satwe infer that [“H”, “W ”, “M√”] = [L2 (γi ), W0,i , M i i i isfies (2.1) with “β” ≥ βiδ 12 2. Since (2.2) also shows that for any δ \{0} there exists a µ ˜ δ with h˜ ˜δi ∈ M µδi , wiδ iL2 (γi ) 6= 0, we wiδ ∈ W0,i i δ = dimM ˜ δ , and so the proof follows from an conclude that dimW0,i i application of Lemma 2.1(a). We assume that for 1 ≤ i ≤ N , δ ∈ I there exist 0 < hδi ∈ L∞ (γi ) such that δ −1 δ kwiδ kH 1 (γi ) < ∼ k(hi ) wi kL2 (γi ) ,

δ wiδ ∈ W0,i ,

(A.3)

and inf

δ wiδ ∈W0,i

1 k(hδi )−1 (w −wiδ )kL2 (γi ) +kwiδ kH 1 (γi ) < ∼ kwkH 1 (γi ) , w ∈ H0 (γi ).

(A.4) If the finite element partition underlying is shape regular and quasi-uniform, both uniformly in δ, then with hδi being the constant function defined as the maximum of the diameters of the elements, (A.3) is the well-known inverse inequality, and (A.4) is guaranteed δ contains all continuous funcby the minimal requirement that W0,i tions that vanish at ∂γi and that are piecewise linear with respect to the partition. Yet, for partitions that, instead of quasi-uniform, are only locally quasi-uniform, meaning that any two elements that have non-empty intersection have uniformly comparable diameters, (A.3) and (A.4) are also valid, now with hδi being defined as the piecewise constant function equal to the diameter of the underlying element. See, e.g., [11,24]. As expected, we will need that the βiδ , defined in (1.4), satisfy δ W0,i

βiδ > ∼ 1,

(A.5)

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i.e., uniformly in δ ∈ I and i. Then from Lemma 2.1 and Proposition 2.2 we know that there exist mortar projectors Πiδ for which δ kΠiδ kL2 (γi )→L2 (γi ) < ∼ 1. For the case that not all hi are constant functions, we make the additional assumption that these projectors can be selected so that δ −1 2 (A.6) k(hδi )−1 Πiδ wkL2 (γi ) < ∼ k(hi ) wkL2 (γi ) , w ∈ L (γi ).

δ is shape regular Again assuming that the partition underlying Wi,0 and locally quasi-uniform, (A.6) is valid when the Πiδ are local in the sense that Πiδ v vanishes on an element T whenever v vanishes on all elements that have distance to T less than some absolute multiple of diam(T ). δ , . . . , wδ } for W δ , and a collecRemark 2.3 Assume a basis {wi,1 i,0 i,L tion {m ˜ δi,1 , . . . , m ˜ δi,L } containing Miδ in its span, both which are uniP 2 formly L2 (γi )-stable, with which we mean that k L ℓ=1 cℓ wi,ℓ kL2 (γi ) h PL 2 2 ℓ=1 |cℓ | kwi,ℓ kL2 (γi ) , and analogously for the other collection. Suppose further that both collections consist of local functions, with which we mean that the support of any of these functions consists of a connected union of a (uniformly) bounded number of underlying ele

ments. Finally, suppose that Rδi :=

δ ,m ˜ δi,ℓ′ iL2 (γ hwi,ℓ

i)

δ k kwi,ℓ ˜ δi,ℓ′ kL2 (γ L2 (γ ) km i

i)

1≤ℓ,ℓ′ ≤L

is (uniformly) bounded invertible, and that its inverse is local in δ , supp m ˜ δi,ℓ′ ) is the sense that ((Rδi )−1 )ℓ,ℓ′ = 0 whenever dist(supp wi,ℓ δ ) larger than some absolute multiple of the maximum of diam(supp wi,ℓ δ and diam(supp m ˜ i,ℓ′ ). Then from Lemma 2.1(b) we know that such local, and uniformly L2 (γi )-bounded mortar projectors exist. Lemma 2.4 Πiδ : L2 (γi ) → L2 (γi ) and Πiδ : H01 (γi ) → H01 (γi ) are bounded, uniformly in δ and i. Proof We already have seen that the first statement is a consequence δ , by (A.3), (A.6) and (A.4) of (A.5). Since Πiδ is a projector onto W0,i for w ∈ H01 (γi ) we have kΠiδ wkH 1 (γi ) < ∼

δ wiδ ∈W0,i

< ∼

δ wiδ ∈W0,i

< ∼

δ wiδ ∈W0,i

inf

kΠiδ (w − wiδ )kH 1 (γi ) + kwiδ kH 1 (γi )

inf

k(hδi )−1 Πiδ (w − wiδ )kL2 (γi ) + kwiδ kH 1 (γi )

inf

k(hδi )−1 (w − wiδ )kL2 (γi ) + kwiδ kH 1 (γi )

< kwk 1 , H (γi ) ∼


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which shows the second statement. Finally, for completeness, we mention the following technical assumption:  There exist Jkδ : H 1 (Ωk ) → Xkδ with kJkδ kH 1 (Ωk )→H 1 (Ωk ) < ∼ 1, δ δ δ (A.7) such that whenever v|∂Ωk = vk |∂Ωk for some vk ∈ Xk ,  δ δ then (Jk v)|∂Ωk = vk |∂Ωk .

For any given Xkδ , the existence of such Jkδ can be demonstrated as in [24]. Under the assumptions we made, and using Lemma 2.4, the error estimates proven in [20] easily generalize to the present more general setting. See [3,18,12] for preceding or related work. We will need 1 ∂u ∈ (H 2 (γi ))′ . A sufficient condition is the mild regularity that a ∂n γ i

assumption u ∈ H 1+ε (Ω) for some arbitrary ε > 0, since then the 1 1 ∂u trace theorem shows that a ∂n ∈ H − 2 +ε (γi ) ⊂ (H 2 (γi ))′ . γ i

(uδ , λδ )

Theorem 2.5 With u and being the solutions of (1.1) and 1 ∂u (1.2), respectively, assume that a ∂nγ ∈ (H 2 (γi ))′ , 1 ≤ i ≤ N . Then i

K X k=1

ku−uδ k2H 1 (Ωk ) < ∼ inf

vδ ∈V δ

K X k=1

ku −

(2.3) v δ k2H 1 (Ωk )

+ inf

µδ ∈M δ

N X i=1

ka

∂u , − µδ k2 1 (H 2 (γi ))′ ∂nγi

where inf

vδ ∈V δ

K X

k=1 N X i=1

ku −

v δ k2H 1 (Ωk )

< inf ∼ xδ ∈X δ

K nX k=1

ku − xδ k2H 1 (Ωk ) +

(2.4)

o 1 1 k(hδi )− 2 (u − xδ )|Ωm(i) k2L2 (γi ) + k(hδi )− 2 (u − xδ )|Ωs(i) k2L2 (γi ) .

With nγi being the outward normal on γi from the master side, for the Lagrange multiplier we have N X i=1

ka

∂u − λδ k2 1 2 (γ ))′ ∂nγi (H00 i < ∼

K X k=1

(2.5)

ku − uδ k2H 1 (Ωk ) + inf

µδ ∈M δ

N X i=1

ka

∂u . − µδ k2 1 2 (γ ))′ ∂nγi (H00 i

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Proof Using (A.2), (2.3) follows from Strang’s second lemma as in δ has [20, proof of Theorem 2.3]. Because of (A.7), any wiδ ∈ W0,i δ δ an extension to a xδs(i) ∈ Xs(i) with kxδs(i) kH 1 (Ωs(i) ) < , ∼ kwi k 21 H00 (γi )

see, e.g., [25, Lemma 5.1]. Now, following [20, proof Lemma 2.2], one derives that

inf

vδ ∈V δ

K X k=1

ku − v δ k2H 1 (Ωk ) < ∼ inf

xδ ∈X δ

K X k=1

(2.6)

ku − xδ k2H 1 (Ωk ) +

N X i=1

kΠiδ [u − xδ ]γi k2

1

2 (γ ) H00 i

.

