A Bddc Algorithm for the Stokes Problem with Weak 2 Galerkin Discretizations

1 A BDDC ALGORITHM FOR THE STOKES PROBLEM WITH WEAK 2 GALERKIN DISCRETIZATIONS

3 XUEMIN TU∗ AND BIN WANG†

4 Abstract. The BDDC (balancing domain decomposition by constraints) methods have been 5 applied successfully to solve the large sparse linear algebraic systems arising from conforming finite 6 element discretizations of second order elliptic and Stokes problems. In this paper, the Stokes 7 equations are discretized using the weak Galerkin method, a newly developed nonconforming finite 8 element method. A BDDC algorithm is designed to solve the linear system such obtained. Edge/face 9 velocity interface average and mean subdomain pressure are selected for the coarse problem. The 10 condition number bounds of the BDDC preconditioned operator are analyzed, and the same rate 11 of convergence is obtained as for conforming finite element methods. Numerical experiments are 12 conducted to verify the theoretical results.

13 Key words. Discontinuous Galerkin, HDG, weak Galerkin, domain decomposition, BDDC, 14 Stokes, Saddle point problems, benign subspace

15 AMS subject classiﬁcations. 65F10, 65N30, 65N55

16 1. Introduction. Numerical solution of saddle point problems using non over- 17 lapping domain decomposition methods have long been an active area of research; see, 18 e.g., [28, 15, 11, 10, 18, 29, 30, 16, 33, 17, 34, 27]. The Balancing Domain Decomposi- 19 tion by Constraints (BDDC) algorithm is an advanced variant of the non-overlapping 20 domain decomposition technique. It was first introduced by Dohrmann [5], and the 21 theoretical analysis was later given by Mandel and Dohrmann [20]. In this theoretical 22 development, optimal condition number bound was obtained for the BBDC opera- 23 tors proposed for symmetric positive definite systems. Nonetheless, the variational 24 form of the incompressible Stokes problem is a saddle point problem [3], and the dis- 25 cretization by finite element methods lead to symmetric indefinite matrices. Thus, 26 the conventional theory usually fails to apply. In the first attempt to apply BDDC to 27 the incompressible Stokes problem by Li and Widlund [18], the approach via benign 28 spaces was used to reduce the Stokes system to a symmetric positive definite problem, 29 and optimal convergence result was obtained as for the elliptic case. However, this 30 method was proposed and analyzed with discontinuous pressure approximation, and 31 there is a big class of mixed finite element spaces featuring continuous pressure, e.g., 32 the Taylor-Hood finite elements. Later, Li and Tu proposed a class of non-overlapping 33 domain decomposition algorithms for continuous finite element pressure space, which 34 were proved and numerically verified to be scalable [16, 33, 17, 34]. Earlier, S´ısteketˇ 35 al. applied a parallel BDDC pre-conditioner based on the corner constraints to the 36 Stokes flow using Taylor-Hood finite element [41]. They numerically demonstrated the 37 promising speedup property of their BDDC pre-conditioner as applied to benchmark 38 test problems of real-life relevance, even though optimal scalability was not achieved. 39 As the property of the discretized system to be solved is dependent on the nu- 40 merical methods used, the BDDC algorithms have been extended to the second or- 41 der elliptic problem with mixed and hybrid formulations, hybridizable discontinuous

∗Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045- 7594, U.S.A, E-mail: [email protected] This work was supported in part by the NSF under Contracts DMS-1419069 and DMS-1723066. †Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045- 7594, U.S.A, E-mail: [email protected] This work was supported in part by the NSF under Contract DMS-1419069. 1

This manuscript is for review purposes only. 42 Galerkin (HDG) methods [29, 30, 35]. In this study, we design BDDC pre-conditioners 43 for trending non-conforming finite element methods, in particular, the weak Galerkin 44 (WG) methods. The WG methods are a class of nonconforming finite element meth- 45 ods, which were first introduced for second order elliptic problems by Wang and Ye 46 [36]. The idea of WG is to introduce weak functions and their weak derivatives as 47 distributions, which can be approximated by polynomials of different degrees on dif- 48 ferent support. For example, for second order elliptic problems, weak functions have 49 the form of v = {v0, vb}, where v0 is defined inside each element and vb is defined on 50 the boundary of the element. v0 and vb can both be approximated by polynomials. 51 The gradient operator is approximated by a weak gradient operator, which is further 52 approximated by polynomials. These weakly defined functions and derivatives make 53 the WG methods highly flexible in terms of approximating functions and finite ele- 54 ment partition of the domain. The same weak concepts have been extended to other 55 differential operators such as divergence and curl, which appears in applications like 56 Stokes [38] and Maxwell [25] equations respectively. 57 As most finite element methods, the WG methods result in a large number of 58 degrees of freedom and therefore require solving large linear systems with condition 59 number deteriorating with the refinement of the mesh. Efficient fast solvers for the 60 resulting linear system are necessary. However, relatively few attempts on designing 61 fast solvers for the WG methods can be found in the literature; see [4]. An effective 62 implementation of WG methods is to reduce the unknown variables to those associated 63 with element boundaries through a Schur-complement approach. It can be further 64 reduced to the subdomain interface. The subdomain interface problem can then be 65 solved using the conjugate gradient method preconditioned with a BDDC algorithm. 66 It is necessary to impose edge or face average constraints across the interface. By 67 a change of variable [19, 14], the primal constraint on edge or face average can be 68 converted to an explicit variable. The reduced system for the primal variables will 69 be the coarse problem to solve. The BDDC preconditioner can be built based on 70 such designed coarse problem, and thus be used as a preconditioner for the conjugate 71 gradient method. 72 In a recent study [35], the authors proved the condition number bound of the 73 BDDC preconditioned operator arising from elliptic problems with hybridizable dis- 74 continuous Galerkin (HDG) discretizations. In this paper, a BDDC algorithm is fur- 75 ther developed for weak Galerkin discretization with reduced polynomial basis func- 76 tions. As in [35], we first establish the connection between the hybridized Raviart- 77 Thomas(RT) method and the WG discretization and obtain the condition number 78 estimate of the BDDC algorithm applied to the elliptic problem with the WG dis- 79 cretization. We then consider the BDDC algorithms for the saddle point problem 80 arising from the WG discretization for the incompressible Stokes problem. In [26], a 81 similar saddle point problem is obtained by the HDG discretization for incompressible 82 Stokes flow, where the resulting system is solved by an augmented Lagrange approach. 83 An additional time dependent problem is introduced and solved by a backward-Euler 84 method. Here, we solve the saddle point problem from WG discretization directly us- 85 ing the BDDC methods. To the best of our knowledge, this is the first attempt for fast 86 solvers applied to the Stokes problems with this type nonconforming discretization. 87 There are many works on preconditioning the saddle point problems resulting from 88 mixed finite element discretizations, such as [1, 2]. In those works, the original saddle 89 point problems are reformulated to positive definite problems under specially defined 90 inner products. In this paper, a benign subspace idea is used as in [18, 29, 40]. In the 91 benign subspace approach, the positive definite system is obtained by carefully choos- 2

