TWO-LEVEL OVERLAPPING SCHWARZ ALGORITHMS FOR A STAGGERED DISCONTINUOUS GALERKIN METHOD

FOR A STAGGERED DISCONTINUOUS GALERKIN METHOD

ERIC T. CHUNG^∗, HYEA HYUN KIM^†,ANDOLOF B. WIDLUND^‡

TR2011-943

Abstract. Two overlapping Schwarz algorithms are developed for a discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems in both two and three dimensions. The discontinuous Galerkin formulation is based on a staggered discretization introduced by Chung and Engquist [13] for the acoustic wave equation. Two types of coarse problems are introduced for the two-level Schwarz algorithms. The first is built on a nonoverlapping subdomain partition, which allows quite general subdomain partitions, and the second on introducing an additional coarse triangulation that can also be quite independent of the fine triangulation. Condition number bounds are established and numerical results are presented.

Key words. domain decomposition, elliptic problems, preconditioned conjugate gradients, discontinuous Galerkin methods, staggered grid, overlapping Schwarz algorithms

AMS(MOS) subject classifications. 65F10, 65N30, 65N55

1. Introduction. Two-level overlapping Schwarz algorithms are developed for the fast and stable solution of a staggered discontinuous Galerkin method applied to second order elliptic problems. Discontinuous Galerkin methods allow test functions which are discon- tinuous across element boundaries and this feature makes them more suitable for modeling problems with discontinuous coefficients, singularities, multiscales and multiphysics. Since the first work, by Reed and Hill [27], for hyperbolic equations, discontinuous Galerkin meth- ods have been applied to various problems and the field has become an active research area, see, e.g., [19, 16, 28, 9, 5]. The design of the flux condition across the inter-element boundary determines the accuracy of the discontinuous Galerkin approximation and the properties of the resulting linear system.

In relatively recent works by Engquist and the first author [12, 13, 15, 14], a staggered discontinuous Galerkin method is developed and analyzed. A second order problem is writ- ten as a system of first order with two unknownsU andu. To approximateU andu, each

∗Department of Mathematics, The Chinese University of Hong Kongtschung@math.cuhk.edu.hk, http://www.math.cuhk.edu.hk/tschung, AND The work of this author was supported by Hong Kong RGC General Research Fund Project 401010,

†Department of Applied Mathematics, Kyung Hee University, Korea hhkim@khu.ac.kr or hyeahyun@gmail.com, http://hhkim.khu.ac.kr. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea Korean Government(MOST) (2011-0023354).

‡Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, N.Y. 10012, USA widlund@cims.nyu.edu, http://www.cs.nyu.edu/cs/faculty/widlund. This work was sup- ported in part by the U.S. Department of Energy under contracts DE-FG02-06ER25718 and in part by National Science Foundation Grant DMS-0914954. Part of the work of this author was also supported by the Institute of Mathematical Sciences and the Department of Mathematics of the Chinese University of Hong Kong.

1

triangle, in a given triangulation, is subdivided and discontinuous functionsU_handuhare built for the resulting triangulation so that on each interelement boundary one of these func- tions is continuous and the other discontinuous. In addition, we require these functions to satisfy a certain inf-sup stability. Using them, a conservative inter-element flux condition is then obtained straightforwardly. Such a flux condition preserves symmetry of the model problem and results in an optimal order of approximation. Moreover, the use of this staggered approximation provides locally and globally conservative schemes.

For elliptic problems, the resulting linear system arising from the staggered discontin- uous Galerkin formulation is symmetric and positive definite after eliminating one set of variables locally. To the best of our knowledge, no discontinuous Galerkin formulation has previously been developed which is symmetric and positive definite without introducing an additional penalty term. However, one disadvantage of the staggered discontinuous Galerkin method is that the resulting linear system is relatively large and less sparse than those from other discontinuous Galerkin formulation, because the test functions are built after a further subdivision of the given triangulation and are also partially continuous. Therefore, a fast and stable solver for the staggered discontinuous Galerkin formulation is quite desirable to increase its applicability for real world problems.

There have been previous studies that address fast and stable solvers for discontinu- ous Galerkin methods. In the works by Feng and Karakashian [21, 22], two-level additive Schwarz methods were developed for second order elliptic problems and fourth order prob- lems, and in the work by Lasser and Toselli [26] overlapping Schwarz preconditioners were developed for advection-diffusion problems. A more general framework of Schwarz pre- conditioners was studied in [1, 2, 3, 4] including multiplicative Schwarz preconditioners and hp-discontinuous Galerkin formulation. In the work by Dryja, Galvis, and Sarkis [20], BDDC methods were applied to discontinuous Galerkin formulations of elliptic problems with dis- continuous coefficients, where the finite element functions are continuous inside each subdo- main and discontinuous across the subdomain boundaries only. Recently, two-level additive Schwarz preconditioners have also been studied by Barker et al [6]. In their work, algorithms are developed and analyzed for several types of coarse problems and their performance com- pared for these different choices.