From (A.3), and (A.6), kΠiδ kL2 (γi )→L2 (γi ) < ∼ 1, using interpolation < k(hδ )− 12 wδ k 2 (cf. [26, ex. in §1.15.3]) we infer that kwiδ k 12 i ∼ i L (γi ) H00 (γi ) δ −1 δ , and k(hδ )−1 Π δ wk < for any wiδ ∈ W0,i L2 (γi ) ∼ k(hi ) wkL2 (γi ) for any i i w ∈ L2 (γi ), respectively, so that (2.4) is valid. The bound (2.5) for the Lagrange multiplier follows from [3, Lemma 2.7], where the condition

inf

sup

µδ ∈M δ \{0} w∈ ˜ δ \{0} ˜ X

P b(w, ˜ µδ )2 / 1≤i≤N kµδ k2 1 2 (γ ))′ (H00 i X X δ δ 2 k[w ˜ ]γi k2 1 kw ˜ kH 1 (Ωk ) + 1≤i≤N

1≤k≤K

2 (γ ) H00 i

>1 ∼

can be shown as in [20, Lemma 2.4]. Corollary 2.6 For 1 ≤ k ≤ N and δ ∈ I, let 1 < tk 6∈ N + 12 and 0 < hδk ∈ L∞ (Ωk ) be such that hδi := hδs(i) |γi ∈ L∞ (γi ) and (A.3), (A.4) and (A.6) are valid, and 1

inf kxk − xδk kH 1 (Ωk ) + k(hδk )− 2 (xk − xδk )|∂Ωk kL2 (∂Ωk )

xδk ∈Xkδ

(2.7)

< khδ ktk∞−1 kxk k tk xk ∈ H∗1 (Ωk ) ∩ H tk (Ωk ), H (Ωk ) , ∼ k L (Ωk ) ts(i) −1 δ ts(i) −1 (γi ). inf kµi − µδi kL2 (γi ) < ∼ khi kL∞ (γi ) kµi kH ts(i) −1 (γi ) , µi ∈ H δ δ

µi ∈Mi

(2.8)


Assume that the solution u of (1.1) is in solution (uδ , λδ ) of (1.2) satisfies K X k=1

ku − uδ k2H 1 (Ωk ) + h

N X i=1

ka

k=1 H

tk (Ω ). k

Then the

∂u < − λδ k2 1 ∼ 2 (γ ))′ ∂nγi (H00 i

1 + max khδm(i) /hδs(i) kL∞ (γi ) 1≤i≤N

QK

11

K iX k=1

(2.9)

2tk −2 2 khδk kL ∞ (Ω ) kukH tk (Ω ) . k k

Proof The proof follows easily from the estimates (2.3) and (2.4), , an application of the trace theorem, and ≤ k · k 12 k · k 21 ′ (H (γi ))

(H00 (γi ))′

inf µδ ∈M δ kµi − µδi k i

1

(H 2 (γi ))′

i

3

−1 < khδ kts(i) ∼ s(i) L∞ (Ωs(i) ) kµi kH ts(i) − 23 (γ ) for any i

µi ∈ H ts(i) − 2 (γi ), which is a consequence of (2.8).

Assuming Xkδ contains all continuous, piecewise polynomials of degree ⌈tk − 1⌉ that vanish on ∂Ωk ∩ ∂Ω with respect to a shape regular and locally quasi-uniform finite element partition, with hδk being the piecewise constant function equal to the diameter of the underlying element (2.7) is easily verified using a nodal, or, when ⌊tk ⌋ ≤ d2 , a modified Clément interpolator as introduced in [24]. Similarly, (2.8) is satisfied when Miδ contains all piecewise, or continuous piecewise polynomials of degree ⌈ts(i) − 2⌉ with respect to the underlying partition inherited from the slave side. Q K tk Corollary 2.6 requires that u ∈ k=1 H (Ωk ), which, in case mink tk > 2, even for a convex polygon Ω generally can only be guaranteed when the right-hand side f vanishes at the corners of Ω. For less smooth u, using suitably locally refined partitions better results may be expected than those reflected by Corollary 2.6, that shows no benefits of local refinements. From the theory of nonlinear approximation, see, e.g. [14], in combination with the bounds from Theorem 2.5, we infer that we may hope for the same asymptotic behaviour of the error as function of the number of degrees of freedom under the relaxed assumption that u is the product space QK t k k=1 Wp (Ωk ), for certain p < 2, or, more precisely, that it is in such a product space with the Sobolev spaces Wptk (Ωk ) replaced by certain tk (Ω ). A treatment of this topic is beyond the scope Besov spaces Bp,q k of this paper. If the finite element partitions from slave and master side match on the interface boundaries ∂γi (and there underly the same finite element space), and tk = t for all 1 ≤ k ≤ K, then starting from (2.6), and by using Lemma 2.4, (2.9) can be proven without the penalty

12


factor 1 + max1≤i≤N khδm(i) /hδs(i) kL∞ (γi ) . This has been shown in [18] for t = 2 and d ≤ 3 by selecting xδ as the nodal interpolant of u, but it extends to arbitrary d and t > 1 by taking xδ as a quasi-interpolant of u that on ∂γi only depends on u|∂γi ∈ L1 (∂γi ) (cf. [24]), and by 1

2 (γi ) and the interpolation space using that for each fixed ε > 0 , H00 1

+ε

1

[H02 (γi ), H 2 −ε (γi )]1/2 agree as sets and have equivalent norms. This “matching condition” on ∂γi is readily met in the two-dimensional case, where the interface boundaries consist of isolated points, but it is an inconvenient restriction in higher dimensions. In the remainder of this section, assuming quasi-uniform meshes on each subdomain without imposing the matching condition, we will demonstrate that the penalty factor 1+max1≤i≤N khδm(i) /hδs(i) kL∞ (γi ) is much too pessimistic. In Theorem 2.8, we will see that only the log of the quotient of the mesh sizes at master and slave sides, assuming that this quotient is larger than 1, may enter the upper bound of the error. Although, when having quasi-uniform meshes on each subdomain, for each interface the slave and master sides can be selected such that the mesh size at the master side is less than or equal to that at the slave side, it shows that a different choice of master and slave sides cannot do much harm. Moreover, it suggests that also for locally refined meshes, where uniform boundedness of khδm(i) /hδs(i) kL∞ (γi ) cannot be guaranteed, the penalty factor from Corollary 2.6 is likely too pessimistic, in any case when locally the mesh sizes along the interfaces do not vary too strongly. A generalization of our result to locally refined meshes might be possible by using sufficiently smooth mesh indicator functions hδk as used in [12]. However, the estimation of the unavoidable fractional Sobolev norms of functions that contain as factors powers of the non-constant functions hδk will require quite some technicalities. It is well-known that the interpolation space [H01 (γi ), L2 (γi )]θ is equal to H01−θ (γi ) for θ ∈ [0, 12 ), and that it is equal to H 1−θ (γi ) for θ ∈ ( 12 , 1], meaning that corresponding spaces are equal as sets and have equivalent norms. The space [H01 (γi ), L2 (γi )]1/2 is denoted 1/2

1/2

as H00 (γi ), and it is strictly contained H0 (γi ) with a strictly finer topology. In the following we will consider these interpolation spaces for θ ↓ 12 . We cannot expect that the aforementioned norm equivalences hold uniformly in θ ∈ ( 12 , 1] which motivates our following lemma.


13

Lemma 2.7 For 1 ≤ i ≤ N and θ ∈ [0, 1]\{ 12 }, we have kvk[H01 (γi ),L2 (γi )]θ

sup v∈[H01 (γi ),L2 (γi )]θ \{0}

kvkH 1−θ (γi )

< 1 − θ −1 . ∼ 2

Proof We may think of γi as being a polytope in Rd−1 . For any function v on γi , let v˜ denotes its extension on Rd−1 with zero. As follows v kH 1−θ (Rd−1 ) from [21, Lemma 11.3], we have kvk[H01 (γi ),L2 (γi )]θ < ∼ k˜ 2 1 uniformly in u ∈ [H0 (γi ), L (γi )]θ and θ ∈ [0, 1]. From [16, Lemma 1.3.2.6 and Theorem 1.4.4.4] it follows that for any θ ∈ [0, 1]\{ 12 }, there exists a constant Cθ > 0 such that k˜ v kH 1−θ (Rd−1 ) ≤ Cθ kvkH 1−θ (γi ) for all v ∈ [H01 (γi ), L2 (γi )]θ . Inspection of the proof of this Theorem 1.4.4.4 reveals that Cθ is a result of the application of Hardy’s inequality, and that it can be bounded by some absolute multiple of 1 − θ −1 . 2 Theorem 2.8 For 1 ≤ k ≤ N and δ ∈ I, let 1 < tk 6∈ N + 12 and hk > 0 be a constant such that, with hδi := hδs(i) |γi , (A.3), (A.4) and (A.6) are valid, and inf kxk −xδk kH 1 (Ωk )

(2.10)

xδk ∈Xkδ

< (hδ )tk −1 kxk k tk ∼ k H (Ωk ) ,

xk ∈ H∗1 (Ωk ) ∩ H tk (Ωk ),

δ ts(i) −1 inf kµi − µδi kL2 (γi ) < kµi kH ts(i) −1 (γ ) , ∼ (hi ) i

µi ∈ H ts(i) −1 (γi ).