This manuscript is for review purposes only. 92 ing the coarse components in the BDDC preconditioner. Therefore, the formulation, 93 implementation, and rate of convergence of the BDDC algorithm for Stokes are very 94 similar to those for the positive definite systems arising from the elliptic problems. 95 We prove that the condition number bound for the Stokes problem with the WG 96 discretization is as good as for the conforming discretization. We note that the WG 97 discretization has been extended to polytopal meshes, [37, 24, 23, 38]. With the 98 development of the domain decomposition methods for irregular subdomain shapes, 99 [6, 13, 7, 39, 8, 9], we believe that the BDDC algorithms proposed in this paper can 100 be extended to polytopal meshes as well. But we will restrict ourselves to the stan- 101 dard finite element triangulation here and leave the complete analysis and numerical 102 verification for more general polytopal meshes in the future study. 103 The rest of the paper is organized as follows. In Section 2, we introduce some 104 notations for relevant Hilbert spaces. In Section 3, we introduce the Stokes problem 105 and its weak Galerkin discretization. In Section 4, we reduce the linear system to 106 an interface problem. Then, we introduce the BDDC preconditioner for the interface 107 problem in Section 5, and give some auxiliary results in Section 6. In Section 7, we 108 provide an estimate for the condition number of the BDDC preconditioned system. 109 Finally, results from numerical experiments are presented and discussed in Section 8.

110 2. Notations for some relevant Spaces. Let Ω ∈ Rd (d = 2, 3) be a bounded 111 open set with Lipschitz continuous boundary. The Sobolev space Hk (Ω) for any 112 integer k ≥ 1 is a Hilbert space with inner product

X α α 113 (u, v)Hk(Ω) = (D u, D v)L2(Ω) , |α|≤k

114 where the multi-index notation for derivatives ∂|α|u 115 Dαu = ∂α1 ··· ∂αn u = x1 xn α α ∂ 1 x1 ··· ∂ n xn

116 with α = (α1, ··· , αn) and |α| = α1 + ··· + αn. Correspondingly, we can deﬁne the 117 induced norm k·kHk(Ω) Z 2 X α 2 118 kukHk(Ω) = (u, u)Hk(Ω) = |D u| dx, |α|≤k Ω

119 and a semi-norm Z 2 X α 2 120 |u|Hk(Ω) = |D u| dx. |α|=k Ω

121 The space H0 (Ω) coincides with L2 (Ω), which is the space of square integrable 122 functions on Ω, i.e., Z 2 123 L2 (Ω) = u : |u| dx < ∞ . Ω

124 The inner product and induced norm of L2 (Ω) are given by: Z Z 2 2 125 (u, v)L2(Ω) = uvdx; kukL2(Ω) = |u| dx. Ω Ω 3

This manuscript is for review purposes only. 126 We deﬁne the subspace of L2 (Ω) with zero mean to be Z 2 2 127 L0 (Ω) = u : u ∈ L (Ω) , udx = 0 . Ω

128 Let HΩ be the diameter of Ω. We have the following scaled norm for the Sobolev 129 space H1(Ω): Z 2 1 2 2 1 2 2 130 kuks,H1(Ω) = 2 kukL2(Ω) + |u|H1(Ω) = 2 kukL2(Ω) + |∇u| dx, HΩ HΩ Ω

131 with ∂ ∂ 132 ∇ = grad = , ··· , . ∂x1 ∂xn

133 The subspace of H1 (Ω) with vanishing boundary values is denoted by

1 1 134 H0 (Ω) = v ∈ H (Ω) : v = 0 on ∂Ω .

135 The spaces introduced above can be extended to spaces of vector-valued functions in 136 a straightforward way. 137 Also, we recall that the space H (div; Ω) is deﬁned as the set of vector-valued 138 functions on Ω such that both the functions and their divergence are square integrable; 139 i.e.,

n d o 140 H (div; Ω) = v : v ∈ L2 (Ω) , ∇ · v ∈ L2 (Ω) .