In our work, we will develop a two-level overlapping Schwarz preconditioner for the staggered discontinuous Galerkin formulation [13] applied to elliptic problems. In all the previous works on two-level Schwarz preconditioners for the discontinuous Galerkin formu- lation, each subdomain is assumed to be an element of a coarse regular partition or the union of a few such elements. Our algorithm, in contrast, allows for a quite general subdomain partition without such an assumption. Two types of coarse problems are introduced. The first one is related only to the subdomain partition where each subdomain is obtained as the union of elements provided in the problem domain. On each face, which is the common

part of two subdomain boundaries, we introduce a face-based finite element function; its value is one on the given face and zero on the rest of the subdomain interface. For these interface values, the values in the interior of each subdomain are determined by minimiz- ing a certain discrete energy norm. By using these face-based functions in the construc- tion of the coarse problem, we can prove that the condition number can be bounded by C(1 +H/δ)(1 +H^2−dmaxFij|θ_F^c_ij|²_H¹_(Ω)), wheredis the dimension,His the subdomain diameter,δthe overlapping width,Ca positive constant independent of any mesh parameters, andθ_F^c_ij(x)a continuous, face-based finite element function described in Section 4. We note that our result can be applied to quite general subdomain partitions, where each subdomain satisfies a Poincar´e-inequality and a starlike property.

The second type of coarse problem is obtained by introducing an additional coarse tri- angulation. In this case, the subdomains again need not be a union of coarse triangles. With the less strong assumption that the diameter of each subdomain is comparable to those of the coarse triangles which intersect it, we can prove a condition number bound ofC(1 +H/δ).

The rest of this paper is organized as follows. In Section 2, the staggered discontinuous Galerkin formulation is introduced for a model elliptic problem and in Sections 3 and 4, our first two-level Schwarz algorithm is developed and analyzed. In Section 5, the algorithm with the second type of the coarse problem is introduced and analyzed. In Section 6, numerical experiments are reported for the proposed algorithms. Throughout this paper,Cdenotes a generic positive constant, which is independent of any mesh parameters.

2. The Staggered Discontinuous Galerkin formulation.

2.1. Variational form. We consider a scalar, elliptic model problem in a bounded do- mainΩ⊂R^dwithd= 2or3:

(2.1) findu∈H₀¹(Ω)such that

−∇ ·(ρ(x)∇u(x)) =f(x) ∀x∈Ω,

whereρ(x)≥ρ0>0withρ0a constant. The domainΩis subdivided into potentially many subdomainsΩi, which may have quite irregular boundaries. In the following, we will used to denote the dimension ofΩ. In our description of the algorithm, we will primarily discuss the case ofd= 3. The coefficient function can be discontinuous inΩ, but will be assumed to vary only moderately in each subdomain. An equivalent variational formulation is obtained by integrating by parts:

(2.2) findu∈H₀¹(Ω)such that

(ρ(x)∇u,∇v)L²(Ω)= (f, v)L²(Ω) ∀v∈H₀¹(Ω).

By introducing an additional unknown, namelyU :=ρ∇u, we can recast this problem, and obtain a suitable framework for our DG discretization, also known as a two-unknown or a

saddle point problem:

(2.3)

find(u,U)∈H₀¹(Ω)×L²(Ω)such that

(ρ(x)⁻¹U,V)L²(Ω)−(∇u,V)L²(Ω)= 0 ∀V ∈L²(Ω), (U,∇v)L²(Ω)= (f, v)L²(Ω) ∀v∈H₀¹(Ω).

2.2. The Staggered Discontinuous Galerkin discretization. Following Chung and Engquist [12, 13], we first define an initial triangulationTu. Thus, the domainΩis trian- gulated using a set of tetrahedra in 3D and triangles in 2D.Fuwill denote the set of all faces in this triangulation andF_u⁰the subset of all interior faces, i.e., the set of faces inFuthat are not embedded in∂Ω.

For each tetrahedron, we select an interior pointν and denote this tetrahedron byS(ν).

We then further subdivide each tetrahedron into4sub-tetrahedra by connecting the pointνto the4vertices of the tetrahedron. The resulting triangulation is denoted byT.We will denote byFpthe set of all the new faces obtained by the second subdivision and setF :=Fu∪ Fp

andF⁰:=F_u⁰∪ Fp.

For each faceκ∈ Fu, we denote byR(κ)the union of the two sub-tetrahedra sharing the faceκ. Ifκis a boundary face, thenR(κ)is just the one tetrahedron having this face. See Figure 1 for an illustration of this concept in two dimensions.

•

• S(ν1)

S(ν2)

R(κ) κ ν1

ν2

FIG. 1. Triangulation in 2D.

We define a unit normal vectorn_κfor each faceκ∈ Fas follows: Ifκ∈ F\F⁰, then n_κis the unit normal vector ofκpointing towards the outside ofΩ. Ifκ∈ F⁰,an interior face, we then fixn_κ as one of the two possible unit normal vectors onκ; when it is clear which face is being considered, we will simplify the notation and useninstead ofn_κ.

We are now ready to introduce our finite element spaces. Letk ≥0be a non-negative integer. Letτ∈ T and letP^k(τ)be the space of polynomials of degree less than or equal to konτ.

We first introduce our discrete scalar field space:

LocallyH¹(Ω)-conforming finite element space for the scalar field:

(2.4) Sh:={v|v_|τ ∈P^k(τ), ∀τ ∈ T; vcontinuous acrossκ∈ F_u⁰; v_|∂Ω= 0}.