µδi ∈Miδ

Assume that the solution u of (1.1) is in solution (uδ , λδ ) of (1.2) satisfies K X k=1

ku − uδ k2H 1 (Ωk ) +

N X i=1

ka

QK

k=1 H

tk (Ω ). k

Then the

∂u − λδ k2 1 2 (γ ))′ ∂nγi (H00 i

K i2 X < 1 + log max hδ /hδ (hδk )2tk −2 kuk2H tk (Ωk ) . ∼ m(i) s(i) 1≤i≤N

h

k=1

Proof Let us consider (2.6). We have kwiδ k

1

2 (γ ) H00 i

< (hδ )− 12 kwδ k 2 ∼ i i L (γi )

δ . So by interpolation and the reiteration theorem, for any wiδ ∈ W0,i δ δ < (hδ )−ε kwδ k 1 we have kwiδ k 21 ∼ i i [H0 (γi ),L2 (γi )] 1 +ε for any wi ∈ W0,i H (γ ) 00

i

2

and ε ∈ [0, 12 ]. Furthermore, by Lemma 2.4 and interpolation, we have

14


that Πiδ : [H01 (γi ), L2 (γi )]θ → [H01 (γi ), L2 (γi )]θ is bounded uniformly in δ ∈ I and θ ∈ [0, 1]. So by Lemma 2.7, we find that kΠ δi [u − xδ ]γi k

1

2 (γ ) H00 i

δ < (hδ )−ε ε−1 k(u − xδ )|Ω k 1 + k(u − x )| k 1 −ε Ω −ε ∼ i m(i) H 2 s(i) H 2 (γi ) (γi ) δ < (hδ )−ε ε−1 ku − xδ k 1−ε H (Ωm(i) ) + ku − x kH 1−ε (Ωs(i) ) , ∼ i

where, for the last inequality, we have used that, for any η ∈ [0, 12 ), 1 the trace operator is bounded from H 1−ε (Ωk ) → H 2 −ε (∂Ωk ) uniformly in ε ∈ [0, η]. For ε ≤ mink tk − 1, a duality argument using H 1 (Ωk ) as pivot space shows that inf xδ ∈X δ ku − xδk kH 1−ε (Ωk ) < ∼ k

k

(hδk )tk −1+ε kukH tk (Ωk ) . With ζ := ⌈max1≤i≤N hδm(i) /hδs(i) ⌉, from (2.3), (2.5), (2.6), and, to bound the error in the Lagrange multiplier, the trace theorem, we conclude the statement of the theorem with the factor [1 + log(ζ)]2 replaced by ε−2 [1 + ζ ε ]2 . By, when ζ is sufficiently large, taking ε = log(2)/ log(ζ), the proof is completed.

3 Generalized dual Lagrange multipliers 3.1 General framework In this subsection, after recalling the definition and the advantages of having dual Lagrange multiplier bases, we sketch a general procedure for their construction. Hereafter,Q in the following subsections, we will QN δ δ and M δ = M give examples of spaces X δ = K X i that i=1 k=1 k satisfy all conditions we have imposed, and that can be equipped with such bases. Let Xkδ and Miδ be equipped with local, uniformly L2 (Ωk )- or 2 L (γi )-stable bases {xδk,j : j ∈ Dkδ } and {mδi,ℓ : ℓ ∈ Iiδ }, respectively. With δ δ S0,i := {j ∈ Ds(i) : xδs(i),j |∂Ωs(i) \γi = 0, xδs(i),j |γi 6≡ 0}, δ = span{xδ δ we have W0,i s(i),j |γi : j ∈ S0,i }.

We will select the bases for Xkδ and Miδ in such a way that

δ Iiδ ⊂ S0,i

and

hxδs(i),ℓ |γi , mδi,ℓ′ iL2 (γi )

(3.1)

kxδs(i),ℓ |γi kL2 (γi ) kmδi,ℓ′ kL2 (γi )

h =0

ℓ′

if ℓ = ∈ Iiδ , if ℓ = 6 ℓ′ ∈ Iiδ ,


15

i.e., suitably scaled, {mδi,ℓ : ℓ ∈ Iiδ } and {xδs(i),ℓ |γi : ℓ ∈ Iiδ } are biorthogonal bases, which are also called dual to each other, explaining why we speak about dual Lagrange multipliers. Dual Laδ and so grange multipliers were introduced earlier, with Iiδ = S0,i δ δ dimMi = dimW0,i , see [28,23,18,29]. Here we generalize the concept δ . of dual Lagrange multipliers by allowing that dimMiδ < dimW0,i Under the natural assumption that {xδs(i),ℓ |γi : ℓ ∈ Iiδ } is a uniformly L2 (γi )-stable system, note that (3.1) implies that βiδ > ∼ 1. Indeed, Lemma 2.1(b) shows then that this uniform “inf-sup condition” alδ replaced by span{xδ δ ready holds with W0,i s(i),ℓ |γi : ℓ ∈ Ii }. What is more, defining for 1 ≤ k ≤ K, the set of “free” indices in ¯k by Ω ˘ δ = {j ∈ D δ : j 6∈ Ipδ for any 1 ≤ p ≤ N with s(p) = k}, D k k the property (3.1) of having dual Lagrange multipliers implies that   K  N X δ , mδ )  [ X b(x i,ℓ k,j δ ˘δ xδk,j − x : j ∈ D k s(i),ℓ   b(xδs(i),ℓ , mδi,ℓ ) δ k=1

i=1 ℓ∈I i

is a basis for V δ . The availability of such bases opens the way to apply efficient multi-level solvers to the elliptic system (1.3). It is generally observed that such schemes are more efficient than iterative solvers for the saddle point formulation (1.2). Because of the incorporation of homogeneous Dirichlet boundary δ , but not in M δ , the construction of dual conditions at ∂γi in W0,i i Lagrange multipliers {mδi,ℓ : ℓ ∈ Iiδ } requires a special treatment near ∂γi . In our examples, initially we ignore the interface boundary problems, and construct {xδk,j : j ∈ Dkδ } and local, uniformly L2 (γi )δ , for stable systems {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, generally with I¯iδ ) Iiδ := I¯iδ ∩ S0,i which h kxδs(i),ℓ |γi kL2 (γi ) km ¯ δi,ℓ′ kL2 (γi ) if Iiδ ∋ ℓ = ℓ′ , δ δ hxs(i),ℓ |γi , m ¯ i,ℓ′ iL2 (γi ) =0 if Iiδ ∋ ℓ 6= ℓ′ ∈ I¯iδ , (3.2) ¯ δi,ℓ : ℓ ∈ I¯iδ } will be ascf. Figure 3.1. Typically, {xδk,j : j ∈ Dkδ } and {m sembled from local functions defined on the individual elements which in turn are push forwards of one pair of suitable sets of δ-independent functions defined on a reference element. For some m ≥ 2, it will hold that Xkδ contains all continuous piecewise polynomials of degree m−1 that vanish on ∂Ωk ∩ ∂Ω with respect to a shape regular, and locally δ will contain in any quasi-uniform partition. As a consequence, W0,i

16


δ = { } on the Fig. 3.1: Illustration of the sets I¯iδ = { } and S0,i interface γi for the example that will be discussed in §3.2.1.

case all continuous functions that vanish at ∂γi and that are piecewise linear with respect to its underlying partition, meaning that (A.4) is satisfied. At the multiplier side, it will be ensured that there exists a δ : ℓ ∈ I¯δ } ⊂ C(γ )′ , such that collection of functionals {ψi,ℓ i i X δ v 7→ ψi,ℓ (v)m ¯ δi,ℓ is a quasi-interpolator of order m − 2 (3.3) ℓ∈I¯iδ

δ are meaning that it preserves Pm−2 (γi ), where, in addition, the ψi,ℓ δ , ψ δ ∈ L∞ (γ ), local with respect to the partition underlying W0,i i i,ℓ δ δ δ . As a ¯ δi,ℓ k−1 with ψi,ℓ (v) := hψi,ℓ , viL2 (γi ) , and kψi,ℓ kL2 (γi ) h km L2 (γi ) 2 consequence, such a quasi-interpolator defines a uniformly L (γi )bounded mapping onto span{m ¯ δi,ℓ : ℓ ∈ I¯iδ }. In view of (3.1), the problem to be solved is that generally I¯iδ ) Iiδ . Obviously, we cannot simply remove those m ¯ δi,n from {m ¯ δi,ℓ : ℓ ∈ I¯iδ } with n ∈ I¯iδ \Iiδ , since generally the span of the remaining set will not have appropriate approximation properties. Therefore, together with the removal of such an m ¯ δi,n , we will add suitable multiples of it to some of the remaining m ¯ δi,ℓ . For some fixed q ≥ dimPm−2 (γi ), for each ¯ δi,n ) < ¯ δi,ℓnr , supp m n ∈ I¯iδ \Iiδ , we select ℓn1 , . . . , ℓnq ∈ Iiδ with dist(supp m ∼ diam(supp m ¯ δi,n ), such that the problem ) (q q X X n δ δ n δ 2 σr ψi,ℓnr (p) = ψi,n (p), p ∈ Pm−2 (γi ) argmin kσr ψi,ℓnr kL2 (γi ) :

(σrn )1≤r≤q

r=1

r=1

(3.4) δ δ < ¯ i,n kL2 (γi ) , 1 ≤ r ≤ q. We ¯ i,ℓnr kL2 (γi ) /km has a solution with ∼ km δ δ ¯ remove n from Ii and thus m ¯ i,n from the collection {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, and redefine |σrn |

m ¯ δi,ℓnr ← m ¯ δi,ℓnr + σrn m ¯ δi,n ,

1 ≤ r ≤ q.