141 The scaled norm in H (div; Ω) is deﬁned by

2 1 2 2 142 kvk = kvk + k∇ · vk . s,H(div;Ω) 2 L2(Ω) L2(Ω) HΩ

143 3. A Stokes problem and its weak Galerkin Discretization. We con- 144 sider the primary velocity-pressure formulation for the Stokes problem on a bounded 145 polygonal domain Ω, in two dimensions (d = 2), or three dimensions (d = 3), with a 146 Dirichlet boundary condition:

 −4u + ∇p = f in Ω,  147 (3.1) ∇ · u = 0 in Ω,  u = g on ∂Ω,

d d 148 where f ∈ L2 (Ω) , and g ∈ H1/2 (∂Ω) . Without loss of generality, we assume 149 that g = 0. The weak form in the primary velocity-pressure formulation for the Stokes 1 d 2 150 problem seeks u ∈ H0 (Ω) and p ∈ L0 (Ω) such that

( d (∇u, ∇v) − (∇ · v, p) = (f, v) ∀v ∈ H1 (Ω) , 151 (3.2) 0 2 (∇ · u, q) = 0 ∀q ∈ L0 (Ω) .

152 The idea of weak Galerkin finite element scheme [38] is to substitute the standard 153 function and differential operators with the weakly defined counterparts. A weak 4

This manuscript is for review purposes only. 2 154 function over the domain D is defined as v = {v0, vb} such that v0 ∈ L (D) and 2 155 vb ∈ L (∂D). The v0 part represents the value of v in the interior of D, while the 156 vb part represents the value of v on the boundary of D. Note that vb does not bind 157 itself with v0 from the definition. In essence, weak functions relax the continuity 158 property of the standard functions, thus to offer more flexibility in terms of variable 159 representation. Following the notation in [38], we denote by V (D) the space of weak 160 functions over the domain D

2 2 161 V (D) = v = {v0, vb} : v0 ∈ L (D) , vb ∈ L (∂D) ,

162 and the relevant vector-valued weak function space by

d n 2 d 2 do 163 [V (D)] = v = {v0, vb} : v0 ∈ L (D) , vb ∈ L (∂D) ,

164 and

d n 2 d 2 o 165 [V (D)] = v = {v0, vb} : v0 ∈ L (D) , vb · n ∈ L (∂D) .

166 The space of weak gradient or divergence operators will be defined as the dual d 167 space of appropriate Hilbert space, in similar manner as the dual of L2 (D) can 168 be identified with itself by using the L2 inner product as the action of the linear 169 functionals. The following two definitions are [38, Definitions 2.1 and 2.3]: d 170 Definition 1. For any v ∈ [V (D)] , the weak gradient of v is defined as the 1 d 171 linear functional ∇wv in the dual space of [H (D)] whose action on each q ∈ 172 [H1(D)]d×d is given by

173 (∇wv, q)D = − (v0, ∇ · q)D + hvb, q · ni∂D ,

174 where n is the outward normal direction to ∂D. d 175 Definition 2. For any v ∈ [V (D)] , the weak divergence of v is deﬁned as the 1 1 176 linear functional ∇w · v in the dual space of H (D) whose action on each ϕ ∈ H (D) 177 is given by

178 (∇w · v, ϕ)D = − (v0, ∇ϕ)D + hvb · n, ϕi∂D ,

179 where n is the outward normal direction to ∂D. 180 Now, we introduce the weak Galerkin finite element discretization for (3.1) as in 181 [38]. First, we introduce the mesh of the domain, then we will define discontinuous 182 weak Galerkin finite element spaces over the mesh. Let Th be a shape-regular and 183 quasi-uniform triangulation of Ω, and the element in Th denoted by K. For any

184 K ∈ Th, we denote by hK the diameter of K with h = maxK∈Th hK . Define Fh be the i ∂ 185 set of edges/faces of elements K ∈ Th. Fh and Fh are subsets of Fh, which consists of 186 domain interior and boundary edges, respectively. For any domain D, let Pk (D) be 187 the space of polynomials of degree ≤ k on D. Define the weak Galerkin finite element 188 spaces for the velocity variable associated with Th as follows:

n d d o 189 Vk = v = {v0, vb} : {v0, vb} |K ∈ [Pk (K)] × [Pk−1 (e)] , ∀K ∈ Th, e ⊂ ∂K .

190 Note that a function v ∈ Vk has a single value vb on each edge e ∈ Fh. The 191 subspace of Vk with vanishing boundary values on ∂Ω is denoted by 5

This manuscript is for review purposes only. 0 192 Vk = {v = {v0, vb} ∈ Vk : vb = 0 on ∂Ω} .

193 We denote a relevant matrix polynomial function space by

n d×d o 194 Qk−1 = v : v|K ∈ [Pk−1 (K)] , ∀K ∈ Th .

195 For the pressure variable, deﬁne the following ﬁnite element space

2 196 Wk−1 = q : q ∈ L0 (Ω) , q|K ∈ Pk−1 (K) .