We define two norms in the spaceSh,the discreteL²-normkvkXand the discreteH¹− normkvkZ, by

kvk²_X = Z

Ω

v²dx+ X

κ∈F_u⁰

hκ

Z

κ

v²dσ, (2.5)

kvk²_Z = Z

Ω

|∇v|²dx+ X

κ∈Fp

h⁻¹_κ Z

κ

[v]²dσ, (2.6)

wherehκis the diameter ofκand the integral of∇vin (2.6) should be understood as defined elementwise:

Z

Ω

|∇v|²dx=X

τ∈T

Z

τ

|∇(v_|τ)|²dx.

Here we recall that, by definition,v ∈ Shis always continuous across each face ofF_u⁰, but that it can be discontinuous across any face ofFp. In the above definition, the jump[v]across eachκ∈ Fpis defined as

[v] =v1−v2

wherevi =v_|τi andτ1andτ2are the two (sub-)tetrahedra sharingκ. We note that by using norm equivalence and a scaling argument (see also [13, Theorem 3.1]), we can show that there exists a constantC >0, independent ofh, such that

kvk²_L²_(Ω)≤ kvk²_X ≤Ckvk²_L²_(Ω) ∀v∈ Sh. We next introduce a discrete space of vector fields:

LocallyH(div; Ω)-conforming finite element space for the vector field:

(2.7) Vh={V |V_|τ ∈P^k(τ)^d, ∀τ∈ T; V ·nis continuous acrossκ∈ Fp}.

In the spaceVh, we define two norms, the discreteL²-norm and the discreteH(div; Ω)-norm, by

kVk²X′ = Z

Ω

|V|²dx+ X

κ∈Fp

hκ

Z

κ

(V ·n)²dσ, (2.8)

kVk²Z′ = Z

Ω

(∇ ·V)²dx+ X

κ∈F_u⁰

h⁻¹_κ Z

κ

[V ·n]²dσ (2.9)

where the integral of(∇ ·V)²in (2.9) is defined elementwise. We also recall that, by defini- tion,V ∈ Vhhas a continuous normal component across each faceκ∈ Fp.

In the definition above, the jump[V ·n]on eachκ∈ F_u⁰is defined as [V ·n] =V₁·n−V₂·n,

whereV_i=V_|τ

iandτ1andτ2are the two sub-tetrahedra withκas their common face.

One can prove, by an argument used in the proof of [13, Theorem 3.2], that there exists a constantC >0, independent ofh, such that

(2.10) kVk²L²(Ω)≤ kVk²X′ ≤CkVk²L²(Ω) ∀V ∈ Vh. We next define

bh(U, v) = Z

Ω

U· ∇v dx− X

κ∈Fp

Z

κ

U·n[v]dσ

− X

κ∈Fu\F_u⁰

Z

κ

vU·ndσ, U ∈ Vh, v∈ Sh

(2.11)

b^∗_h(u,V) =− Z

Ω

u∇ ·V dx+ X

κ∈Fu⁰

Z

κ

u[V ·n]dσ

+ X

κ∈Fu\F_u⁰

Z

κ

uV ·ndσ, u∈ Sh,V ∈ Vh. (2.12)

We note that whenvanduin the above formulae vanish on∂Ω, the last term in both bh(U, v)andb^∗_h(u,V)vanish.

According to Lemma 2.4 of Chung and Engquist [13], we have (2.13) bh(V, v) =b^∗_h(v,V), ∀(v,V)∈ Sh× Vh. Moreover, the following holds

(2.14) bh(V, v)≤ kvkZkVkX′, ∀(v,V)∈ Sh× Vh. The Staggered Discontinuous Galerkin method reads:

find(uh,U_h)∈ Sh× Vhsuch that (U_h,V)L²

ρ(Ω)−b^∗_h(uh,V) = 0, ∀V ∈ Vh

bh(U_h, v) = (f, v)L²(Ω), ∀v∈ Sh. (2.15)

Here

(U,V)L²

ρ(Ω)= Z

Ω

1

ρ(x)U·V dx.

LetBh andMhare matrices obtained frombh(V, v)and(U,V)L²_ρ(Ω)for functions in (V, v) ∈ Vh× Sand(U,V) ∈ Vh× Vh, respectively. Using thatbh(V, v) = b^∗_h(v,V),

the matrixB_h^T corresponds to the bilinear formb^∗_h(v,U)for(v,U)∈ Sh× Vh. We can then rewrite (2.15) as an algebraic system of equations:

MhU_h−B_h^Tuh= 0, (2.16)

BhU_h=fh. (2.17)

SinceMh is symmetric and positive definite and block diagonal with small blocks, we can eliminateU_hfrom (2.16) to obtain an equation foruh,

(2.18) BhM_h⁻¹B_h^Tuh=fh,

with a matrix which is symmetric and positive definite. We introduce a bilinear form for (u, v)∈ Sh× Sh

a(u, v) :=v^TBhM_h⁻¹B^T_hu and use the notationAto denote the matrixBhM_h⁻¹B_h^T,

(2.19) A:=BhM_h⁻¹B^T_h.

We will develop two two-level overlapping Schwarz algorithms for solving the algebraic system (2.18).