(3.5)


17

Note that the reduced and modified collection {m ¯ δi,ℓ : ℓ ∈ I¯iδ } still satisfies (3.2). Moreover, since before the removal of m ¯ δi,n and the reP δ (p)mδ for any p ∈ P definition of m ¯ δi,ℓnr , we had p = ℓ∈I¯δ ψi,ℓ m−2 (γi ), i,ℓ i from (3.4) we conclude that this property is retained. After the removal of all n ∈ I¯iδ \Iiδ and the corresponding updates of the m ¯ δi,ℓnr , the collection {m ¯ δi,ℓ : ℓ ∈ Iiδ } will be written as {mδi,ℓ : ℓ ∈ Iiδ }, and we conclude that we have obtained (3.1). To give useful error bounds, we have to verify whether the resulting space Miδ := span{mδi,ℓ : ℓ ∈ Iiδ } constructed in this way does have appropriate approximation properties. For any element T from the underlying partition, by construction there exists a ball B(T ) δ with diam(B(T )) < ∼ diam(T ) such that supp ψi,ℓ ⊂ B(T ) for any of the uniformly bounded number of ℓ ∈ Iiδ with supp mδi,ℓ ∩ T 6= ∅. By an application of the Bramble-Hilbert lemma, for any s ∈ [0, m − 1], u ∈ H s (γi ), we have X δ ku − hψi,ℓ , uiL2 (γi ) mδi,ℓ kL2 (T ) ℓ∈Iiδ

=

inf

p∈Pm−2 (γi )

ku − p −

X

ℓ∈Iiδ

δ , u − piL2 (γi ) mδi,ℓ kL2 (T ) hψi,ℓ

s < < ∼ p∈P inf (γ ) ku − pkL2 (B(T )∩γi ) ∼ diam(T ) kukH s (B(T )∩γi ) m−2

i

which yields (2.8) for ts(i) ≤ m. In our examples, the initial set I¯iδ will naturally identify with a set of points on γi , such that each element from the underlying partition contains a subset of q = dimPm−2 (γi ) of these points on which the local interpolation problem to determine p ∈ Pm−2 (γi ) is (uniformly) well-posed. Furthermore, in our first two examples, each m ¯ δi,ℓ from the ¯ δi,ℓ (ℓ) = 1, initial set {m ¯ δi,ℓ : ℓ ∈ I¯iδ } will be continuous at ℓ, with m ¯ δi,ℓ )) km ¯ δi,ℓ kL2 (γi ) h (diam(supp m v 7→

d−1 2

X

, and

v(ℓ)m ¯ δi,ℓ

ℓ∈I¯iδ

will be a quasi-interpolator of order m − 2. In our third and final example, a slightly different situation will occur that we will discuss there. Obviously the functionals v 7→ v(ℓ) are not in L∞ (γi ), but, as δ ∈ L∞ (γ ) outlined in [22, §2.1.1], we can always construct local ψi,ℓ i δ k ¯ δi,ℓ ))− with kψi,ℓ L2 (γi ) h (diam(supp m

d−1 2

δ (p) = p(ℓ) for and ψi,ℓ

18


P δ ¯ δi,ℓ is a quasiall p ∈ Pm−2 (γi ), meaning that v 7→ ℓ∈I¯iδ ψi,ℓ (v)m interpolator of the required type. The set Iiδ will be equal to I¯iδ ∩ γi . Given n ∈ I¯iδ \Iiδ , the selection of a suitable set ℓn1 , . . . , ℓnq ∈ Iiδ now can boil down to the selection of an element T , having empty intersection with ∂γi and with ¯ δi,n ), and inside this element, to the selection dist(n, T ) < ∼ diam(supp m of a Pm−2 (γi )-unisolvant set ℓn1 , . . . , ℓnq of nodal points as above, with which the problem (3.4) has a unique solution. Equipping Pm−2 (γi ) 1 if r = r ′ , n with the basis {pℓn1 , . . . , pℓnq } defined by pℓnr (ℓr′ ) = , this 0 if r 6= r ′ , solution is given by σkn =

δ (p n ) ψi,n ℓr δ (p n ) ψi,ℓ n ℓr

= pℓnr (n).

r

σrn

Note that the are uniformly bounded as required, and that they only depend on the polynomial degree m − 2 and the location of the points n and ℓn1 , . . . , ℓnq . Thinking of rectangular elements, if initially the nodal interpolator preserves Qm−2 (γi ), then, in view of [1], it seems recommendable to impose (3.4) for all p ∈ Qm−2 (γi ) so that this property is retained. Finally we make some comments about how to work efficiently with the resulting dual Lagrange multipliers. Concerning the solution δ δ δ of (1.3), the basis ∪N i=1 {mi,ℓ : ℓ ∈ Ii } for M only enters via the matrices i h i h δ δ δ Mδi,M := b(xδm(i),j , mδi,ℓ ) ) , M := b(x , m , i,S i,ℓ s(i),j δ δ δ δ ˘ ℓ∈Ii ,j∈D m(i)

and

i h Sδi := diag b(xδs(i),ℓ , mδi,ℓ )

˘ ℓ∈Ii ,j∈D s(i)

ℓ∈Iiδ

.

In view of this, instead of directly working with the basis functions mδi,ℓ of which those with supports near ∂γi generally, because of the updates (3.5), have larger supports and have “irregular” definitions, it will be more efficient to apply the procedure outlined below. Let {m ¯ δi,ℓ : ℓ ∈ I¯iδ } be the initial collection satisfying (3.2), i.e., before the removal of any of the functions with indices in I¯iδ \Iiδ and the ¯δ , corresponding updates of q of the remaining functions. Let M i,M δ δ δ δ δ ¯ ¯ be the matrices defined as M , M and S with M and S i i i,S i,M i,S {mδi,ℓ : ℓ ∈ Iiδ } replaced by {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, meaning that these matrices can be assembled via “regular” finite element computations. ¯ δ by deleting all rows The matrix Sδi is now simply obtained from S i δ δ δ ¯ and columns with indices from Ii \Ii . With Ti being the #Iiδ × #I¯iδ


19

matrix containing in its ℓ′ th row the coordinates of mδi,ℓ′ in terms of {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, we have ¯ δi,M Mδi,M = Tδi M

¯ δi,S . and Mδi,S = Tδi M

Fixing some numbering of I¯iδ , where indices from Iiδ precede those ¯ℓ or eℓ being the standard (column) basis from I¯iδ \Iiδ , and with e δ ¯δ #I # I i i ¯ℓ or eℓ is 1 on the position or R , respectively, i.e., e vector of R corresponding to ℓ and is zero elsewhere, in view of (3.5) one infers that q X X X δ T ¯ℓ + Ti = σrn eℓnr )¯ eTn , eℓ e ( ℓ∈Iiδ

n∈I¯iδ \Iiδ k=1

indicating an efficient way to apply Tδi . Having solved uδ from (1.3), one retrieves λδ represented in terms δ δ δ δ δ δ of ∪N i=1 {mi,ℓ : ℓ ∈ Ii } by testing the equation a(u , v ) + b(v , λ ) = f (v δ ) for any v δ of the form xs(i),ℓ for 1 ≤ i ≤ N , ℓ ∈ Iiδ . By splitting the vector into N parts corresponding to the different interfaces, and by applying (Tδi )T to the part corresponding to γi , one obtains a ¯ δi,ℓ : ℓ ∈ I¯iδ }, which is more representation of λδ in terms of ∪N i=1 {m convenient for further processing. 3.2 Examples