197 Denote the discrete weak gradient operator by ∇w,k−1, and the discrete weak 198 divergence operator by (∇w,k−1·), respectively. On the ﬁnite element space Vk, they 199 are deﬁned as follows: for v = {v0, vb} ∈ Vk, on each element K ∈ Th, ∇w,k−1v |K ∈ d×d 200 [Pk−1 (K)] and ∇w,k−1 · v |K ∈ Pk−1 (K) are the unique solutions of the following 201 equations, respectively,

d×d 202 (∇w,k−1v |K , q)K = − (v0,K , ∇ · q)K + hvb,K , q · ni∂K , ∀q ∈ [Pk−1 (K)] ,

203 (∇w,k−1 · v |K , ϕ)K = − (v0,K , ∇ϕ)K + hvb,K · n, ϕi∂K , ∀ϕ ∈ Pk−1 (K) ,

204 where v0,K and vb,K are the restrictions of v0 and vb to K, respectively, (u, w) = R R K 205 K uwdx, and hu, wi∂K = ∂K uwds. To simplify the notation, we shall drop the 206 subscript k − 1 in the notation ∇w,k−1 and (∇w,k−1·) for the discrete weak gradi- 207 ent and the discrete weak divergence operators. We denote the L2 inner product 208 over the triangulation as a summation over each element of the triangulation, for X X 209 example,(∇ u, ∇ w) = (∇ u, ∇ w) ,(∇ · v, q) = (∇ · v, q) . w w Th w w K w Th w K K∈Th K∈Th 2 2 d d 2 210 Let Q0 be the L projection from L (K) onto [Pk (K)] , and Qb be the L 2 d d 211 projection from L (e) onto [Pk−1 (e)] , for e ∈ Fh. We write the corresponding 212 projection operator for the weak function as Qh = {Q0,Qb}. Next, we deﬁne three 213 bilinear forms as below

214 (3.3) s (v, w) = P h−1 hQ v − v ,Q w − w i , K∈Th K b 0 b b 0 b ∂K a (v, w) = (∇ v, ∇ w) + s (v, w) , w w Th b (v, q) = (∇ · v, q) . w Th

215 The discrete problem resulting from the WG discretization can then be written 0 216 as: ﬁnd uh = {u0, ub} ∈ Vk and ph ∈ Wk−1 such that

( 0 a (uh, v) − b (v, ph) = (f, v0) , ∀v = {v0, vb} ∈ V , 217 k b (uh, q) = 0, ∀q ∈ Wk−1.

0 0 0 218 We introduce the following operators: A : Vk → Vk , B : Vk → Wk−1, by

219 (3.4) (Auh, v) = a (uh, v) , (Buh, q) = −b (uh, q) .

220 Using these operators, the matrix form of the weak Galerkin scheme can be rep- 221 resented as 6

This manuscript is for review purposes only. T AB uh f 222 = . B 0 ph 0

223 At element level, for each K, given the edge component vb of the velocity and 224 the pressure p, the interior component v0 of the velocity can be uniquely determined. 225 Namely, v0 can be eliminated in each element independently. We thus obtain the 226 reduced system of vb and p only with considerable smaller size but diﬀerent sparsity 227 pattern (denser than the full system) as below

T Auu B uh,b fu 228 pu = b . Bpu Cpp ph fp

229 Throughout the rest of the paper, we will work with the reduced system such 230 obtained.

231 4. Reduced Subdomain Interface Problem. We decompose Ω into N non 232 overlapping subdomain Ωi with diameters Hi, i = 1,. . . , N, and set H = maxi Hi . 233 We assume that each subdomain is a union of shape-regular coarse triangles and that 234 the number of such elements forming an individual sudomain is uniformly bounded. 235 We deﬁne edges/faces as open sets shared by two subdomains. Two nodes belong to 236 the same face when they are associated with the same pair of subdomains. Let Γ be 237 the interface between the subdomains. The set of the interface nodes Γh is deﬁned as 238 Γh := (∪i6=j ∂Ωi,h ∩ ∂Ωj,h) \ ∂Ωh, where ∂Ωi,h is the set of nodes on ∂Ωi and ∂Ωh is 239 that of ∂Ω. We assume the triangulation of each subdomain is quasi-uniform. 240 We decompose the discrete velocity and pressure spaces Vk and Wk−1 into:

241 Vk = VI ⊕ VbΓ,Wk−1 = WI ⊕ W0.

(i) 242 Here, VI and WI are products of subdomain interior velocity spaces VI and subdo- (i) 243 main interior pressure spaces WI , respectively; i.e.,

N N Y (i) Y (i) 244 VI = VI ,WI = WI . i=1 i=1

(i) 245 The elements of VI are supported in the subdomain Ωi and vanishes on its (i) 246 interface Γi, while the elements of WI are restrictions of the pressure variables to R (i) 247 Ωi which satisfy p = 0. VΓ is the subspace of edge functions on Γ , and W0 Ωi I b (i) 248 is the subspace of W with constant values p0 in the subdomain Ωi that satisfy PN (i) 249 i=1 p0 m (Ωi) = 0, where m (Ωi) is the measure of the subdomain Ωi. 250 We denote the space of interface edge velocity variables of the subdomain Ωi by (i) QN (i) 251 VΓ , and the associated product space by VΓ = i=1 VΓ ; generally edge functions (i) 252 in VΓ are discontinuous across the interface. We deﬁne the restriction operators RΓ : (i) 253 VbΓ → VΓ to be an operator which maps functions in the continuous global interface (i) 254 edge variable space VbΓ to the subdomain component space VΓ . Also, RΓ : VbΓ → VΓ (i) 255 is the direct sum of RΓ . We denote the spaces of the right-hand-side interior load 256 vectors fI and interface load vectors fΓ by FI and FΓ, respectively. Similar notation (i) (i) 257 conventions apply to the spaces FeΓ, FbΓ, FbΠ, F∆ , FΓ , and F0. We will use them 258 throughout this paper without further explanation. 7

This manuscript is for review purposes only. 259 With the decomposition of the solution space, the global Stokes problem can be 260 written as follows: ﬁnd (uI , pI , uΓ, p0) ∈ VI ,WI , VbΓ,W0 such that