In the design of the first preconditioner, we will build coarse basis functions related to a nonoverlapping subdomain partition ofΩsimilar to that of [17]. Let{Ωi}be a nonoverlap- ping partition ofΩ. For a given partition, we introduce local finite element spaces,

Vh,i:=Vh|Ωi, Sh,i:=Sh|Ωi,

which are the restrictions ofVhandShto the subdomainΩi. Associated with (Vh,i,Sh,i), we introduce local bilinear formsbh,iandb^∗_h,iby

bh,i(U, v) = Z

Ωi

U· ∇v dx− X

κ∈FpT Ωi

Z

κ

U·n[v]dσ (2.20)

b^∗_h,i(u,V) =− Z

Ωi

u∇ ·V dx+ X

κ∈F_u⁰T Ωi

Z

κ

u[V ·n]dσ (2.21)

+ X

κ∈F_u⁰T

∂Ωi

Z

κ

uV ·n_idσ,

wheren_iis the unit normal to∂Ωionκ. It can be seen easily that (2.22) bh,i(V, v) =b^∗_h,i(v,V), and that

bh(V, v) =X

i

bh,i(V|Ωi, v|Ωi), b^∗_h(v,V) =X

i

b^∗_h,i(v|Ωi,V|Ωi).

LetBiandB_i^∗be the matrices associated to the bilinear formsbh,iandb^∗_h,i, respectively, i.e.,

hBiV|Ωi, v|Ωii=bh,i(V|Ωi, v|Ωi) and

hB_i^∗v|Ωi,V|Ωii=b^∗_h,i(v|Ωi,V|Ωi).

Hereh·,·idenotes thel²-inner product. Using (2.22), we have B^∗_i =B^T_i .

By introducingMi, the matrix associated to the bilinear form hMiU|Ωi,V|Ωii= (U|Ωi,V_Ω

i)_L²

ρ(Ωi)

andRi, the restriction fromShtoSh,i, we can rewrite (2.15) as MiU_i−B_i^TRiu= 0, i= 1,· · ·, N, (2.23)

X

i

R^T_i BiU_i=X

i

R^T_i fi, (2.24)

whereU_iis the restriction ofUtoΩiandfiis given by hfi, v|Ωii= (f, v)L²(Ωi).

SinceMiare invertible, by (2.23) and (2.24), we can obtain the algebraic equation (2.18) by assembling of local matrices:

(2.25) X

i

R^T_iBiM_i⁻¹B_i^TRiu=X

i

R^T_i fi.

Here we note thatu ∈ Vh, where functions can be discontinuous across each faceκ∈ Fp. We introduce the notationAifor

Ai=BiM_i⁻¹B^T_i . and we introduce a bilinear form defined onSh,i× Sh,i, (2.26) ai(ui, vi) =hAiui, vii.

3. A two-level overlapping Schwarz algorithm. We consider a nonoverlapping parti- tions ofΩ, which is denoted by{Ωi}. The nonoverlapping partition can be obtained from the original triangulationTuprovided forΩ, e.g., by using a mesh partitioner; the subdomains in the resulting partition may then have quite irregular boundaries. The interfaceΓis defined by

(∪i6=k∂Ωi∩∂Ωk)\∂Ω, andΓ^his the set of nodes that belong to the boundaries of at least two substructures.

We introduce an overlapping partition{Ω^′_j}ofΩand each subregionΩ^′_j is associated with finite element spacesVh(Ω^′_j)andS_h⁰(Ω^′_j), which are the restrictions ofVh andSh to the subregionΩ^′_j. Here the superscript0indicates that the functions inS_h⁰(Ω^′_j)vanish on the boundary ofΩ^′_j.

A bilinear form is introduced for(u, v)∈ S_h⁰(Ω^′_j)× S_h⁰(Ω^′_j), by aΩ^′_j(u, v) :=v^TBh,Ω^′_jM_Ω⁻¹′

jB_h,Ω^T ′ ju,

whereBh,Ω^′_j is the matrix obtained frombh(U, v)for(U, v)∈ Vh(Ω^′_j)× S_h⁰(Ω^′_j)andM_Ω⁻¹′ j

is the inverse of the weighted mass matrix obtained from (U,V)L²_ρ(Ω) where (U,V) ∈ Vh(Ω^′_j)× Vh(Ω^′_j).

To simplify the presentation, we will use the notationV_j^′to denoteS_h⁰(Ω^′_j)and introduce the trivial extension by zero

R^T_j : V_j^′→Sh. A projectionPj,related to the subregionΩ^′_j, is defined by

Pj=R_j^TP_j^′, whereP_j^′is obtained from

aΩ^′_j(P_j^′u, v) =a(u, R^T_jv), ∀v∈V_j^′.

We now construct the coarse spaceV0based on the nonoverlapping partition{Ωi}. Let Fijdenote the common face (edge) of two subdomainsΩiandΩjin three (two) dimensions.

On eachFij, we define a face (edge)-based functionθ^(k)_Fij(x)as follows. Forx ∈ ∂Ω^h_i its value is given by

θ_F^(k)_ij(x) =

( 1, x∈F^h_ij 0, x∈Γ^h\F^h_ij.

We extend these interface values to the interior by a minimal energy extension with respect to the seminormai(vi, vi)^1/2 defined in (2.26). Here we use the superscriptkto stress that Shis defined by piecewise polynomials of orderk. Forx∈Ω^h_j, we defineθ^(k)_Fij(x)similarly.