Q δ In this subsection we will give three examples of spaces X δ = K k=1 Xk Q δ and M δ = N i=1 Mi that satisfy all conditions (A.2)-(A.7) we have imposed, and, moreover, that will be equipped with dual Lagrange 3 multiplier bases. In our examples Ω = ∪K k=1 Ωk ⊂ R will be a polyhedron, and thus so are the subdomains Ωk . In case of piecewise tensor product polynomial approximation on the slave domain, dual Lagrange multiplier bases can simply be made by taking tensor products of such bases on one-dimensional interfaces. Our examples concern non-tensor product approximations, where the construction is less straightforward. Examples of dual Lagrange multiplier spaces for Mortar finite elements can also be found in [28,23,30,19], and in particular for three dimensional domains, in [8,31,18] dealing with linear tetrahedral elements, and in [20] dealing with quadratic serendipity hexahedral elements. Here, we consider quadratic and cubic tetrahedral elements on possibly locally refined partitions. In [20], we constructed dual Lagrange multipliers for a modified quadratic serendipity hexahedral finite element space that along the interface was enriched with bubble functions. Here, by taking an non-standard multiplier space, it will turn out that the addition of these bubble functions

20


can be avoided. The existence of dual Lagrange multipliers depends δ as much on the choice of the basis for Xs(i) as it does on the basis δ δ δ for Mi . Apart that we allow that dimMi < dimW0,s(i) , another key point of our constructions will be that we do not simply resort on the δ . nodal basis for Xs(i) δ ) 3.2.1 Quadratic tetrahedral elements For an 1 ≤ i ≤ N , let (Ts(i) δ∈I be a family of shape regular, locally quasi-uniform and conforming δ δ partitions of Ωs(i) into tetrahedra T . Let Ds(i),v and Ds(i),m denote the set of vertices and midpoints of edges, respectively, which are δ , and let D δ δ δ not on ∂Ω of T ∈ Ts(i) s(i) = Ds(i),v ∪ Ds(i),m . We define Q δ Xs(i) = C(Ωs(i) ) ∩ H∗1 (Ωs(i) ) ∩ T ∈T δ P2 (T ), i.e., m = 3, and equip s(i)

δ }. Note that with these it with the nodal basis {xδs(i),j : j ∈ Ds(i) δ is given by D δ definitions, and because of (A.1), the set S0,i s(i) ∩ γi . For δ ∈ I, let △δi be the subdivision of γi into triangles △ obtained δ by intersecting all T ∈ Ts(i) with γi , and let I¯iδ denote the set of all vertices Q of △ ∈ △δi . We define {m ¯ i,ℓ : ℓ ∈ I¯iδ } as the nodal basis for C(γi ) ∩ △∈△δ P1 (△). Now we apply a basis transformation at the i primal side. For any j ∈ I δ = I¯δ ∩ S δ , which is equal to D δ ∩ γi , i

i

0,i

s(i),v

we redefine

xδs(i),j ← xδs(i),j +

1 12

X

xδs(i),˜ .

(3.6)

δ {˜ ∈Ds(i),m :j,˜ ∈△ for some △∈△δi }

Then, as we will verify below, we have h kxδs(i),ℓ |γi kL2 (γi ) km ¯ δi,ℓ′ kL2 (γi ) if Iiδ ∋ ℓ = ℓ′ , δ δ hxs(i),ℓ |γi , m ¯ i,ℓ′ iL2 (γi ) =0 if Iiδ ∋ ℓ 6= ℓ′ ∈ I¯iδ , (3.7) δ δ δ i.e., (3.2) is valid. Obviously {xs(i),j : j ∈ Ds(i) }, {m ¯ i,ℓ : ℓ ∈ I¯iδ } δ } are uniformly L2 (Ωs(i) )- or L2 (γi )-stable and {xδs(i),j |γi : j ∈ S0,s(i) collections of local functions. P The mapping v 7→ ℓ∈I¯δ v(ℓ)m ¯ δi,ℓ is a quasi-interpolator of order i m − 2 = 1. The general procedure outlined in §3.1 to remove the degrees of freedom associated to n ∈ I¯iδ \Iiδ , while retaining the local approximation properties reads in this case as follows: For any vertex ˜ ∈ △δ that has empty n of a △ ∈ △δi that is on ∂γi , select a △ i ˜ < diam(△). Remove n from I¯δ intersection with ∂γi and dist(△, △) ∼ i and thus m ¯ δi,ℓ from {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, and with (µ1 , µ2 , µ3 ) being the barycentric coordinates of n with respect to the vertices ℓ1 , ℓ2 , ℓ3 of


21

˜ update m ¯ δn , 1 ≤ r ≤ 3. After the removal of all ¯ δℓr + µr m △, ¯ δℓr ← m δ δ n ∈ I¯i \Ii , we end up with a set {mδi,ℓ : ℓ ∈ Iiδ } of (generalized) dual Lagrange multipliers, which spans a space Miδ that has the required local approximation properties. To show (3.7), let φlr , r = 1, · · · , 3 and φqr , r = 1, · · · , 6 be the linear and quadratic finite element nodal basis functions on a referˆ with vol(△) ˆ = 1, respectively, using a numbering of ence triangle △ the points as in see Figure 3.2. A direct computation gives that 3

6

5

1

4

2

Fig. 3.2: Numbering of the vertices and midpoints of edges on the reference triangle.

h

hφqr , φlr′ iL2 (△) ˆ

Hence, if we define

i

1≤r≤6,1≤r ′ ≤3

=



2 −1 −1 8 4 8

T

 1   −1 2 −1 8 8 4  .  60  −1 −1 2 4 8 8

1 φ˜q1 := φq1 + (φq4 + φq6 ), 12 and similarly φ˜q2 and φ˜q3 using symmetry in the barycentric coordinates, we obtain i h 1 q l ˜ = I, hφr , φr′ iL2 (△) ˆ 18 1≤r,r ′ ≤3

which, in view of the definition (3.6), by using affine bijections beˆ and all △ ∈ △δ easily leads to (3.7). tween the reference triangle △ i The basis transformation (3.6) was made as a result of the above construction of the basis transformation on the reference triangle. Yet, note that not all local basis transformations give rise to corresponding global basis transformations. Essential properties of the modified basis functions on the reference triangle are symmetry in

22


the barycentric coordinates, and the fact that a basis function associated to a point, here a vertex or a midpoint of an edge, vanishes on all edges that do not contain that point. Finally, for the case that the family of triangulations (△δi )δ∈I is not quasi-uniform, we will verify the existence of mortar projectors Πiδ that are local, so that (A.6) is valid. Following the approach from Remark 2.3, we extend {mδi,ℓ : ℓ ∈ Iiδ } to a suitable collecδ }. It is easy to construct functions m ∈ P (△) ˆ tion {mδi,ℓ : ℓ ∈ S0,i r 2 ˆ that are symmetric in for r = 4, 5, 6 on the reference triangle △ q ˜ the barycentric coordinates, and, with φr = φqr for r = 4, 5, 6, sat 0 if r ′ 6= r ∈ {1, . . . , 6}, By lifting m4 , m5 , m6 isfy hφ˜qr , mr′ iL2 (△) ˆ = 1 if r ′ = r. to all △ ∈ △δi by using affine bijections, and by connecting those pairs that are associated to the same edge, and finally by leaving out those functions associated to edges on ∂γi , we obtain a collecδ \I δ } on γ . The union tion of auxiliary functions {mδi,ℓ : ℓ ∈ S0,i i i δ } is a uniformly L2 (γ )-stable set of local functions. {mδi,ℓ : ℓ ∈ S0,i i δ into I δ and S δ \I δ , the matrix With respect to the partition of S0,i i 0,i i hxδi,ℓ ,mδi,ℓ′ |γi iL2 (γ ) Dδi,1 0 δ i , Ri := kxδ | k is of the form δ Aδi Dδi,2 i,ℓ γi L2 (γi ) kmi,ℓ′ kL2 (γi ) ℓ,ℓ′ ∈S δ 0,i

with Dδi,1 , Dδi,2 being diagonal matrices that are uniformly bounded invertible, and Aδi being a uniformly sparse and bounded matrix. We infer that Rδi is uniformly bounded invertible with a local inverse, showing that a uniformly L2 (γi )-bounded local mortar projector exists. As an alternative for our definition Miδ = span{mδi : i ∈ Iiδ }, one δ }, may use such auxiliary functions to define Miδ = span{mδi : i ∈ S0,i with which one gets into the situation of having quasi-dual Lagrange δ = dimM δ . This approach, howmultipliers as in [20], and dimW0,i i ever, leads to a more complicated coupling between master and slave sides. In the cases where we have tested both approaches, the discretization errors were almost equal. 3.2.2 Quadratic serendipity hexahedral elements For an 1 ≤ i ≤ N , δ ) let (Ps(i) δ∈I be a family of shape regular, and conforming partitions of Ωs(i) into parallelepipeds P . So here, in order not to be forced to handle cases with hanging nodes, we restrict ourselves to quasiδ δ uniform partitions. Let Ds(i),v and Ds(i),m denote the set of vertices and midpoints of edges, respectively, which are not on ∂Ω of P ∈ δ , and let D δ δ δ δ Ps(i) s(i) = Ds(i),v ∪Ds(i),m . We define Xs(i) as the quadratic


