Adaptive ACMS: A robust localized Approximated Component Mode ...

arXiv:1709.04044v1 [math.NA] 12 Sep 2017

ADAPTIVE ACMS: A ROBUST LOCALIZED APPROXIMATED COMPONENT MODE SYNTHESIS METHOD ALEXANDRE L. MADUREIRA AND MARCUS SARKIS Abstract. We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous L∞ coefficients. The methods are of Galerkin type and follows the Variational Multiscale and Localized Orthogonal Decomposition–LOD approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, however it deteriorates with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space H 1 .

1. Introduction Consider the problem of finding the weak solution u : Ω → R of (1)

− div A ∇ u = f

in Ω,

u = 0 on ∂Ω,

where Ω ⊂ R2 , and is an open bounded domain with polygonal boundary ∂Ω, the symmetric tensor A ∈ [L∞ (Ω)]2×2 sym is uniformly positive definite and bounded, and f is part of the given data. For almost all x ∈ Ω let the positive constants amin and amax be such that (2)

amin |v|2 ≤ a− (x)|v|2 ≤ A(x) v · v ≤ a+ (x)|v|2 ≤ amax |v|2

for all v ∈ R2 ,

where a− (x) and a+ (x) are the smallest and largest eigenvalues of A(x). Let ρ ∈ L∞ (Ω) be chosen by the user and such that ρ(x) ∈ [ρmin , ρmax ] almost everywhere for some positive constants ρmin and ρmax . Consider g such that f = ρg, 1

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ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

and then the ρ-weighted L2 (Ω) norm kgkL2ρ (Ω) := kρ1/2 gkL2(Ω) = kf kL21/ρ (Ω) is finite. The introduction of the weight ρ is to balance u and f with respect to the tensor A. For instance, the choice ρ(x) = amin implies that the global Poincar´e inequality constant CP,G defined in (10) is O(1) while ρ(x) = a− (x) is related to local weighted Poincar´e inequalities found in the literature; see Remark 1. For v, w ∈ H 1 (Ω) let a(v, w) =

Z

A ∇ v · ∇ w dx, Ω

and denote by (·, ·) the L2 (Ω) inner product. Although the solution u of (1) in general only belongs to the Sobolev space H 1 (Ω), a priori error analyses of multiscale methods established on the literature rely on solution regularity; see [2, 10, 22–24, 31, 32, 37, 38, 65, 67] and references therein. Recently methods that do not rely on solution regularity were introduced: generalized finite element methods [3], the rough polyharmonic splines [58], the variational multiscale method [39, 40], and the Localized Orthogonal Decomposition (LOD) [50–52]. These methods are based on splitting approximation spaces into fine and multiscale subspaces, and the numerical solution of (1) is sought in the latter. We note that these works were designed for the low-contrast case, that is, amax /amin not large. We note that for a class of coefficients A, that is, when local Poincar´e inequality constants are not large, the LOD methodology works [34, 63]; see Remark 1. We point out that all the above methods are based on overlapping techniques. On the other side, there exist several domain decomposition solvers which are optimal with respect to mesh and contrast. All of them are based on extracting coarse basis functions from local generalized eigenvalue problems. For non-overlapping domain decomposition based on FETI-DP [28, 45] and BDDC [14], the technique named adaptive choice of primal constraints was introduced to obtain robustness with respect to contrast and we refer [4, 9, 15, 16, 42–44, 53–55, 57, 70]. For overlapping domain decomposition we refer [17, 26, 30, 42, 69] and some of of this ideas were incorporated in [11, 12] to obtain a discretizations that depend only logarithmically with respect to contrast. In [49] we introduced the Localized Spectral Decomposition–LSD method for mixed and hybrid-primal methods [64], that is, we re-frame the LOD version in [52] into the nonoverlapping domain decomposition framework, and consider the Multiscale Hybrid Method– MHM [1, 32, 33], which falls in the BDDC and FETI-DP classes, and then explore adaptive choice of primal constraints to generate the multiscale basis functions. We obtain a discretization that is robust with respect to contrast.

ADAPTIVE ACMS

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In this paper we consider Approximated Component Mode Synthesis–ACMS methods [6, 7, 13, 35–37, 41]; these methods require extra solution regularity and do not work for high contrast. The goal here is to develop a discretization that has optimal a priori error approximation, assuming minimum regularity on the solution, A and ρ. In order to consider the LOD approach with Galerkin-Ritz projection, we consider conforming primal iterative substructuring techniques [5, 8, 19, 20, 46, 47, 56, 66, 71] rather than BDDC and FETI-DP methods. We show that for our purpose, generalized eigenvalue problem based on series sums A + B is more appropriated than parallel sums (A−1 + B −1 )−1 usually found in BDDC and FETI-DP methods for optimal convergence. Two versions are under consideration here, both of Galerkin type and based on edges and local harmonic extensions. The first method is simpler and converges at optimal rates, even under minimal regularity of the solution, but its properties deteriorate if the contrast of the coefficients increase. We then modify the method, incorporating solutions of specially designed local eigenfunction problems, yielding optimal convergence rate even under the high contrast case. The remainder of the this paper is organized as follows. Section 2 describes the substructuring decomposition into interior and interface unknowns, while our methods for low and high contrast coefficients are considered in Sections 3 and 4, respectively. In Section 5 we consider how to deal with elementwise problems. 2. Substructuring Formulation We start by defining a partition of Ω by a triangular finite element regular mesh TH with elements of characteristic length H > 0. Let ∂TH be the mesh skeleton, and NH the set of nodes on ∂TH \∂Ω. Consider Th a refinement of TH , in the sense that every (coarse) edge of the elements in TH can be written as a union of edges of Th . We assume that h < H. Let Nh be the set of nodes of Th on the skeleton ∂TH \∂Ω; thus all nodes in Nh belong to edges of elements in TH . For v ∈ H 1 (Ω) let |v|2H 1 (Ω) = kA1/2 ∇ vk2L2 (Ω) , A

|v|2H 1 (T ) = A

X

kA1/2 ∇ vk2L2 (τ ) ,

τ ∈T

where T ⊂ TH denotes a given set of elements. Let Vh ⊂ H01 (Ω) be a the space of continuous piecewise linear functions related to Th . For the sake of reference, let uh ∈ Vh such that a(uh , vh ) = (ρg, vh ) for all vh ∈ Vh .