We then extend it by zero to the rest ofΩas an element ofSh. We can now obtain the space of coarse basis functions,

V0=spann

θ^(k)_Fij(x), ∀Fij

o. The projectionP0is then defined by

a(P0u, v) =a(u, v), ∀v∈V0

and the two-level overlapping Schwarz operator is given by Pas=

N

X

j=0

Pj.

4. Estimate of the condition number. We will now provide a bound of the condition number of our first two-level overlapping Schwarz algorithm. See [29, Chapter 3] for this algorithm and theory in the standard conforming case.

For the upper bound, we obtain

a(Pasu, u)≤(1 +Nc)a(u, u),

whereNc is the number of colors required to color the overlapping subregions in{Ω^′_j}so that no two of them have the same color.

For the lower bound, we will prove that for some decomposition ofu∈ Sh, u=u0+

N

X

j=1

R^T_juj,

withu0∈V0anduj∈V_j^′,the following inequality holds a(u0, u0) +

N

X

j=1

aΩ^′_j(uj, uj)≤C0²a(u, u).

The condition number ofPasis then bounded by

κ(Pas)≤(1 +Nc)C₀².

In our theory, we need an assumption on the nonoverlapping subdomain partition{Ωi}. A domainΩis starlike if there exits ax0∈Ωand a constantc >0such that

(4.1) (x−x0)·n≥cHΩ, ∀x∈∂Ω,

wherenis the unit normal to∂Ωatx.

ASSUMPTION4.1. Each subdomainΩisatisfies the Poincar´e inequalities and the star- like property, and the number of tetrahedra along each edge ofΩiis proportional to(H/h)^d−2. With the above assumption on each subdomain in the nonoverlapping subdomain partition, we will prove that

C₀²≤C

1 +H

δ 1 +H^2−dmax

Fij

|θ^c_Fij|²_H¹_(Ω)

, whereθ_F^c_ij is a linear conforming face function with the boundary values

(4.2) θ_F^c_ij(x) =

( 1, x∈F_ij^h 0, x∈Γ^h\F_ij^h,

and which minimizes theH¹-seminorm on a spaceVh. HereVhis the space of linear con- forming finite element functions on the given initial triangulationTu. We note thatθ^c_Fij is needed only for the theory.

We recall the following properties forbh(V, v)andb^∗_h(v,V)(see [12, 13]):

|bh(V, v)| ≤ kVk_Z^′kvkX, (4.3)

|bh(V, v)| ≤ kVk_X^′kvkZ, , (4.4)

and

(4.5) inf

V∈Vh

sup

v∈Sh

b^∗_h(v,V) kvkXkVk_Z^′ ≥β,

(4.6) inf

v∈Sh

sup

V∈Vh

bh(V, v) kVk_X^′kvkZ

≥β,

whereβis a positive constant independent ofhandH. We note that using (4.4) and (4.6), we obtain foru∈Sh

c(ρ)β²kuk²_Z ≤a(u, u)≤C(ρ)kuk²_Z,

wherec(ρ) andC(ρ) are positive constants depending on ρ(x). Similarly, we obtain for ui∈Sh,i

(4.7) cβ²ρikuik²_Zi≤ai(ui, ui)≤Cρikuik²_Zi, where

kuik²_Zi:=

Z

Ωi

|∇ui|²dx+ X

κ∈FpT Ωi

h⁻¹_κ Z

κ

[ui]²ds

andc andC are positive constants, which do not depend onρ(x). Here we assume that ρ(x) =ρiforxinΩiwhereρiis a positive constant.

Foru∈H₀¹(Ω), we have

kukZ =|u|_H¹_(Ω),

and for allu∈H₀¹(Ω)

(4.8) c(ρ)β|u|²_H¹_(Ω)≤a(u, u)≤C(ρ)|u|²_H¹_(Ω). We list some auxiliary results which will be useful in our analysis.

• Poincar´e(-Friedrichs) inequalities (Brenner [8]) (4.9) kvk²_L2(Ω)≤C X

τ∈T

|v|²_H1(τ)+X

κ∈F

h⁻¹_κ Z

κ

[v]²ds+ Z

Ω

v ds 2!

.

(4.10) kvk²_L2(Ω)≤C X

τ∈T

|v|²_H1(τ)+X

κ∈F

h⁻¹_κ Z

κ

[v]²ds+ Z

Γ

v ds 2!

.

• Trace inequality (Feng and Karakashian [22, Lemma 3.6]) (4.11)

kvk²_L²_(∂Ω)≤C H_Ω⁻¹kvk²_L²_(Ω)+HΩ

X

τ∈T

|v|²_H¹_(τ)+X

κ∈F

h⁻¹_κ Z

κ

[v]²ds

!!

.

LetΩδ be the thin layer ofΩwhich consists ofx∈Ωsuch that dist(x, ∂Ω)≤δ.

• Generalized Poincar´e inequality (Feng and Karakashian [22, Lemma 3.7]) (4.12)

kvk_L²_(Ωδ)≤Cδ H_Ω⁻¹kvk²_L²_(Ω)+HΩ

X

τ∈T

|v|²_H¹_(τ)+X

κ∈F

h⁻¹_κ Z

κ

[v]²ds

!!

.