23

δ , i.e., m = 3, with serendipity finite element space with respect to Ps(i) zero boundary conditions on ∂Ωs(i) ∩ ∂Ω, and equip it with the nodal δ }. Note that the set S δ is given by D δ ∩ γ . basis {xδs(i),j : j ∈ Ds(i) i 0,i s(i)

For δ ∈ I, let δi be the subdivision of γi into parallelograms δ with γ , and let I¯δ denote the set obtained by intersecting all P ∈ Ps(i) i i δ of all vertices of ∈ i . We apply the following basis transformation δ , which is equal to at the primal side. For any j ∈ Iiδ = I¯iδ ∩ S0,i δ Ds(i),v ∩ γi , we redefine xδs(i),j ← xδs(i),j +

1 5

X

xδs(i),˜ .

(3.8)

δ {˜ ∈Ds(i),m :j,˜ ∈ for some ∈δi }

We will construct an L2 (γi )-stable collection {m ¯ i,ℓ : ℓ ∈ I¯iδ } of local functions such that h kxδs(i),ℓ |γi kL2 (γi ) km ¯ δi,ℓ′ kL2 (γi ) if Iiδ ∋ ℓ = ℓ′ , δ δ hxs(i),ℓ |γi , m ¯ i,ℓ′ iL2 (γi ) =0 if Iiδ ∋ ℓ 6= ℓ′ ∈ I¯iδ , (3.9) δ } and i.e., such that (3.2) is valid. Obviously {xδs(i),j : j ∈ Ds(i) δ {xδs(i),j |γi : j ∈ S0,s(i) } are uniformly L2 (Ωs(i) )- or L2 (γi )-stable collections of local functions. To show (3.9) and to construct {m ¯ i,ℓ : ℓ ∈ I¯iδ }, using the numbering of the points as in Figure 3.3, let φbr for r = 1, · · · , 4, and φsi for i = 1, · · · , 8 denote the bilinear and serendipity finite element nodal ˆ = (0, 1)2 , respectively. A basis functions on the reference square 4

7

3

8

6

1

5

2

Fig. 3.3: Numbering of the vertices and midpoints of edges on the reference square. direct computation gives that i h T 1 0 −1 −1 −1 4 2 2 4 , hφsr , φb1 iL2 () = ˆ 36 1≤r≤8

24


where the other entries follow using symmetry. Because of the missing interior degree of freedom, for the serendipity element we are not able to construct a biorthogonal system by only making a transformation at the ‘primal’ side. Therefore, for some α 6= 1, we set φb + α(φb3 − φb2 − φb4 ) φ˜b1 := 1 , 1−α and define φ˜b2 , φ˜b3 and φ˜b4 analogously. The corresponding collection of global functions {m ¯ δi,ℓ : ℓ ∈ I¯iδ } is now formally defined by m ¯ δi,ℓ | = −1 φ˜b1 ◦ F for any ∈ δi that contains ℓ as a vertex, and where F ˆ and mapping point 1 onto ℓ, and is an affine bijection between m ¯ i,ℓ is zero elsewhere. Note that the m ¯ δi,ℓ are not continuous, and that {m ¯ δi,ℓ : ℓ ∈ I¯iδ } does not span the space of all continuous piecewise P ¯ δi,ℓ does bilinears with respect to δi . Yet, although v 7→ i∈I¯δ v(ℓ)m i not preserve Q1 (γi ), since φ˜b1 + φ˜b2 = φb1 + φb2 , φ˜b2 + φ˜b3 = φb2 + φb3 , φ˜b3 + φ˜b4 = φb3 + φb4 , and φ˜b1 + φ˜b2 + φ˜b3 + φ˜b4 = φb1 + φb2 + φb3 + φb4 it does preserve P1 (γi ), i.e., it is a quasi-interpolator of order m − 2 = 1. Defining, for some θ, φ˜s1 := φs1 + θ(φs5 + φs8 ), ˜s ˜s ˜s i φ2 , φ3 and φ4 , a direct computation shows that hand analogously is diagonal if and only if α = 16 and θ = 51 , in hφ˜sr , φ˜br′ iL2 () ˜ ′ 1≤r,r ≤4

1 I. With α = 61 substituted into which case this matrix is equal to 20 δ δ the definition of {m ¯ i,ℓ : ℓ ∈ I¯i }, we conclude (3.9). The procedure outlined in §3.1 to remove the degrees of freedom associated to n ∈ I¯iδ \Iiδ reads here as follows: For any vertex n of a ˜ ∈ δ that has empty intersection ∈ δi that is on ∂γi , select a i < ˜ ¯ δi,ℓ with ∂γi and dist(, ) ∼ diam(). Remove n from I¯iδ and thus m from {m ¯ δi,ℓ : ℓ ∈ I¯iδ }, and with (µ1 , µ2 , µ3 ) being the barycentric co˜ update ordinates of n with respect to three vertices ℓ1 , ℓ2 , ℓ3 of , δ δ δ ¯ n , 1 ≤ r ≤ 3. After the removal of all n ∈ I¯iδ \Iiδ , ¯ ℓr + µr m m ¯ ℓr ← m we end up with a set {mδi,ℓ : ℓ ∈ Iiδ } of (generalized) dual Lagrange multipliers, which spans a space Miδ that has the required local approximation properties. δ ) 3.2.3 Cubic tetrahedral elements For an 1 ≤ i ≤ N , let (Ts(i) δ∈I and δ (△i )δ∈I be families as in §3.2.1 of partitions of Ωs(i) into tetrahedra, δ and of γi into triangles, respectively. We define Xs(i) = C(Ωs(i) ) ∩


25

Q ¯ δ = C(γi ) ∩ Q H∗1 (Ωs(i) ) ∩ T ∈T δ P3 (T ), and M i △∈△δi P2 (△), i.e., s(i) m = 4. ¯ δ , and a modification of To define a basis {m ¯ δi,ℓ : ℓ ∈ I¯iδ } for M i δ } for X δ the nodal basis {xδs(i),j : j ∈ Ds(i) s(i) , which only affects δ δ ˆ {x |γ : j ∈ S }, it is sufficient to consider a reference triangle △ s(i),j

0,i

i

ˆ = 1. Using a numbering of the nodal points as indicated with vol(△) in Figure 3.4, let {φc1 , . . . , φc10 } and {φq1 , . . . , φq6 } be the nodal bases 3

3

6

8 5

6 9

5

10 1

4

7

2

1

4

2

Fig. 3.4: Numbering of nodal points at primal and multiplier side for the cubic tetrahedral case. ˆ and P2 (△), ˆ respectively. We define transformed bases by for P3 (△) 20 φ˜c1 = 60φc1 + (φc4 + φc9 − φc6 − φc7 − φc10 ), 9 5 c c ˜ φ4 = 10(φ4 + φc7 ) − φc10 , φ˜c10 = φc10 , φ˜c7 = φc4 − φc7 , 6 and 1 φ˜q1 = φq1 − (φq4 + φq6 ), 8

φ˜q4 = φq4 ,

(3.10)

and the others by permuting barycentric coordinates. All these basis functions at primal and multiplier side can be naturally associated to either a vertex, an edge (at the primal side two at each edge) or the ˆ where they vanish on the opposite edge, on the other center of △, two edges, or on all edges, respectively, and finally, where they are either symmetric or anti-symmetric in permutations of the barycentric

26


coordinates. A direct computation shows that   I 0  0 0 −1  8 .  [hφ˜cr , φ˜qr′ iL2 (△) ˆ ]1≤r,r ′ ≤6 =  1 −8 0 0 I  0 − 18 0