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ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

We assume that uh approximates u well, but we remark that this quantity is never computed; the analyses show that our numerical schemes yield good approximations for uh . H Assume the decomposition uh = uB h + uh in its bubble and harmonic components, and let

VhB = {vh ∈ Vh : vh = 0 on ∂τ , τ ∈ TH }, H B B B VhH = {uH h ∈ Vh : a(uh , vh ) = 0 for all vh ∈ Vh },

i.e., VhH = (VhB )⊥a . It follows immediately from the definitions that (3)

H H H H a(uH h , vh ) = (ρg, vh ) for all vh ∈ Vh ,

B B B B a(uB h , vh ) = (ρg, vh ) for all vh ∈ Vh .

The problem related to uB h is local and uncoupled and treated in Section 5. H We now proceed to approximate uH h . Note that any function in Vh is uniquely determined

by its trace on the boundary of elements in TH . Let Λh = {vh |∂TH : vh ∈ VhH } ⊂ H 1/2 (∂TH ), and the local discrete-harmonic extension operator T : Λh → VhH such that, for µh ∈ Λh , (4)

(T µh )|∂TH = µh ,

and

a(T µh , vhB ) = 0 for all vhB ∈ VhB .

Define the bilinear forms sτ , s : Λh × Λh → R such that, for µh , νh ∈ Λh , Z X sτ (µh , νh ). sτ (µh , νh ) = A ∇ T µh · ∇ T νh dx for τ ∈ TH , s(µh , νh ) = τ

τ ∈TH

Let λh = uh |∂TH . Then uH h = T λh and (5)

s(λh , µh ) = (ρg, T µh ) for all µh ∈ Λh . 3. The Low-Contrast Multiscale Case

We now propose a scheme to approximate (5) based on LOD techniques. Define the e h ⊂ Λh by fine-scale subspace Λ e h ∈ Λh : λ eh (xi ) = 0 for all xi ∈ NH }. e h = {λ Λ

ms ms e e Let the multiscale space Λms h ⊂ Λh be such that Λh ⊥s Λh and Λh = Λh ⊕ Λh . Our

ms numerical method is defined by λms h ∈ Λh such that

(6)

ms ms ms ms s(λms h , µh ) = (ρg, T µh ) for all µh ∈ Λh ,

ms H and we set ums h = T λh as an approximation for uh .

To make the definition of Λms h explicit, let the coarse-scale space ΛH ⊂ Λh be the trace of piecewise continuous linear functions on the ∂TH triangulation. Thus, a function λH ∈ ΛH i #NH is uniquely determined by its nodal values and is linear on each edge. A basis {θH }i=1 for

ADAPTIVE ACMS

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i ΛH can be obtained by imposing that θH be continuous and piecewise linear on ∂TH and i i θH (xj ) = δij for all xj ∈ NH . The support of θH is on all edges of elements τ ∈ TH for which P#NH i xi ∈ τ¯. Let µH = i=1 µH (xi )θH , then we note that if µH (xi ) = 0 for all xi ∈ NH , then e h. µH (x) = 0 for all x ∈ Nh . Hence, Λh = ΛH ⊕ Λ

e h be defined by For each K ∈ TH and νh ∈ Λh , let P K : Λh → Λ

(7)

e h, s(P K νh , µ eh ) = sK (νh , µ eh ) for all µ eh ∈ Λ

e h be such that and P : Λh → Λ (8)

P νh =

X

P K νh .

K∈TH ms It follows from the above that Λms h = {(I − P )θH : θH ∈ ΛH }. A basis for Λh is defined by i ms λms = (I − P )θH ∈ Λms i h , and by construction, λi (xj ) = δij for all xj ∈ NH .

An alternative to (6) is to find λH ∈ ΛH such that
(9) s (I − P )λH , (I − P )µH = (ρg, T (I − P )µH ) for all µH ∈ ΛH ,

and then λms h = (I − P )λH . We name it as ACMS–NLOD (Approximated Component Mode Synthesis Non-Localized Orthogonal Decomposition ) method. Albeit being well-defined, the method (9) is not “practical”, in the sense that the operators and P K and P are nonlocal, and computing (7) is as hard as solving (1). To circumvent that, we use the fact that the solutions of (7) actually decay exponentially to zero away from K. That allows the definition of a local approximation P K,j for P K , having support at a patch of width j around K. Next, before proving the exponential decay, we investigate the convergence rates for the ideal nonlocal solution ums h . In what follows, γ1 , γ2 , etc denotes positive constants that do not depend on A, f , ρ, h and H, depending only on the shape regularity of elements on Th and TH . Let aτmax τ τ κ = max κ , κ = τ , aτmax = sup a+ (x), aτmin = inf a− (x), x∈τ τ ∈TH amin x∈τ ρτmax = sup ρ(x) and ρτmin = inf ρ(x). x∈τ

x∈τ

Let us introduce the global Poincar´e’s inequality constant CP,G which is the smallest constant such that for all µh ∈ Λh (10)

kT µh kL2ρ (Ω) ≤ CP,G |T µh |HA1 (Ω) .