We note that these results hold for any piecewise polynomial functionugiven in terms of a partitionT withF, the set of all interior faces (edges) inT. In our case,Fis the union ofFpandF_u⁰.Γis a measurable subset of∂Ωwith a positive(d−1)-dimensional measure.

HΩandhκdenote the diameter of the domainΩandκ, respectively.

The inequalities in (4.9) and (4.10) hold for anyΩwhich satisfy the standard Poincar´e(- Friedrichs) inequalities. The inequalities in (4.11) and (4.12) hold for any bounded polyhedral domain which is starlike. The constantCin (4.11) depends on the constantcappearing in (4.1), the definition of the starlike property. We note thatΩneed not be convex. The result in (4.12) is a general version of Lemma 3.10 in [29]. In our theory, these results will be applied to each subdomainΩi.

For a given functionu∈Sh, we consider

(4.13) u0(x) =X

ij

uFijθ^(k)_Fij(x),

whereuFij is the average ofuoverFij, i.e.,

(4.14) uFij =

R

Fiju(x(s))ds R

Fij1ds .

We note thatθ^(k)_Fij(x(s)) = 1onFij, while the coarse basis functionθ^c_Fij(x)of the standard conforming finite elements vanishes at the boundary of the face, and thatθ^(k)_Fij satisfies

X

Fij⊂∂Ωi

θ^(k)_Fij(x) = 1for allx∈Ωi.

LetI^hbe an interpolant ofv∈ H¹(T), which is a space of piecewiseH¹-functions in T, toShwhich satisfies

Z

κ

(I^hv−v)q ds= 0, ∀q∈P^k(κ), ∀κ∈ F_u⁰, Z

τ

(I^hv−v)q dx= 0, ∀q∈P^k−1(τ), ∀τ ∈ T. (4.15)

We note thatI^hvsatisfies, see [13],

(4.16) |I^hv|H¹(τ)≤C|v|H¹(τ), ∀v∈H¹(τ).

We prove the following lemmas, which will be used in our analysis.

LEMMA4.2. Forv∈H¹(T), we have

kI^hvkZ ≤CkvkZ.

Proof. We will show that

kI^hv−vkZ ≤CkvkZ.

By the definition ofk · kZ-norm and the inequality (4.16), it suffices to prove that X

κ∈Fp

h⁻¹_κ Z

κ

[I^hv−v]²ds≤Ckvk²_Z.

For a givenκ∈ Fp, letτ1andτ2be the two (sub-)tetrahedra which shareκ.Letwi be the restriction ofI^hv−vtoτifori= 1,2. By a trace inequality, we obtain

Z

κ

[I^hv−v]²ds≤C X

i=1,2

(h⁻¹_τi kwik²_L²_(τi)+hτi|wi|²_H¹_(τi)).

From (4.15),wihas a zero average over any faceκu∈ F_u⁰ofτiand by applying a Poincar´e(- Friedrichs) inequality

kwik²_L²_(τi)≤Ch²_τi|wi|²_H¹_(τi), and the following bound is obtained

Z

κ

[I^hw−v]²ds≤C X

i=1,2

hτi|I^hv−v|²_H¹_(τi).

Combining the above inequality with (4.16), we obtain X

κ∈Fp

h⁻¹_κ Z

κ

[I^hv−v]²ds≤CX

τ∈T

|v|²_H¹_(τ).

LEMMA 4.3. With the assumption that the number of tetrahedra along each edge ofΩi

is proportional to(H/h)^d−2, the coarse basis function satisfies,

kθ^(k)_Fijk²_Z ≤C(H^d−2+|θ_F^c_ij|²_H¹_(Ω)),

for allk≥0, whereθ_F^c_ij(x)is the standard linear conforming face coarse basis function.

Proof. We assumek≥1and later extend the result to the casek= 0. LetVij be the set of all triangles in the initial triangulationTuthat have a non-empty intersection with∂Fij. We defineθ˜^(k)_Fij(x)by

θ˜^(k)_Fij(x)|τ =

( θ_F^(k)_ij(x)|τ, τ ∈Vij

θ_F^c_ij(x)|τ, otherwise.

Since the number of triangles inVijis proportional toH/hand to a constant ford= 3and d= 2, respectively,

|θ˜^(k)_Fij −θ^c_Fij|²_H¹_(τ)≤Ch^d−2 for aτonVijandκinFpT

τand 1

hk[˜θ^(k)_Fij−θ^c_Fij]k²_L²_(κ)≤Ch^d−2, we obtain

kθ˜_F^(k)_ij−θ_F^c_ijk²_Z≤CH^d−2.

We note thatθ˜^(k)_Fij(x)has the same boundary data asθ^(k)_Fij(x). Using that|θ_F^c_ij|1,Ω=kθ^c_FijkZ

and thatθ_F^(k)_ij|Ωiminimizes the normai(·,·)^1/2, which is equivalent toρik · kZi, we obtain kθ^(k)_Fijk²_Z≤ kθ˜^(k)_Fijk²_Z

≤C

H^d−2+kθ^c_Fijk²_Z

=C

H^d−2+|θ_F^c_ij|²_1,Ω . (4.17)

Fork= 0, we considerI^hθ⁽¹⁾_Fij, whereI^his the interpolant from a piecewiseH¹-function inT toShwithk= 0. Using the stability of the interpolant of Lemma 4.2, we obtain

kθ_F⁽⁰⁾_ijk²_Z≤ kI^hθ_F⁽¹⁾_ijk²_Z≤Ckθ_F⁽¹⁾_ijk²_Z.