Although this matrix is thus not diagonal, the fact that, with respect to a splitting of the degrees of freedom into those associated to vertices and those associated to edges, it is block lower triangular with diagonal blocks equal to identity matrices which will allow for the construction of local dual Lagrange multiplier bases. By lifting {φ˜c1 , . . . , φ˜c10 } and {φ˜q1 , . . . , φ˜q6 } to all △ ∈ △δi using affine bijections, and by gluing them continuously over the interfaces, and, at the primal side, by leaving out those functions associated to vertices or edges on ∂γi , we end up with the modified collection δ }, and a basis {m ¯ δ . Here as ˘ δi,ℓ : ℓ ∈ I¯iδ } for M {xδs(i),j |γi : j ∈ S0,i i δ and I¯δ a natural index set I¯iδ , we use the union of the sets I¯i,v i,m of δ all vertices and midpoints of edges of △ ∈ △i , respectively. Defining δ = I¯δ ∩ γ , I δ δ δ δ δ ¯δ Ii,v i i,v i,m = Ii,m ∩ γi and Ii = Ii,v ∪ Ii,m , we write S0,i as δ \I δ ), where I δ and I δ are the index sets for those xδ Iiδ ∪ (S0,i i i,v i,m s(i),j |γi c c c c c c resulting from the functions φ˜1 , φ˜2 , φ˜3 and φ˜4 , φ˜5 , φ˜6 , respectively, and δ \I δ is some index set for those xδ where S0,i i s(i),j |γi resulting from the c c c c ˜ ˜ ˜ ˜ functions φ7 , φ8 , φ9 , φ10 . For ℓ ∈ Iiδ , we have X hxδs(i),ℓ |γi , m ˘ δi,ℓ iL2 (γi ) = dδℓ := vol(△), {△∈△δi :△∋ℓ}

δ , and and for any ℓ′ ∈ I¯i,v δ δ ℓ ∈ γi ∩ I¯i,opp (ℓ′ ) := {ℓ ∈ I¯i,m :∃△ = △ℓ,ℓ′ ∈ △δi with ℓ′ , ℓ ∈ △

and ℓ is on the edge opposite to ℓ′ },

see Figure 3.5, we have 1 ˘ δi,ℓ′ iL2 (γi ) = − vol(△ℓ,ℓ′ ), hxδs(i),ℓ |γi , m 8 whereas hxδs(i),ℓ |γi , m ˘ δi,ℓ′ iL2 (γi ) = 0 for all other ℓ ∈ Iiδ , ℓ′ ∈ I¯iδ . Defining the transformed basis {m ¯ δi,ℓ : ℓ ∈ I¯iδ } by ( 1 P vol(△ ′ ) δ δ , m ˘ δi,ℓ′ + ℓ∈I¯δ (ℓ′ ) 8 dδ ℓ,ℓ m ˘ i,ℓ if ℓ′ ∈ I¯i,v δ i,opp (3.11) m ¯ i,ℓ′ := ℓ δ , if ℓ′ ∈ I¯i,m m ˘ δi,ℓ′


27

we find hxδs(i),ℓ |γi , m ¯ δi,ℓ′ iL2 (γi )

=

dδℓ if Iiδ ∋ ℓ = ℓ′ , 0 if Iiδ ∋ ℓ 6= ℓ′ ∈ I¯iδ ,

so that (3.2) is valid by dδℓ h kxδs(i),ℓ |γi kL2 (γi ) km ¯ δi,ℓ kL2 (γi ) . δ , m Concerning implementation, since for ℓ ∈ I¯i,v ¯ δi,ℓ has a much ¯δ larger support than m ˘ δi,ℓ , instead of computing the matrices M i,M ¯ δ mentioned in §3.1, it is much more efficient to write them and M i,S as the composition of the basis transformation defined by (3.11), that ˘ δ and M ˘ δ one gets can be applied at low cost, and the matrices M i,M i,S ˘ δi,ℓ : ℓ ∈ I¯iδ }. by replacing {m ¯ δi,ℓ : ℓ ∈ I¯iδ } by {m Finally, we have to remove m ¯ δi,ℓ for ℓ ∈ ∂γi . Let {¯ ni,ℓ : ℓ ∈ I¯iδ } be ¯ δ . As discussed in §3.1, there exists a collection the nodal basis for M i of functions {ψ˘i,ℓ : ℓ ∈ I¯iδ }, which is local with respect to △δi , and with , such that ψ˘i,ℓ (p) = p(ℓ) for p ∈ P2 (γi ). With, ni,ℓ k−1 kψ˘i,ℓ kL2 (γi ) < ∼ k¯ L2 (γi ) ′ ′ δ δ , I¯δ ¯δ for ℓ ∈ I¯i,m i,adj (ℓ) := {ℓ ∈ Ii,v : ℓ, ℓ are on one edge of a △ ∈ △i }, see Figure 3.5, we define  δ , ˘ if ℓ ∈ I¯i,v   ψi,ℓ X X 1 vol(△ℓ,ℓ′ ) ˘ δ . 8 ψi,ℓ = ψ˘i,ℓ + 1 ψi,ℓ′ if ℓ ∈ I¯i,m ψ˘i,ℓ′ − 8 dδℓ   δ δ δ ′ ¯ ′ ¯ ′ ¯ ℓ ∈Ii,adj (ℓ)

{ℓ ∈Ii,v :ℓ∈Ii,opp (ℓ )}

From (3.10) and (3.11), we infer that X X v 7→ ψi,ℓ (v)m ¯ i,ℓ = ψ˘i,ℓ (v)¯ ni,ℓ , ℓ∈I¯iδ

ℓ∈I¯iδ

which is thus a uniformly L2 (γi )-bounded quasi-interpolator of order m − 2 = 2. Now for any n ∈ I¯iδ \Iiδ , vertex or midpoint of a △ ∈ △δi , we select ˜ ˜ < diam(△), such that a △ ∈ △δi with dist(△, △) ∼ ˜ := (I¯δ ∩ △) ˜ ∪ ∪ ¯δ I¯iδ (△) i ℓ∈I

˜ {ℓ

i,m ∩△

′

δ δ ∈ I¯i,v : ℓ ∈ I¯i,opp (ℓ′ )} ⊂ γi ,

see Figure 3.5. Using that, because of the locally quasi-uniform parδ for ℓ ∈ I δ (△) ˜ have comparable L2 (γi )-norms, we solve tition, all ψi,ℓ i (σℓn )ℓ∈I¯δ (△) ˜ from i    X  X n 2 δ n δ argmin | : |σ (p) = ψ (p), p ∈ P (γ ) ψ σ . 2 i i,n ℓ ℓ i,ℓ  (σℓn )ℓ∈I¯δ (△) ˜  δ δ i

˜ ℓ∈I¯i (△)

˜ ℓ∈I¯i (△)

28


˜ △

ℓ ℓ′

δ δ ˜ Fig. 3.5: I¯i,opp (ℓ′ ), I¯i,adj (ℓ) and I¯iδ (△) δ : ℓ ∈ I¯δ (△)} ˜ ⊃ span{ψ˘i,ℓ : ˜ = span{ψ˘δ : ℓ ∈ I¯δ (△)} Since span{ψi,ℓ i i i,ℓ ˜ and the transformation from {ψ δ : ℓ ∈ I¯δ (△)} ˜ ℓ ∈ I¯iδ ∩ △}, to i i,ℓ δ δ ˘ ˜ ¯ {ψi,ℓ : ℓ ∈ Ii (△)} is uniformly bounded invertible, we infer this ¯δ problem has a unique solution with |σℓn | < ∼ 1. We remove n from Ii δ δ δ and thus m ¯ i,n from the collection {m ¯ i,ℓ : ℓ ∈ I¯i }, and redefine

m ¯ δi,ℓ ← m ¯ δi,ℓ + σℓn m ¯ δi,n ,

˜ ℓ ∈ I¯iδ (△).

After removal of all n ∈ I¯iδ \Iiδ , the obtained collection {mδi,ℓ : ℓ ∈ Iiδ } of (generalized) dual Lagrange multipliers spans a space Miδ that has the required local approximation properties. Finally, for the case that the family of triangulations (△δi )δ∈I is not quasi-uniform, with the aid of Remark 2.3, the existence of uniformly L2 (γi )-bounded and local mortar projectors Πiδ can be verified in a similar way as in §3.2.1. 4 Numerical Results In this section, we present some numerical results illustrating the performance of the quadratic tetrahedral and quadratic serendipity hexahedral mortar finite elements from §3.2.1 and §3.2.2. For the primal variable, we give the discretization errors in the k · k1 := P 1 2 2 [ K k=1 k · kH 1 (Ωk ) ] -norm, as also in the k · kL2 (Ω) -norm. Using exclusively quasi-uniform partitions and having piecewise smooth solutions in our numerical examples, with h denoting the largest diameter of any underlying element, Corollary 2.6 shows that the k · k1 -norm of the error is of order h2 . Using duality arguments, one can prove, see e.g. [9], that the error in L2 (Ω)-norm is of order h3 . Since the -norms are not so easy to compute, we measure the disk · k 12 (H00 (γi ))′