Let us also introduce the local Poincar´e’s inequality constant cP,L = maxτ ∈TH cτP,L , where the cτP,L are the smallest constants such that (11)

kT µ eh kL2ρ (τ ) ≤ cτP,L H|T µ eh |HA1 (τ )

e h. for all µ eh ∈ Λ

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ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

Lemma 1. Let τ ∈ TH and cτP,L as in (11). Then, an upper bound for cτP,L is given by (cτP,L )2

(12)

ρτmax ≤ γ1 (1 + log(H/h)) τ . amin

Proof. Using that µ eh vanishes at the NH nodes, we have [71] kT µ eh k2L2ρ (τ ) ≤ γ1 H 2 ρτmax kT µ eh k2L∞ (τ )

eh |2H 1 (τ ) ≤ γ1 H 2 (1 + log(H/h)) ≤ γ1 H 2 (1 + log(H/h)) ρτmax |T µ

ρτmax |T µ eh |2H 1 (τ ) . A aτmin



Lemma 2. Given µh ∈ Λh let IH µh ∈ ΛH be the Lagrange NH -nodal linear interpolation on ∂TH . Then (13)


|T IH µh |2H 1 (Ω) ≤ γ2 κ 1 + log(H/h) |T µh |2H 1 (Ω) . A

A

Proof. Let TI be defined by (4) with A = I, the identity matrix. It follows [71] for each τ ∈ TH that
|T IH µh |2H 1 (τ ) ≤ |TI IH µh |2H 1 (τ ) ≤ aτmax |TI IH µh |2H 1 (τ ) ≤ γ2 aτmax 1 + log(H/h) |TI µh |2H 1 (τ ) A A

τ ≤ γ2 amax 1 + log(H/h) |T µh |2H 1 (τ ) ≤ γ2 κτ 1 + log(H/h) |T µh |2H 1 (τ ) . A



We know extend the Face Lemma [71, Subsection 4.6.3] to variable coefficients. Lemma 3. Let τ ∈ TH , e an edge of ∂τ and χe be the characteristic function of e being e h we have identically equal to one on e and zero on ∂τ \e. Then given µ eh ∈ Λ
eh |2H 1 (τ ) . |T χe µ eh |2H 1 (τ ) ≤ γ3 κτ 1 + log(H/h)2 |T µ A

A

Proof. We have


2 eh |2H 1 (τ ) ≤ γ3 aτmax 1 + log(H/h) |TI µ eh |2H 1 (τ ) eh |2H 1 (τ ) ≤ aτmax |TI χe µ |T χe µ eh |2H 1 (τ ) ≤ |TI χe µ A A
2
2 ≤ γ3 aτmax 1 + log(H/h) |T µ eh |2H 1 (τ ) ≤ γ3 κτ 1 + log(H/h) |T µ eh |2H 1 (τ ) . A

ms e Theorem 4. Let λh = uh |∂TH , and λms h solution of (6). Then λh − λh ∈ Λh and ms 1 (Ω) ≤ cP,L HkgkL2 (Ω) . |uH h − u h | HA ρ



ADAPTIVE ACMS

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e Proof. First note that λh − λms h ∈ Λh since it follows from the Galerkin orthogonality that

ms ms ms s(λh − λms e’s inequality we obtain h , µh ) = 0 for all µh ∈ Λh . Using the local Poincar´ ms ms ms ms ms 2 |uH h − uh |H 1 (Ω) = s(λh − λh , λh − λh ) = s(λh − λh , λh ) = ρg, T (λh − λ ) A




ms 1 (Ω) , ≤ kgkL2ρ(Ω) kT (λh − λms h )kL2ρ (Ω) ≤ cP,L HkgkL2ρ (Ω) |T (λh − λh )|HA

and the result follows.



3.1. Decaying Low-Contrast. We next prove exponential decay of P K νh for K ∈ TH . Denote T1 (K) = {K},

Tj+1 (K) = {τ ∈ TH : τ ∩ τ j 6= ∅ for some τj ∈ Tj (K)}.

The following estimate is fundamental to prove exponential decay. e h . Then, for any Lemma 5. Assume that K ∈ TH and νh ∈ Λh , and let φ˜h = P K νh ∈ Λ integer j ≥ 1,

|T φ˜h |2H 1 (TH \Tj+1 (K)) ≤ 9α|T φ˜h |2H 1 (Tj+1 (K)\Tj (K)) , A

A

2

where α = γ3 κ(1 + log H/h) . e h such that νeh |∂τ = φ˜h if τ ∈ TH \Tj+1(K), and νeh = 0 on the remaining Proof. Choose νeh ∈ Λ

edges. We obtain

νh , νh ) − |T φ˜h |2H 1 (TH \Tj+1 (K)) = sK (e A

X

sτ (e νh , φ˜h ) = −

τ ∈Tj+1 (K)\Tj (K)

≤

X

X

sτ (e νh , φ˜h )

τ ∈Tj+1 (K)\Tj (K)

τ ∈Tj+1 (K)\Tj (K)

|T νeh |HA1 (τ ) |T φ˜h |HA1 (τ ) ,

where we used that s(e νh , φ˜h ) = sK (e νh , νh ) = 0 since the support νeh does not intersect with K. For each edge e of ∂τ , let χe be the characteristic function of e being identically equal to one on e and zero on ∂τ \e. For τ ∈ Tj+1 (K)\Tj (K), X |T (χe νeh )|2H 1 (τ ) ≤ 9γ3 κτ (1 + log H/h)2 |T φ˜h |2H 1 (τ ) , |T νeh |2H 1 (τ ) ≤ 3 A

A

A

e⊂∂τ

where we have used the Face Lemma [71, Subsection 4.6.3].



e h . Then, for any Corollary 6. Assume that K ∈ TH and νh ∈ Λh and let φ˜h = P K νh ∈ Λ

integer j ≥ 1,

j |T φ˜h |2H 1 (TH \Tj+1 (K)) ≤ e− 1+9α |T φ˜h |2H 1 (TH ) , A

where α is as in Lemma 5.