 T T      AII BII AbΓI 0 uI fI  T  BII CII Bb 0 pI  fpI  261 (4.1)  ΓI  = .  T  u   f  AbΓI BbΓI AbΓΓ Bb0Γ  Γ  Γ  p 0 0 0 Bb0Γ 0 0

262 The lower left block in (4.1) is zero, because the bilinear form b(uh, ϕ) does not 0 263 explicitly relate to uI and pI for any uh ∈ Vk and ϕ ∈ W0. The leading two-by-two 264 block of the matrix above can be rewritten into a block diagonal form with each block 265 corresponding to an independent subdomain problem. And the global problem can 266 be assembled from the subdomain problems, deﬁned as below

 (i) (i)T (i)T   (i)  (i) AII BII AbΓI 0 uI fI (i) (i) (i)T (i) B C Bb 0  p  f (i) 267 (4.2)  II II ΓI   I  =  pI  .  (i) (i) (i) (i)T   (i)  (i) AbΓI BbΓI AbΓΓ Bb0Γ  uΓ  fΓ  (i) (i) 0 0 Bb0Γ 0 p0 0

(i) (i) 268 We can eliminate the subdomain interior variables uI and pI in each subdo- 269 main independently, and assemble the global interface problem from the subdomain 270 interface problems. 271 Definition 3. (Schur complement of the Stokes problem) Deﬁne the subdomain (i) (i) (i) 272 Schur complement SΓ for the Stokes problem as follows: given uΓ ∈ VΓ , determine (i) (i) (i) 273 SΓ uΓ ∈ FΓ such that

 (i) (i)T (i)T   (i)   AII BII AΓI uI 0 274 (4.3)  (i) (i) (i)T   (i) = 0 . BII CII BΓI  pI    (i) (i) (i) (i) S(i)u(i) AΓI BΓI AΓΓ uΓ Γ Γ 275 The global interface problem can then be written as: to ﬁnd (uΓ, p0) ∈ VbΓ,W0 , 276 such that

uΓ gΓ 277 (4.4) Sb = . p0 0

Here the global interface matrix Sb is deﬁned as

" T # SbΓ Bb Sb = 0Γ , Bb0Γ 0

PN (i)T (i) (i) PN (i) (i) 278 where SbΓ = i=1 RΓ SΓ RΓ , Bb0Γ = i=1 B0Γ RΓ , and  −1  " (i) (i)T # " (i) # N (i)T  (i) h i A B f  279 P (i) (i)T II II I gΓ = i=1RΓ fΓ − AΓI BIΓ (i) (i) (i) .  BII CII fpI  8

This manuscript is for review purposes only. Fig. 1. Left: VbΓ for the subdomains Ω1 and Ω2; Right: VeΓ for the subdomains Ω1 and Ω2, where the midpoint of each edge is chosen as the primal unknown. The red dots are the primal variable VΠ and the black dots are the dual variable V∆.

VbΓ VeΓ

V∆

Ω1 Ω2 Ω1 Ω2 VΠ

280 The operator SbΓ is symmetric positive definite but Sb is symmetric indefinite. In 281 what follows, we will propose a BDDC preconditioner, and show that the precondi- 282 tioned operator is positive definite when restricted to a proper subspace. A precon- 283 ditioned conjugate gradient method can then be used to solve the global interface 284 problem.

285 5. The BDDC Preconditioner. The BDDC (Balancing Domain Decomposi- 286 tion by Constraints) algorithm is a variant of the two-level Neumann-Neumann type 287 preconditioner. It was introduced and analyzed by Dohrmann, Mandel, and Tezaur 288 [5, 20, 21] for standard ﬁnite element discretization of elliptic problems. The BBDC 289 preconditioner consists of local solvers for the subdomain problems and the artisti- 290 cally designed global coarse-level problem. The coarse level problem is assembled 291 from primal variables, such as edge/face averages across the subdomain interface on 292 which the continuity constraints are enforced. In contrast to earlier versions of balanc- 293 ing Neumann-Neumann methods, the BDDC methods do not need to solve singular 294 systems and the algorithms demonstrate good scalability for parallel computation. 295 In order to introduce the BDDC preconditioner, we ﬁrst introduce a partially 296 assembled interface space VeΓ by

N ! Y (i) 297 VeΓ = VbΠ ⊕ V∆ = VbΠ ⊕ V∆ . i=1

298 Here, VbΠ is the continuous, coarse level, primal interface edge velocity space. The 299 variables in this space are called the primal unknowns, and each primal unknown is 300 shared by the adjacent subdomains. The remaining interface velocity variables live (i) 301 in the complimentary dual space V∆ . This space is the direct sum of the V∆ , which 302 are spanned by basis functions with vanishing value at the primal degrees of freedom. 303 The functions in V∆ are generally discontinuous, see Figure 1. Thus, in the space 304 VeΓ, we relax the continuity constraints across the interface at the dual variables but 305 retain the continuity at the primal variables, which makes all the component linear 306 systems in the preconditioner nonsingular. 307 308 We need to introduce several restriction, extension, and scaling operators between (i) (i) 309 diﬀerent spaces. RΓ : VeΓ → VΓ restricts functions in the space VeΓ to the components (i) (i) (i) (i) 310 VΓ of the subdomain Ωi. RΓ : VeΓ → VΓ is the direct sum of RΓ . R∆ : VbΓ → V∆ (i) 311 maps the functions from VbΓ to V∆ , its dual subdomain components. RΓΠ : VbΓ → VbΠ 312 is a restriction operator from VbΓ to its subspace VbΠ. ReΓ : VbΓ → VeΓ is the direct sum 9