cretization errors in the flux across the interfaces in a mesh-dependent


29

Lagrange multiplier norm, that for tetrahedral elements is defined by kµk2δ :=

N X X

i=1 △∈△δ i

diam(△)kµ|△ k2L2 (△) ,

and that has an analogous definition for the hexahedral elements with △δi , △ replaced by δi , . Theoretically, the asymptotic rates of errors in the weighted Lagrange multiplier norm are of order h2 for quadratic mortar finite elements, see [20]. However, under the assumption that the errors in the primal variable are equally distributed, which is the case here because of the smooth solution, the asymptotic rates can be shown to be of order h5/2 , see [30]. In our examples, we combine tetrahedral and hexahedral elements on the different subdomains. We point out that different Lagrange multipliers should be used depending on the hexahedral or tetrahedral partition on the slave side of the interface. Starting with some partition into tetrahedra or hexahedra on each subdomain, we create sequences of partitions by uniform dyadic refinements. We use I = N0 as index set for δ, where then δ refers to the level of refinement, that is, the number of uniform dyadic refinement steps that has been applied. For solving the arising linear systems, we have used the multigrid method applied to a reformulation of (1.2) as a positive definite system on the product space X δ as introduced in [31]. This multigrid method has a level independent convergence rate and is of optimal computational complexity, see [31,29]. Our implementation is based on the finite element toolbox ug, [2]. In all our examples, we observe convergence rates for ku−uδ kL2 (Ω) , 5

ku − uδ k1 , and kλ − λδ kδ that approach the values 8, 4, and 2 2 ≈ 5.66, respectively, that when having smooth solutions are expected for dyadic refinements and quadratic/linear approximation for the primal/Lagrange multiplier variables. In our first example, we consider −∆u = f in Ω, where Ω is composed of five cubes Ω1 := (0, 1)3 , Ω2 := (1, 2) × (0, 1)2 , Ω3 := (0, 1)2 ×(1, 2), Ω4 := (−1, 0)×(0, 1)2 and Ω5 := (0, 1)2 ×(−1, 0). Here, subdomain Ω1 , which has an initial partition into 27 hexahedra, is the slave subdomain and the others are master subdomains. In each of the master subdomains, the initial partition consists of 6 tetrahedra. The right hand side f and the Dirichlet boundary conditions are chosen such that the exact solution is given by u(x, y, z) = e−0.25(x

2 +y 2 +z 2 )

(cos(5x + z) + 3 sin(4y + z)).

30


In Figure 4.1, the decomposition of the domain, the initial finite element partitions, and the isolines of the solution at the interface z = 1 are shown. The discretization errors are given in Table 4.1.

Fig. 4.1: Decomposition of the domain and initial partitions (left), isolines of the solution at the plane z = 1 (right) for Example 1.

Table 4.1. Discretization errors for Example 1. δ 0 1 2 3

# elem. 51 408 3264 26112

ku − uδ kL2 (Ω) 3.24e-1 5.02e-2 6.5 6.79e-3 7.4 8.52e-4 8.0

ku − uδ k1 5.75e-1 1.70e-1 3.4 4.70e-2 3.6 1.19e-2 3.9

kλ − λδ kδ 3.20e+0 8.44e-1 3.8 1.60e-1 5.3 2.94e-2 5.4

In our second example, we consider a domain and problem from [20]. Here, Ω := (0, 2) × (0, 1) × (0, 2) is decomposed into four subdomains Ω1 := (0, 1)3 , Ω2 := (0, 1)2 × (1, 2), Ω3 := (1, 2) × (0, 1)2 and Ω4 := (1, 2) × (0, 1) × (1, 2), with Ω2 and Ω3 being slave subdomains, and Ω1 and Ω4 being master subdomains. On both Ω2 and Ω3 the initial partition consists of 27 hexahedra, and on both Ω1 and Ω4 it consists of 6 hexahedra. The problem for this example is given by a reaction-diffusion equation −div(a∇u) + u = f

in

Ω,

where a = 1 in Ω1 and Ω4 , and a = 10 in Ω2 and Ω3 . The right hand side f and the Dirichlet boundary conditions are chosen such that the exact solution is given by 2 −y 2 −(z−1)2

u(x, y, z) = (x − 1)y(z − 1)e−(x−1)

cos(2 x + 2 y + 2 z)/a.


31

Note that u has a jump in the normal derivative across the interface, whereas the flux is continuous. In Figure 4.2, the decomposition of the domain together with the initial finite element partitions, the isolines of the solution on the plane y = 12 , and the flux of the exact solution at the interface x = 1 are shown. The discretization errors

Fig. 4.2: Decomposition of the domain and initial partitions (left), isolines of the solution at the plane y = 12 (middle) and flux of the exact solution at the interface x = 1 (right) for Example 2. are given in Tables 4.2. Table 4.2. Discretization errors for Example 2. δ 0 1 2 3

ku − uδ kL2 (Ω) 7.94e-2 1.69e-2 4.7 2.35e-3 7.2 2.73e-4 8.6

# elem. 66 528 4224 33792

ku − uδ k1 1.57e-1 4.99e-2 3.1 1.48e-2 3.4 3.87e-3 3.8

kλ − λδ kδ 5.19e-2 4.11e-3 13 6.26e-4 6.5 1.04e-4 6.0

Our third example is given by −∆u = f in Ω, where the nonconvex Ω is composed of eight cubes Ω1 := (0, 1)3 , Ω2 := (1, 2) × (0, 1)2 , Ω3 := (2, 3) × (0, 1)2 , Ω4 := (0, 1)2 × (1, 3), Ω5 := (2, 3) × (0, 1) × (1, 3), Ω6 := (0, 1)2 × (3, 4), Ω7 := (1, 2) × (0, 1) × (3, 4) and Ω8 := (2, 3) × (0, 1) × (3, 4). Here, subdomains Ω1 , Ω3 , Ω6 and Ω8 are taken as slave subdomains and the others are master subdomains. The initial partitions consist of tetrahedra for Ω1 and Ω3 , and of hexahedra for the other subdomains, and they are illustrated in Figure 4.3. The right hand side f and the Dirichlet boundary conditions are chosen such that the exact solution is given by 1

u(x, y, z) = e− 4 (x+y+z) (sin(5xy) + cos(5yz))(x + y 2 + z 2 ). The isolines of this solution at the interface z = 1 can also be found in Figure 4.3. Finally, the discretization errors are given in Table 4.3.

32


Fig. 4.3: Decomposition of the domain and initial partitions (left), isolines of the solution at the plane z = 1 (right) for Example 3. Table 4.3. Discretization errors for Example 3. δ 0 1 2 3

# elem. 426 3408 27264 218112

ku − uδ kL2 (Ω) 6.94e-1 1.43e-1 4.9 1.98e-2 7.2 2.63e-3 7.5

ku − uδ k1 9.13e-1 4.18e-1 2.2 1.10e-1 3.8 2.85e-2 3.9

kλ − λδ kδ 1.07e+1 5.79e+0 1.8 1.34e+0 4.3 1.71e-1 7.8

In our last example, we consider an example in linear elasticity. We point out that the theory can easily be extended to this case with standard arguments. For this example, we take the domain and the problem from [18]. Here, the computational domain Ω, which is a beam, is decomposed into three subdomains Ω1 , Ω2 and Ω3 with Ω1 := (0, 50) × (0, 10) × (0, 2), Ω2 := (0, 50) × (3, 7) × (2, 11) and Ω3 := (0, 50) × (0, 10) × (11, 13). The decomposition of the domain, is given in the left picture of Figure 4 with an initial partition. Here, Ω2 is the slave subdomain, so that, although the decomposition is geometrically nonconforming, condition (A.1) is satisfied. Our linear elasticity problem is −∇ · σ(u) = 0

in Ω

with u = 0 on ΓD and σ(u)n = f on ΓN , where σ(u) is a second order tensor defined as 1 ∇u + [∇u]T σ(u) = 2µε(u) + λtrace ε(u)I, ε(u) = 2

with µ = 8.2, λ = 10 (kg/cm3 ), and I is the identity matrix of size 3 × 3. Here, ΓD is the part of the boundary of Ω with x = 0 and x = 50 so that the left and the right sides of each subdomain are fixed, and ΓN := ∂Ω\ΓD . The function f = (f1 , f2 , f3 ) on ΓN is given as f1 = f2 = 0, and ( −0.35 if 22 ≤ x ≤ 28 and z = 13 f3 = 0 otherwise


33

so that a constant vertical force is applied from the top boundary (z = 13) of Ω. We have computed the solution with quadratic finite elements. The resulting deformation of the beam is given in the right picture of Figure 4.

Fig. 4.4: Decomposition of the domain with initial partition (left) and the distorted mesh after refining once (right)

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