A

8

ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

Proof. Using Lemma 5 we have |T φ˜h |2H 1 (TH \Tj+1 (K)) ≤ 9α|T φ˜h |2H 1 (TH \Tj (K)) − 9α|T φ˜h|2H 1 (TH \Tj+1 (K)) A

A

A

and then |T φ˜h |2H 1 (TH \Tj+1 (K)) ≤ A

1 9α |T φ˜h |2H 1 (TH \Tj (K)) ≤ e− 1+9α |T φ˜h |2H 1 (TH \Tj (K)) , A A 1 + 9α

and the theorem follows.



Remark 1. The α in this paper is estimated as the worst case scenario. For particular cases of coefficients A and ρ, sharper estimated for α can be derived using weighted Poincar´e inequalities techniques and partitions of unity that conform with A in order to avoid large energies on the interior extensions [19, 21, 37, 59–62, 68]; see [34, 63] for examples. Inspired by the exponential decay stated in Corollary 6, we define the operator P j as follows. First, for a fixed K ∈ TH , let e K,j = {e eh : T µ Λ µh ∈ Λ eh = 0 on TH \Tj (K)}. h

e K,j such that Given µh ∈ Λh , define then P K,j µh ∈ Λ h

e K,j , s(P K,j µh , µ eh ) = sK (µh , µ eh ) for all µ eh ∈ Λ h

and let

P j µh =

(14)

X

P K,j µh .

K∈TH

We define the approximation λjH ∈ ΛH of λH by (15)


s (I − P j )λjH , (I − P j )µH = (ρg, T (I − P j )µH ) for all µH ∈ ΛH ,

and then let λms,j = (I − P j )λjH and ums,j = T λms,j . We name the scheme as ACMS–LOD h h h (Approximated Component Mode Synthesis Localized Orthogonal Decomposition) method. We now analyze the approximation error of the method, starting by a technical result essential to obtain the final estimate. Let cγ be a constant depending only on the shape regularity of TH such that (16)

X

τ ∈TH

for all v ∈ H 1 (TH ).

|v|2H 1 (τ ) ≤ (cγ j)2 |v|2H 1(TH ) ,

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Lemma 7. Consider νh ∈ Λh and the operators P defined by (8) and P j by (14) for j > 1. Then j−2

|T (P − P j )νh |2H 1 (TH ) ≤ (9cγ jα)2 e− 1+9α |T νh |2H 1 (TH ) . A

A

P

˜ h be Proof. Let ψ˜h = (P − P j )νh = K∈TH (P K − P K,j )νh . For each K ∈ TH , let ψ˜hK ∈ Λ such that ψ˜hK |e = 0 if e is a face of an element of Tj (K) and ψ˜hK |e = ψ˜h |e , otherwise. We obtain |T ψ˜h |2H 1 (TH ) =

(17)

A

X X

sτ (ψ˜h − ψ˜hK , (P K − P K,j )νh ) + sτ (ψ˜hK , (P K − P K,j )νh ).

K∈TH τ ∈TH

See that the second term of (17) vanishes since X X sτ (ψ˜K , P K νh )∂τ = 0. sτ (ψ˜K , (P K − P K,j )νh )∂τ = τ ∈TH

τ ∈TH

For the first term of (17), as in Lemma 5, X

X

sτ (ψ˜h − ψ˜hK , (P K − P K,j )νh )∂τ ≤

τ ∈TH

|T (ψ˜h − ψ˜hK )|HA1 (τ ) |T (P K − P K,j )νh |HA1 (τ )

τ ∈Tj+1 (K)

≤ 3α1/2 |T ψ˜h |HA1 (Tj+1 (K)) |T (P K − P K,j )νh |HA1 (Tj+1 (K)) . e K,j be equal to zero on all faces of elements of TH \Tj (K) and equal to P K νh Let νhK,j ∈ Λ h

otherwise. Using Galerkin best approximation property and Corollary 6 we obtain

|T (P K − P K,j )νh |2H 1 (Tj+1 (K)) ≤ |T (P K − P K,j )νh |2H 1 (TH ) ≤ |T (P K νh − νhK,j )|2H 1 (TH ) A

A

A

≤ 9α|T P

K

νh |2H 1 (TH \Tj−1 (K)) A

j−2 − 1+9α

≤ 9αe

K

|T P νh |2H 1 (TH ) . A

We gather the above results to obtain X j−2 |T ψ˜h |2H 1 (TH ) ≤ 9αe− 2(1+9α) |T ψ˜h |HA1 (Tj+1 (K)) |T P K νh |HA1 (TH ) A

K∈TH

j−2 − 2(1+9α)

≤ 9αe

cγ j|T ψ˜h |HA1 (TH )

X

K∈TH

We finally gather that |T P

K

νh |2H 1 (TH ) A

K

K

K

= s(P νh , P νh )∂TH = sK (P νh , νh ) =

Z

|T P

K

νh |2H 1 (TH ) A

1/2

.