This manuscript is for review purposes only. (i) † 313 of RΓΠ and R∆ . We deﬁne the positive scaling factor δi (x) as follows:

† 1 314 δi (x) = , x ∈ ∂Ωi,h ∩ Γh, card (Ix)

315 where Ix is the set of indices of the subdomains that have x on their boundaries, and 316 card (Ix) counts the number of the subdomain boundaries to which x belongs. It is † † 317 clear that δ (x)’s provide a partition of unity, i.e., P δ (x) = 1, for any x ∈ Γ . i i∈Ix i h † (i) 318 We note that δi (x) is constant on each edge. Multiplying each row of R∆ with the (i) 319 scaling factor gives us RD,∆. The scaled operators ReD,Γ is the direct sum of RΓΠ and (i) 320 RD,∆. 321 The partially assembled Schur complement SeΓ, deﬁned on the interface velocity 322 space VeΓ, can be represented as follows: given uΓ ∈ VeΓ, SeΓuΓ ∈ FeΓ satisﬁes

 (1) (1)T (1)T (1)T   (1)  0  A B A ··· A u II II ∆I eΠI I 0 B(1) C(1) B(1)T ··· B(1)T   (1)    II II ∆I eΠI  pI   (1)  (1) (1) (1) (1)T   (1)  SeΓuΓ  323 A∆I B∆I A∆∆ ··· AeΠ∆  u∆  =  ∆  .  . . . .     .   ......   .   .   . . . . .   .   .  (1) (1) (1)   AeΠI BeΠI AeΠ∆ ··· AeΠΠ uΠ SeΓuΓ Π PN (i)T (i) (i) (i) (i)T (i) (i) (i)T (i) (i) 324 Here, AeΠΠ = i=1 RΠ AΠΠRΠ , AeΠI = RΠ AΠI , AeΠ∆ = RΠ AΠ∆, and BeΠI = (i)T (i) 325 RΠ BΠI . 326 Based on this deﬁnition, we can also obtain SeΓ from subdomain Schur comple- (i) 327 ments SΓ by assembling with respect to the global degrees of freedom of the primal 328 interface velocities, i.e.,

T 329 (5.1) SeΓ = RΓ SΓRΓ.

(i) 330 Here, we denote the direct sum of SΓ by SΓ. The global interface Schur operator SbΓ 331 on the continuous interface velocity space VbΓ can be obtained by further assembling 332 with respect to the dual interface variables, i.e.,

T T 333 (5.2) SbΓ = ReΓ SeΓReΓ = RΓ SΓRΓ.

T 334 We note that, for any xΓ ∈ VeΓ with xΓ SeΓxΓ = 0, xΓ has to be a constant on each 335 subdomain. Due to the continuity of the primal components of xΓ and the Dirichlet 336 boundary condtion of (3.1), xΓ has to be zero and therefore SeΓ is symmetric positive 337 definite. 338 Correspondingly, we define an operator Be0Γ, which maps the partially assembled 339 interface velocity space VeΓ into F0, the space of right-hand sides corresponding to W0. (i) 340 Be0Γ can be obtained from the subdomain operators B0Γ by assembling with respect to PN (i) (i) 341 the primal interface velocity part, i.e., Be0Γ = i=1 B0Γ RΓ . Similarly, the operator 342 Bb0Γ can be obtained from the partially assembled operator Be0Γ by further assembling 343 with respect to the dual interface velocity variables on the subdomain interfaces, i.e., 344 Bb0Γ = Be0ΓReΓ. By the definition, we have that the Be0Γ has a full row rank since Bb0Γ 345 does. 10

This manuscript is for review purposes only. 346 Let

" T # ReD,Γ SeΓ Be0Γ 347 (5.3) ReD = , Se = . I Be0Γ 0

348 Due to the positive deﬁniteness of SeΓ and the full row rank of Be0Γ, Se is invertible and 349 we can deﬁne the preconditioner for solving the global interface Stokes problem as

−1 T −1 350 M = ReDSe ReD.

351 Note that ReD,Γ is of full rank and that the preconditioner is nonsingular. The pre- 352 conditioned BDDC algorithm is then of the form: to ﬁnd (uΓ, p0) ∈ VbΓ,W0 , such 353 that

uΓ gΓ 354 (5.4) M −1Sb = M −1 . p0 0

R (i) 355 We require that u · ni = 0, for all the dual interface velocity variables ∂Ωi ∆ (i) (i) 356 u∆ ∈ V∆ , with ni the unit outward normal of ∂Ωi; see [18, 29]. We will refer to this 357 assumption as the divergence free constraint for the dual velocity variables. When the 358 Conjugate Gradient (CG) method is used to solve the preconditioned system (5.4), the 359 divergence free constraint can ensure the CG iterations will be in a special subspace 360 where the preconditioned operator is positive deﬁnite and therefore the CG method 361 can be applied. In order to satisfy this constraint, we choose the primal variables 362 which are spanned by subdomain interface edge/face basis functions with constant 363 values on these edges/faces for two/three dimensions. We change the variables so 364 that the degree of freedom of each primal constraint is explicit; see [19, 14]. The 365 dual space is correspondingly spanned by the remaining interface degrees of freedom 366 with zero average values over the interface edge/face. This constraint is critical to the 367 design of the preconditioner, as we will see more details in Section 6. 368 At the end of this section, we discuss the implementation of the preconditioner. 369 The main operation is the product of Se−1 with a vector, which requires solving a 370 coarse problem related to the primal variables we choose and independent subdomain 371 Stokes problems with Neumann type boundary conditions. The size of the coarse 372 problem will increase with the increasing of the number of the subdomains and it can 373 be a bottleneck of the algorithm. The multilevel extension of the algorithms can be 374 explored as in [32, 31, 22].