The above inequality combined with the result fork≥1shows that the result also holds for the casek= 0.

LEMMA4.4. With the assumption that the subdomainsΩisatisfy the Poincar´e inequality and starlike property, theu0in (4.13) satisfies

a(u0, u0)≤Ca(u, u) (1 +H^2−dmax

Fij

kθ^c_Fijk²_Z).

HereCdepends on the Poincar´e and the starlike parameters of the subdomains.

Proof. We consider

a(u−u0, u−u0) =X

i

ai((u−u0)|Ωi,(u−u0)|Ωi).

LetRibe the restriction toΩi. ThenRi(u−u0) = (u−u0)|Ωi. Each term above is bounded by

ai(Ri(u−u0), Ri(u−u0))≤2ai(Riu, Riu) + 2ai(Riu0, Riu0)

≤C



ai(Riu, Riu) + X

Fij⊂∂Ωi

u²_Fijai(RiθFij, RiθFij)





≤C



ai(Riu, Riu) + X

Fij⊂∂Ωi

u²_FijρikθFijk²_Z



. (4.18)

Here we use the inequalities in (4.7).

For the term,u²_Fij, we obtain by applying (4.11) toΩi

(4.19) Z

Fij

u²ds≤C



H



|u|²_H¹_(Ωi)+ X

κ∈ΩiTFp

h⁻¹_κ k[u]k²_L²_(κ)



+ 1

Hkuk²_L²_(Ωi)



.

Using the fact thatu−u0 is invariant to a shift by a constant and applying the Poincar´e- inequality (4.9) to the bound above , we obtain

(4.20) u²_Fij ≤CH^2−d



|u|²_H¹_(Ωi)+ X

κ∈ΩiT Fp

h⁻¹_κ k[u]k²_L²_(κ)



.

Combining (4.18) with (4.20), we get

ai(Ri(u−u0), Ri(u−u0))≤C



ai(Riu, Riu) + X

Fij⊂∂Ωi

ρikRiuk²_ZiH^2−dmax

Fij

kθFijk²_Z





and by the boundρikRiuk²_Zi≤Cai(Riu, Riu), see (4.7), we finally obtain (4.21) a(u0, u0)≤C

1 +H^2−dmax

Fij

kθFijk²_Z

a(u, u).

We now turn to the bounds for the local components. Let{θj}be a partition of unity provided for{Ω^′_j}and whereθj ∈R^T_jV_j^′with|∇θj| ≤C/δand letuj=I^h(θj(u−u0))∈ R_j^TV_j^′, whereI^hinterpolates intoShas defined in (4.15).

We obtain

THEOREM 4.5. Foru ∈ Sh, when subdomainsΩi satisfy Assumption 4.1 there is a partitionu=PN

j=0ujwhich satisfies a(u0, u0) +

N

X

j=1

aΩ^′_j(uj, uj)≤C

1 +H

δ 1 +H^2−dmax

Fij

|θ^c_Fij|²_H¹_(Ω)

a(u, u), whereCdepends on the Poincar´e and starlike parameters of the subdomains and the number of colorsNc, andθ^c_Fij(x)is the standard linear conforming coarse basis function defined in (4.2).

Proof. We letw=u−u0and then let

uj=I^h(θjw)∈Sh. We consider

a(uj, uj)≤CX

i

ρikujk²_Zi ≤CX

i

ρikθjwk²_Zi

≤CX

i

ρi





X

τ∈TT Ω^′_jT

Ωi

|θjw|²_H1(τ)+ X

κ∈FpT Ω^′_jT

Ωi

h⁻¹_κ Z

κ

[w]²ds





≤CX

i

ρi





X

τ∈TT Ω^′_jT

Ωi

k∇θjwk²_L2(τ)+ X

τ∈TT Ω^′_jT

Ωi

|w|²_H1(τ) (4.22)

+ X

κ∈FpT Ω^′_jT

Ωi

h⁻¹_κ Z

κ

[w]²ds



. We consider the first term in (4.22):

X

τ∈TT Ω^′_jT

Ωi

k∇θjwk²_L2(τ)≤C 1 δ²

X

τ∈TT Ω^′_j,δT

Ωi

kwk²_L²_(τ)=C 1

δ²kwk²_L²_(Ω′ j,δ

TΩi)

≤C 1 δ²δ



H_Ω⁻¹′

jkwk²_L²_(Ω′ j

TΩi)+HΩ^′_j



 X

τ∈Ω^′_jT Ωi

|w|H¹(τ)+ X

κ∈FpT Ω^′_jT

Ωi

h⁻¹ Z

κ

[w]²ds







. (4.23)

HereΩ^′_j,δis the union ofτ ∈ T where∇θjdoes not vanish, and the bound (4.12) is applied toΩ^′_j,δT

Ωi.