A ∇(T P K νh ) · ∇ T νh dx

K

K

and from Cauchy–Schwarz, |T P νh |HA1 (TH ) ≤ |T νh |HA1 (K) , we have X |T P K νh |2H 1 (TH ) ≤ |T νh |2H 1 (TH ) . A

A

K∈TH



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ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

ms,j Theorem 8. Define uH = T (I − P j )λjH , where λjH is as in (15). Then h by (3) and let uh  j−2 ms,j |uH |HA1 (TH ) ≤ H cP,L + cP,G [γ2 κ (1 + log(H/h))]1/2 cγ j 2 9αe−([ 2(1+9α) −log(1/H)) kgkL2ρ (Ω) . h −u

Proof. First, from the triangle inequality, ms,j ms,j ms ms 1 (T ) + |u |uH |HA1 (TH ) ≤ |uH |HA1 (TH ) , h − uh h − u h | HA h − uh H

and for the first term we use Theorem 4. For the second term, we first define uˆms,j = h P ms j i i λh (xi )T (I − P )θH , and then X j ms i ums ˆms,j = (P − P j ) λms h −u h (xi )T θH = T (P − P )T IH λh , h i

where IH is as in Lemma 2. Relying on the Galerkin best approximation we gather from Lemma 7 that j−2

ms,j 2 2 ˆms,j |2H 1 (TH ) ≤ (cγ j)2 (9α)2 e− (1+9α) |T IH λms |ums |H 1 (TH ) ≤ |ums h −u h − uh h |H 1 (TH ) . h A

A

Since

ums h

=

T λms h

A

the result follow from Lemma 2 and the global Poincar´e’s inequality (10).  4. The High-Contrast Multiscale Case

The main bottle-neck in dealing with high-contrast coefficients is that α becomes too large, therefore j has to be large as well, cf. Theorems 4 and 8. Furthermore, the large local Poincar´e inequality constant cτP,L deteriorates the a priori error estimate Theorem 4. To deal e h by a subspace Λ△ ⊂ Λ e h by removing a subspace spanned with these two issues, we replace Λ h

by some eigenfunctions associated to an appropriated generalized eigenvalue problem to each

edge of the mesh TH . On the space Λ△ h , it is possible to remove the dependence on the

coefficients. In order to define these generalized eigenvalue problems, we first introduce some notation. ee = Λ e h |e and Λτ = Λh |∂τ , Let e be an edge shared by the elements τ and τ ′ , and denote Λ h h e that is, the restriction of functions on Λh to e and Λh to ∂τ , respectively. Note that a e e vanishes at the end-points of e; it is thus possible to extend continuously function µ ee ∈ Λ h

h

T e e → Λτ . by zero. Let us denote this extension by Reτ :Λ h h e e ′ e ′ τ e e e e e , by Let us define See : Λh → (Λh ) , where (Λh ) is the dual space of Λ h τ e T T (e µeh , See νeh )e = (Re,τ µ eeh , S τ Re,τ νehe )∂τ

where (·, ·)e is the L2 (e) inner product and Z τ τ τ (µh , S νh )∂τ = A ∇ T µτh · ∇ T νhτ dx τ

ee , for all µ eeh , νehe ∈ Λ h for all µτh , νhτ ∈ Λτh .

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c τ τ In a similar fashion, define Seτc e , See c and Sec ec , related to the degrees of freedom on e = ∂τ \e.

We remind that e is an open edge, not containing its endpoints. τ Let us introduce Mee by τ e (˜ µeh , Mee ν˜h )e τ and define Sbee =H

−2

=

Z

τ

T T ρ (T Re,τ µ ˜eh ) (T Re,τ ν˜he ) dx

τ τ Mee + See , where H is the target precision of the method, that can

be set by the user. Define also

and it is easy to show that (18)

τ τ τ τ −1 τ Seee = See − See Se c e , c (Sec ec )

τ e (e νhe , Seee νeh ) ≤ (νh , S τ νh ) for all νh ∈ Λτh such that Re,τ νh = νehe ,

e e is such that Re,τ νh (xi ) = νee (xi ) for all nodes where the restriction operator Re,τ : Λh → Λ h h xi ∈ Ne := (Nh \NH ) ∩ e.

In what follows, to take into account high contrast coefficients, we consider the following e e e ), where αe ≥ αe ≥ αe ≥ generalized eigenvalue problem: Find eigenpairs (αie , ψ˜h,i ) ∈ (R, Λ 1 2 3 h

e > 1, such that if the edge e is shared by elements τ and τ ′ we solve · · · ≥ αN e

(19)

τ ′ ˜e τ τ ′ ˜e τ + Sbee (Sbee + Seee )ψh,i = αie (Seee )ψh,i .

τ τ′ The eigenfunctions µ ˜eh,i are chosen to be orthonormal with respect to (·, (Sbee + Sbee )·)e . e,△ e,Π e e e e Now we decompose Λh := Λh ⊕ Λh where for a given αstab > 1,

(20)

e e,△ := span{˜ Λ µeh,i : αie < αstab }, h

e e,Π := span{˜ Λ µeh,i : αie ≥ αstab }. h

τ τ Note that we added H −2 Mee to define Sbee . This is necessary otherwise we might have a

few modes that would make the local Poincar´e’s inequality constant in (24) too large. We remark that αstab is defined by the user to replace α in the proof of Lemma 5. To define our ACMS–NLSD (Approximated Component Mode Synthesis Non-Localized Spectral Decomposition ) method for high-contrast coefficients,let (21)

e Π = {˜ eh : µ e e,Π for all e ∈ ∂TH }, Λ µh ∈ Λ ˜ h |e ∈ Λ h h

e △ = {˜ eh : µ e e,△ for all e ∈ ∂TH }. Λ µh ∈ Λ ˜ h |e ∈ Λ h h

e△ Note that Λh = ΛΠ h ⊕ Λh , where

0 eΠ ΛΠ h = Λh ⊕ Λh

12

ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

and Λ0h is the set of functions on Λh which vanish on all nodes of Nh \NH . Denote X (νhτ , S τ µτh )∂τ . (νh , Sµh )∂TH = τ ∈TH

We now introduce the ACMS–NLSD multiscale functions. For τ ∈ TH , consider the e △ as follows: Given µh ∈ Λh , find P τ,△ µh ∈ Λ e △ and define P △ operators P τ,△ , P △ : Λh → Λ h h such that (22)

(e νh△ , SP τ,△ µh )∂TH = (e νh△ , S τ µh )∂τ

e △, for all νeh△ ∈ Λ h

P△ =

X

P τ,△ .