375 6. Some Auxiliary Results. We adopt the convention that C denotes a generic 376 constant independent of the mesh size h and subdomain size H. In general, its value 377 may vary at diﬀerent instances. 378 First, we list two useful results. For shape regular partition Th as detailed in [38], 379 the following trace and inverse inequalities hold; see[37]. 380 Lemma 4. (Trace Inequality) There exists a constant C such that

2 −1 2 2 381 (6.1) kgke ≤ C hK kgkK + hK k∇gkK ,

1 382 where g ∈ H (K), and K is an element of Th with e as an edge/face. 11

This manuscript is for review purposes only. 383 Lemma 5. (Inverse Inequality) There exists a constant C = C(k) such that −1 384 (6.2) k∇gkK ≤ C(k)hK kgkK , ∀K ∈ Th

385 for any piecewise polynomial g of degree k on Th. 386 We collect a few results of the weak Galerkin ﬁnite element scheme, which will 387 be used in our analysis of the BDDC preconditioner. Note that the discrete weak 0 388 velocity function space Vk is a normed linear space with a triple-bar norm given by 389 [38, (4.1)] 2 2 2 390 (6.3) |||v||| = P k∇ vk + P h−1 kQ v − v k . K∈Th w K K∈Th K b 0 b ∂K 391 392 Lemma 6. For the weak Galerkin scheme described in Section 3, the following 393 results hold: 2 2 394 1. For any v = {v , v } ∈ V , we have P k∇v k ≤ C|||v||| ; 0 b k T ∈Th 0 T 0 2 395 2. For any v ∈ Vk , a (v, v) = |||v||| ; 0 396 3. For any v, w ∈ Vk , |a (v, w)| ≤ |||v||||||w|||; 0 397 4. For any v = {v0, vb} ∈ Vk , ρ ∈ Wk−1, |b (v, ρ)| ≤ C|||v||| kρkL2 ; b(v, ρ) 398 5. For any ρ ∈ Wk−1, supv∈V 0 ≥ β kρk 2 , where β is positive constant k |||v|||w L 399 independent of the mesh size h. 400 Proof. The ﬁrst result is in [38, Lemma A.2]; the second and third results give the 401 coercivity and boundedness property of the bilinear form a (·, ·), which are proved in 402 [38, Lemma 4.1]. The fourth result is the boundedness property of the bilinear form 403 b (·, ·). This can be proved as follows.

X 404 |b (v, ρ)| = (∇w · v, ρ)K K∈Th

X 405 = (− (v0, ∇ρ)K + hvb · n, ρi∂K ) K∈Th

X 406 = ((∇ · v0, ρ)K − h(Qbv0 − vb) · n, ρi∂K ) K∈Th !1/2 !1/2 X 2 X 2 407 ≤ C k∇v0kL2(K) kρkL2(K) K∈Th K∈Th !1/2 !1/2 X −1 2 X 2 408 + C hK kvb − Qbv0kL2(∂K) hk kρkL2(∂K) K∈Th K∈Th

410409 ≤ C|||v|||kρkL2 , 411 where we use the deﬁnition of weak divergence for the second equality, and integration 412 by parts for the third equality. We use the Cauchy-Schwarz inequality for the fourth 413 inequality. Part (1) of Lemma 6, the deﬁnition of the triple-bar norm (6.3), and the 414 trace inequality (6.1) and the inverse inequality (6.2) for the last inequality. 415 The last result is the discrete inf-sup condition, which is proved in [38, Lemma 416 4.3]. These results also hold for the subdomain Ωi. If follows that the weak Galerkin 417 scheme is well-posed for the global interface problem and local subdomain problems. 12

This manuscript is for review purposes only. 418 We introduce several conceptual tools which will be useful in our analysis of the 419 BDDC preconditioner. 420 Definition 7. (Schur complement of the subdomain elliptic problem) The sub- (i) 421 domain Schur complement for the elliptic problem, denoted by SΓ,E, is deﬁned as (i) (i) (i) (i) (i) 422 follows: given uΓ ∈ VΓ , determine SΓ,EuΓ ∈ FΓ such that " # " # u(i) 0 423 A(i) I = , (i) S(i) u(i) uΓ Γ,E Γ

" (i) (i)T # (i) AII AΓI 424 where A = (i) (i) . AΓI AΓΓ 425 Since the subdomain elliptic problem A(i) is symmetric positive definite [36], the (i) 426 Schur complement SΓ,E is also symmetric positive definite by the inertia of Schur 427 complements [19]. Thus, we can define the norm 2 428 u(i) = u(i)T A(i)u(i) = a u(i), u(i) , for all u(i) ∈ V (i), A(i) 429 and 2 (i) (i)T (i) (i) (i) (i) 430 u = u S u , for all u ∈ V . Γ (i) Γ Γ,E Γ Γ Γ SΓ,E 431 Similarly, the subdomain Schur complements for the Stokes problems, defined in 432 (4.3), are symmetric, positive semi-definite [18]. They are singular for any floating 433 subdomains, by which we mean the boundary of the subdomain does not intersect (i) 434 with the global domain boundary ∂Ω. Thus, we can define the SΓ − seminorms by 2 (i) (i)T (i) (i) (i) (i) 435 u = u S u , for all u ∈ V . Γ (i) Γ Γ Γ Γ Γ SΓ

436 It follows that N 2 2 X (i) 437 T |uΓ|S = uΓ SΓuΓ = uΓ (i) . Γ S i=1 Γ

438 The fully and partially assembled global interface velocity operators SbΓ and SeΓ, 439 given in (5.2) and (5.1), are both symmetric, positive deﬁnite because of the Dirichlet 440 boundary conditions on ∂Ω and the adequacy of the primal continuity constraints 441 for the divergence free condition. In similar way as before, we deﬁne the SbΓ− and 442 SeΓ−norms on the spaces VbΓ and VeΓ, respectively, as below.