For the termkwk²_L2(Ω′ j

TΩi), we use the Poincar´e-Friedrichs inequality (4.10)

kwk²_L²_(Ωi)≤C(Ωi)



 X

τ∈Ωi

|w|²_H1(τ)+ X

κ∈ΩiT Fp

h⁻¹_κ k[w]k²_L2(κ)+ Z

Γ

w ds 2

. By choosing Γ = Fij, we haveR

Γw ds = 0, see (4.13) and (4.14), and from a scaling argument, we obtain

kwk²_L²_(Ωi)≤CH²



 X

τ∈Ωi

|w|²_H1(τ)+ X

κ∈ΩiTFp

h⁻¹k[w]k²_L2(κ)



.

Summing (4.23) overicombined with the above bound and assuming thatHΩ^′_j is com- parable to the diameterHof theΩi, which intersectsΩ^′_j, we obtain

(4.24) X

i

ρi

X

τ∈TT Ω^′_jT

Ωi

k∇θjwk²_L2(τ)≤CH δ

X

i,ΩiT Ω^′_j6=∅

ρi



 X

τ∈TT Ωi

|w|²_H1(τ)+ X

κ∈FpT Ωi

h⁻¹k[w]k²_L2(κ)



. Here we note that the sum on the right hand side runs over only the subdomainsΩi which

intersect the subregionΩ^′_j. Summing (4.22) overjcombined with (4.24), we finally obtain

(4.25) X

j

a(uj, uj)≤C

1 +H δ

X

i

ρikRiwk²_Zi ≤C

1 + H δ

a(w, w),

wherew=u−u0. The bound in Lemma 4.4 then completes the proof.

REMARK4.6. The above result holds for quite general subdomainsΩi, which satisfy the standard Poincar´e(-Friedrichs) inequalities and the starlike property, and has a number of tetrahedra across each edge proportional to(H/h)^d−2. The resulting bound depends on the energy of the linear conforming coarse basis function,θ_F^c_ij(x). In the standard case, when Ωiis tetrahedral (d= 3) and rectangular or triangular (d= 2), we have

|θ^c_Fij|²_H¹_(Ω)≤CH^d−2

1 + logH h

,

whereCis a positive constant independent of any mesh parameters. We also note that for John domainsΩiin two dimensions the above bound was proved in [25]. We refer [7, 23, 10]

for the definition of John domains. John domains satisfy Poincar´e inequalities but they do not in general have the starlike property. Instead of the trace inequality in (4.11), we can apply the Sobolev inequality

u²_Fij ≤max

x∈Ωi

|u(x)|²≤C

1 + logH h

kRiuk²_Zi to get the bound

a(u0, u0) +X

j

aΩ^′_j(uj, uj)≤

1 + logH h

2 1 + H

δ

for the two-dimensional case whenΩi are John domains, see [17]. Here one additional log factor comes from the Sobolev inequality. We refer to some recent works [30, 18] for theory of domain decomposition methods on quite general subdomains.

In the three dimensions, with an assumption thatΩiare Lipschitz, we obtain the follow- ing result:

LEMMA 4.7. For a Lipschitz Ωi in three dimensions, there exists a function θ^c_F ∈ Vh,iT

H¹(Ωi)with the bound

|θ^c_F(x)|²_H¹_(Ωi)≤CH

1 + logH h

.

Proof. LetV = {x ∈ Ωi : dist(x, F) ≤ sinαdist(x, ∂F)}. SinceΩiis a Lipschitz domain, we may selectαso thatF2:=∂V \Fdoes not touch∂Ωi.

Forx∈V,we define

d(x) = d2(x) d1(x) +d2(x),

whered1(x) = dist(x, F)andd2(x) = dist(x, F2), and where we extendd(x)by zero for x∈Ωi\V.We note that the construction of such a functiond(x)was first given by Dohrmann in [18]. Letd∂F(x) =dist(x, ∂F). We will show that forx∈V, there existsc >0such that

d1(x) +d2(x)≥cd∂F(x).

Forx ∈ V, letx1 and x2 be points on F andF2 such that d1(x) = |x−x1| and d2(x) =|x−x2|. Letax2be points on∂F such thatd∂F(x2) =|x2−ax2|. We then have (4.26) d∂F(x)≤ |x−ax2| ≤ |x−x2|+|x2−ax2|.

Sincex2∈F2, we have

|x2−ax2|=d∂F(x2) = 1

sinαd1(x2) and by usingd1(x2)≤ |x2−x1|,we obtain

|x2−ax2| ≤ 1

sinα(|x2−x|+|x1−x|) and from the bound in (4.26) combined with the above, we prove that

(4.27) d∂F(x)≤

1 + 1

sinα

(d1(x) +d2(x)).

We interpolated(x)to the finite element spaceVh,iT

H¹(Ωi)and obtainθ_F^c(x). We note thatθ_F^c(x)vanishes on the boundary of F. The functionθ^c_F(x)satisfies the required boundary condition, i.e., it has value one in the interior of F and zero at the rest of the boundary ofΩi. We will prove that

|θ_F^c|²_H¹_(Ωi)≤CH(1 + log(H/h)).

By the construction, it suffices to consider all tetrahedra coveringV. For each tetrahedron τtouching the boundary ofF, we have

|θ^c_F|²_H¹_(τ)≤Ch,

and using that the number of such tetrahedra isO(H/h),we obtain

(4.28) X

τT

∂F6=∅

|θ^c_F|²_H¹_(τ)≤CH.