τ ∈TH

ms,Π Consider Λms,Π = (I − P △ )ΛΠ ∈ h . The ACMS–NLSD method is defined by: Find λh h

ΛhΠ,ms such that (23)

(νhms,Π , Sλms,Π )∂TH = (ρg, T νhms,Π ) for all νhms,Π ∈ Λhms,Π . h

Note that (νhms,Π , Sλms,Π )∂TH h

=

Z

Ω

A ∇ T νhms,Π

·

∇ T λhms,Π

dx =

Z

Ω

ρgT νhms,Π dx.

Remark 2. In [36, 37], a different, but still local, eigenvalue problems are introduced, aiming to build approximation spaces. The analysis of the method however requires extra regularity of the coefficients, and the error estimate is not robust with respect to contrast. The counterpart of Lemma 1 follows. e△ Lemma 1’. Let µ e△ h ∈ Λh . Then

(24)

1/2 1 (Ω) . kT µ e△ H |T µ e△ h kL2ρ (Ω) ≤ (9αstab ) h | HA

Proof. We have for τ ∈ TH ,

2 −2 H −2 kT µ e△ h kL2ρ (τ ) ≤ 3H

Fixing the edge e of both τ and τ ′ , we have

X

e⊂∂τ

T 2 kT Re,τ µ ee,△ h kL2ρ (τ ) .

2 T 2 −2 T bτ ee,△ )e + (e bτ ′ ee,△)e ee,△ µe,△ µe,△ H −2 kT Re,τ µ ee,△ kT Re,τ ′µ h kL2ρ (τ ′ ) ≤ (e h , See µ h h , See µ h kL2ρ (τ ) + H h

′ ,△ ′ τ τ,△ e,△ eτ e,△ τ ≤ αstab µ eh |2H 1 (τ ′ ) eh |2H 1 (τ ) + |T µ ˜h , (See + Seee )e µh e ≤ αstab |T µ A

from (19), (20) and (18). By adding all τ ∈ TH , the results follows. Now we concentrate on the counterpart of Lemma 2.

A



ADAPTIVE ACMS

13

Lemma 2’. Let µh ∈ Λh and let µh = µΠ e△ h +µ h . Then

1/2 1 (Ω) ≤ (2 + 18αstab ) |T µh |HA1 (Ω) . |T µΠ h | HA

Proof. We have 2 2 2 e△ |T µΠ h |H 1 (Ω) ≤ 2(|T µh |H 1 (Ω) + |T µ h |H 1 (Ω) ). A

A

A

Consider the decomposition

µh |τ = µτ,0 h +

X

e⊂∂τ

where

µτ,0 h

∈

Λ0h

and

µ ˜τ,e h

=

2 |T µ ˜ τ,△ h |H 1 (τ ) A

µ ˜ τ,e,Π h

µ ˜τ,e,△ . h

µ eτ,e h

+ Then X τ,e,△ X τ τ,e,△ T (˜ µh , See µ ˜ h )e . ≤3 |T Re,τ µ ˜τ,e,△ |2H 1 (τ ) = 3 h A

e⊂∂τ

e⊂∂τ

Now we use that, if e is an edge of both τ and τ ′ ,     τ,e,△ eτ τ,e,△ τ,e,△ τ′ τ τ′ e . ≤ α µ ˜ , ( S + S )˜ µ µ ˜τ,e,△ , (S + S )˜ µ stab ee ee ee ee h h h h e

e

e e,△ and Λ e e,Π with respect to In addition, due to the orthogonality condition of the spaces Λ h h τ τ′ e e the inner product (·, (See + See )·)e , and (18), we have     ′ τ,e eτ τ,e,△ τ,e eτ + Seτ ′ )˜ eτ ′ )˜ µ ˜τ,e,△ , ( S ≤ µ ˜ , ( S + S ≤ |T µτh |2H 1 (τ ) + |T µτh |2H 1 (τ ′ ) . µ µ ee ee ee ee h h h h e

A

e

A

Adding all terms together we obtain the result.



We now state the counterpart of the face lemma [71, Subsection 4.6.3]. The lemma follows e τ,e,△ directly from the definition of the generalized eigenvalue problem and properties of Λ h

and (18).

e△ Lemma 3’. Let e be a common edge of τ , τ ′ ∈ TH , and µ e△ ee,△ = h ∈ Λh . Then, defining µ h

Re,τ µ e△ eτ,△ =µ e△ h | and µ h h |∂τ it follows that

′

2 2 T T 2 eτh ,△ |2H 1 (τ ′ ) eτ,△ ee,△ |T Re,τ µ ee,△ h |H 1 (τ ) + |T µ h |H 1 (τ ′ ) ≤ αstab |T µ h |H 1 (τ ) + |T Re,τ ′ µ A

A

A

A




Proof. We have

′

τ e,△ τ e,△ 2 T T 2 µe,△ eh )∂τ + (e µe,△ eh )∂τ ′ ee,△ |T Re,τ µ ee,△ h , See µ h , See µ h |H 1 (τ ′ ) = (e h |H 1 (τ ) + |T Re,τ ′ µ A

A

′

′

τ △ τ △ eτ ee,△ )∂τ + (e eτ ee,△ )∂τ ′ ≤ αstab (e ≤ αstab (e µe,△ µe,△ eh )∂τ ′ . µ△ eh )∂τ + (e µ△ h , See µ h h , See µ h h ,S µ h ,S µ

e△ Theorem 4’. Let λh = uh |∂TH , and λms,Π solution of (23). Then λh − λms h ∈ Λh and h ms,Π 2 |uH |H 1 (Ω) ≤ 9αstab H 2 kgk2L2ρ (Ω) . h − uh A



14

ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

e△ Proof. First note that λh − λms h ∈ Λh since it follows from the Galerkin orthogonality that

s(λh − λms,Π , µms,Π ) = 0 for all µms,Π ∈ Λms h . Using Lemma 1’ we obtain h h h


ms,Π 2 )|HA1 (Ω) , |uH |H 1 (Ω) = ρg, T (λh − λms,Π ) ≤ (9αstab )1/2 H kgkL2ρ (Ω) |T (λh − λms,Π h − uh h A

and the result follows.



4.1. Decaying High-Contrast. We next prove exponential decay of P K,△νh for K ∈ TH . K,△ Lemma 5’. Let µh ∈ Λh and let φ˜△ µh for some fixed element K ∈ TH . Then, for h = P

any integer j ≥ 1, 2 ˜△ 2 |T φ˜△ h |H 1 (TH \Tj+1 (K)) ≤ 9αstab |T φh |H 1 (Tj+2 (K)\Tj (K)) . A

A

Proof. Following the steps of the proof of Lemma 5, we gather that X 2 1 (τ ) , |T νeh |HA1 (τ ) |T φ˜△ |T φ˜△ h | HA h |H 1 (TH \Tj+1 (K)) ≤ A

τ ∈Tj+1 (K)\Tj (K)

e △ is such that νe△ |∂τ = φ˜△ if τ ∈ TH \Tj+1 (K), and νe△ = 0 on the remaining where νeh△ ∈ Λ h h h h

edges. If e is an edge of ∂τ and ∂τ ′ , and χe the characteristic function of e, for τ ∈ Tj+1 (K)\Tj (K) and τ ′ ∈ Tj (K)\Tj−1 (K), then, for µ ee,△ =µ e△ h h |e , X 2 2 |T (χe µ ˜△ ≤ 3 |T (˜ µ△ )| 1 h )|H 1 (τ ) h H (τ ) A

A

e⊂∂τ

and 2 τ e,△ eτ eτ ′ µe,△ )e |T (χe µ ˜△ µe,△ ˜h )e ≤ αstab (˜ µe,△ 1 (τ ) = (e h )|HA h , See µ h , (See + See )˜ h

τ △ τ △ 2 2 ≤ αstab (˜ µ△ ˜h )∂τ + (˜ µ△ ˜h )∂τ ′ = αstab |T µ e△ e△ h ,S µ h ,S µ h |H 1 (τ ′ ) , h |H 1 (τ ) + |T µ A

where we have used (18).

A



Note that now the bound is in terms of Tj+2(K)\Tj (K) rather than Tj+1 (K)\Tj (K). This means that the j in Corollary 6 is replaced below by the integer part of (j + 1)/2. K,△ e △ . Then, for Corollary 6’. Assume that K ∈ TH and νh ∈ Λh and let φ˜△ νh ∈ Λ h = P h

any integer j ≥ 1,

[(j+1)/2]

− 1+9α

2 |T φ˜△ h |H 1 (TH \Tj+1 (K)) ≤ e A

where [s] is the integer part of s.

stab

|T φ˜h |2H 1 (TH ) . A

ADAPTIVE ACMS

15

Proof. Using Lemma 5’ we have 2 ˜△ 2 |T φ˜△ h |H 1 (TH \Tj+1 (K)) ≤ |T φh |H 1 (TH \Tj (K)) A

A

2 ˜△ 2 ≤ 9αstab |T φ˜△ h |H 1 (TH \Tj−1 (K)) − 9αstab |T φh |H 1 (TH \Tj+1 (K)) , A

A

and then 2 |T φ˜△ h |H 1 (TH \Tj+1 (K)) ≤ A

9αstab − 1+9α1 2 ˜△ |2 1 stab |T φ ≤ e |T φ˜△ 1 (T \T h HA (TH \Tj−1 (K)) . h | HA j−1 (K)) H 1 + 9αstab 

Inspired by the exponential decay stated in Corollary 6’, we define the operator P △,j as follows. First, for a fixed K ∈ TH , let e △,K,j = {e e△ : T µ Λ µh ∈ Λ eh = 0 on TH \Tj (K)}. h h

e K,j such that For µh ∈ Λh , define P △,K,j µh ∈ Λ h

e △,K,j , s(P △,K,j µh , µ eh ) = sK (µh , µ eh ) for all µ eh ∈ Λ h

and let

P △,j µh =

(25)

X

P △,K,j µh .

K∈TH Π Finally, define the approximation λΠ,j H ∈ ΛH such that

(26)


△,j △,j Π Π s (I − P △,j )λΠ,j )µΠ )µΠ H = (ρg, T (I − P H ) for all µH ∈ ΛH , H , (I − P

ms,Π,j and then let λhms,Π,j = (I − P △,j )λΠ,j = T λhms,Π,j . We name as ACMS–LSD H and uh

(Approximated Component Mode Synthesis Localized Spectral Decomposition) method. We now analyze the approximation error of the method, starting by a technical result essential to obtain the final estimate. Lemma 7’. Consider νh ∈ Λh and the operators P △ defined by (22) and P △,j by (25) for j > 1. Then [(j−1)/2] − 1+9α

|T (P △ − P △,j )νh |2H 1 (TH ) ≤ (cγ j)2 (9αstab )2 e A

where cγ is as in (16).

stab

|T νh |2H 1 (TH ) , A

16

ALEXANDRE L. MADUREIRA AND MARCUS SARKIS

P K,△ Proof. Let ψ˜h△ = (P △ − P △,j )νh = − P K,△,j )νh . For each K ∈ TH , let K∈TH (P e △ be such that ψ˜K,△|e = 0 if e is an edge of an element of Tj (K) and ψ˜K,△ |e = ψ˜△ |e , ψ˜hK,△ ∈ Λ h h h h

otherwise. We obtain (27) |T ψ˜△ |2 1

h HA (TH )

=

X X

sτ (ψ˜h△ − ψ˜hK,△ , (P K − P K,△,j )νh ) + sτ (ψ˜h△,K , (P K,△ − P K,△,j )νh ).

K∈TH τ ∈TH

See that the second term of (27) vanishes since X X sτ (ψ˜K,△ , P K,△νh )∂τ = 0. sτ (ψ˜K,△, (P K,△ − P K,△,j )νh )∂τ = τ ∈TH

τ ∈TH

For the first term of (17), as in Lemma 5, X

sτ (ψ˜h△ − ψ˜hK,△, (P K,△ − P K,△,j )νh )∂τ

τ ∈TH

≤

X

|T (ψ˜h△ − ψ˜hK,△ )|HA1 (τ ) |T (P K,△ − P K,△,j )νh |HA1 (τ )

τ ∈Tj+1 (K) 1/2 ≤ 3αstab |T ψ˜h△ |HA1 (Tj+1 (K)) |T (P K,△ − P K,△,j )νh |HA1 (Tj+1 (K)) .

e K,△,j be equal to zero on all faces of elements of TH \Tj (K) and equal to Let νhK,△,j ∈ Λ h

P K,△νh otherwise. Using Galerkin best approximation property, Lemma 3’ and Corollary 6’, we obtain |T (P K,△ − P K,△,j )νh |2H 1 (Tj+1 (K)) ≤ |T (P K,△ − P K,△,j )νh |2H 1 (TH ) A

A

≤ |T (P

K,△

νh −

νhK,△,j )|2H 1 (TH ) A

≤ 9αstab |T P

K,△

νh |2H 1 (TH \Tj−1 (K)) A

[(j−1)/2] − 1+9α

≤ 9αstab e

stab

|T P K,△νh |2H 1 (TH ) . A

We gather the above results to obtain [(j−1)/2]

− |T ψ˜h△ |2H 1 (TH ) ≤ 9αstab e 2(1+2αstab ) A

X

|T ψ˜h△ |HA1 (Tj+1 (K)) |T P K,△νh |HA1 (TH )

K∈TH [(j−1)/2]

− 2(1+9α

≤ 9αstab e

˜△ |H 1 (T ) stab ) c j|T ψ γ h A H

X

K∈TH

|T P

K,△

νh |2H 1 (TH ) A

1/2

.

We finally gather that |T P K,△νh |2H 1 (TH ) = s(P K,△νh , P K,△νh )∂TH = sK (P K,△νh , νh ) A Z = A ∇(T P K,△νh ) · ∇ T νh dx, K

ADAPTIVE ACMS

17

and from Cauchy–Schwarz, |T P K,△νh |HA1 (TH ) ≤ |T νh |HA1 (K) , we have X |T P K,△νh |2H 1 (TH ) ≤ |T νh |2H 1 (TH ) . A

A

K∈TH

 ms,Π,j Π,j Theorem 8’. Define uH = T (I − P △,j )λΠ,j h by (3) and let uh H , where λH is as in (26).

Then |uH h

−u

ms,Π,j

|HA1 (TH )

   [(j−1)/2] − 2(1+9α −log(1/H ) 1/2 ) stab ≤ H 3(2αstab ) + cP,G cγ j9αstab e kgkL2ρ (Ω) .

Proof. First, from the triangle inequality, ms,Π ms,Π,j − uhms,Π,j |HA1 (TH ) , |HA1 (TH ) + |ums,Π |uH |HA1 (TH ) ≤ |uH h − uh h − uh h

and for the first term we use Theorem 4’. For the second term, we define uˆhms,Π,j = T (I − P △,j )λms,Π , and then h ums ˆhms,Π,j = T (P − P j )λms,Π . h −u h Relying on the Galerkin best approximation we gather from Lemma 7’ that [(j−1)/2]

− (1+9α

ms,Π,j 2 ˆhms,Π,j |2H 1 (TH ) ≤ (cγ j)2 (9αstab )2 e |ums |H 1 (TH ) ≤ |ums h −u h − uh

stab )

|T λms,Π |2H 1 (TH ) . h A

A

A

ms Since ums e’s inequality (10). h = T λh the result follow from Lemma 1’ and the global Poincar´

 5. Spectral Multiscale Problems inside Substructures To approximate uB h on an element τ ∈ TH , we introduce a multiscale method by first building the approximation space Vτms := Span{ψh1 , ψh2 , · · · , ψhNτ } generated by the following generalized eigenvalue problem: Find the eigenpairs (αi , ψhi ) ∈ (R, VhB (τ )) such that aτ (vh , ψhi ) = λi (ρvh , ψhi )τ where aτ (vh , ψhi )

=

Z

A ∇ vh · τ

and 0 < λ1 ≤ λ2 ≤ · · · λNτ