2 T T T 2 443 kuΓk = u SΓuΓ = u R SΓRΓuΓ = |RΓuΓ| , ∀uΓ ∈ VΓ, SbΓ Γ b Γ Γ SΓ b

2 T T T 2 444 kuΓk = u SΓuΓ = u R SΓRΓuΓ = RΓuΓ , ∀uΓ ∈ VΓ. SeΓ Γ e Γ Γ SΓ e

445 The global interface operator Sb and Se, introduced in (4.4) and (5.3), are sym- 446 metric indefinite on the space VbΓ × W0 and VeΓ × W0, respectively. However, when 447 restricted to the proper subspaces, these operators can be positive semidefinite, and 448 we can thus define a Sb− and Se−seminorms on these subspaces. We call such subspaces 449 as the benign subspaces, and denote them by VbΓ,B × W0 and VeΓ,B × W0, respectively. 450 Specifically, they can be defined as follows. 13

This manuscript is for review purposes only. 451 Definition 8. (Benign subspaces) n o n o 452 VbΓ,B = uΓ ∈ VbΓ|Bb0ΓuΓ = 0 and VeΓ,B = uΓ ∈ VeΓ|Be0ΓuΓ = 0 .

453 If follows that we can deﬁne

2 454 |u| = uT Su, ∀u = (u , p ) ∈ V × W , Sb b Γ 0 bΓ,B 0

2 455 |u| = uT Su, ∀u = (u , p ) ∈ V × W . Se e Γ 0 eΓ,B 0 456 We can show by direct computation that the following facts hold.

2 2 457 |u| = kuΓk , ∀u = (uΓ, p0) ∈ VΓ,B × W0, Sb SbΓ b

2 2 458 |u| = kuΓk , ∀u = (uΓ, p0) ∈ VΓ,B × W0. Se SeΓ e

459 We denote the null space of the Sb−seminorm operator on the space VbΓ,B × W0 by 460 Zb. It is easy to see that this space is comprised of elements u = (0, p0) ∈ VbΓ,B × W0. 461 The following lemma is crucial to the analysis of the preconditioned BDDC op- 462 erator. The proof can be found in [18, 29]. 463 Lemma 9. Under the divergence free constraint for the dual interface velocities, T 464 introduced in Section 5, we have ReDu ∈ VbΓ,B × W0 for any u ∈ VeΓ,B × W0. 465 With the choice of the primal velocity continuity constraints of the BDDC algorithm, 466 the preconditioned BDDC operator M −1Sb is positive deﬁnite on the quotient space, 467 and correspondingly, we can use the preconditioned conjugate gradient method when 468 the iterations are restricted to the quotient space. The design of the BDDC precondi- 469 tioner and the result from Lemma 9 guarantee that the iterations of the preconditioned 470 conjugate gradient method will stay in the quotient subspace if the initialization lies 471 in the quotient subspace [18]. 472 Next we introduce two important extension operators for the trace over the sub- 473 domain boundary. 474 Definition 10. (Discrete harmonic extension) The discrete harmonic extension (i) (i) (i) 475 of γ ∈ VΓ over the subdomain Ωi, denoted by H (γ): VΓ → V , satisﬁes the 476 following:

( 0 a (H (γ) , v) = 0, ∀v = {v0, vb} ∈ V (Ωi) , 477 k H (γ) |∂Ωi = γ.

478 The bilinear form a (·, ·) is deﬁned in (3.3). 479 Definition 11. (Discrete Stokes extension) The discrete Stokes extension of γ ∈ (i) (i) (i) 480 VΓ over the subdomain Ωi, denoted by S (γ): VΓ → V , satisﬁes the following:  a (S (γ) , v) − b (v, P (γ)) = 0, ∀v = {v , v } ∈ V 0 (Ω ) ,  0 b k i 481 b (S (γ) , q) = 0, ∀q ∈ Wk−1 (Ωi) ,  S (γ) |∂Ωi = γ,

482 where P (γ) is the corresponding pressure extension with zero mean value living in the 483 space Wk−1 (Ωi). The bilinear forms a (·, ·) and b (·, ·) are deﬁned in (3.3). 14

This manuscript is for review purposes only. 484 The connection between the discrete harmonic/Stokes extensions and the Schur 485 complements of the corresponding linear systems can be revealed as follows. 486 Remark 1. By deﬁnition, it is clear that

2 2 2 (i) (i) 487 (i) uΓ (i) = H uΓ = inf u , S A(i) (i) (i) (i) (i) A(i) Γ,E u ∈V , u |∂Ωi =uΓ

488 and that

2 2 2 (i) (i) 489 (i) uΓ (i) = S uΓ = inf u . S A(i) (i) (i) (i) (i) (i) (i) A(i) Γ u ∈V , u |∂Ωi =uΓ ,B u =0

(i) 490 For the same edge velocities uΓ over the subdomain boundary ∂Ωi, we have 2 2 (i) (i) 491 H uΓ ≤ S uΓ , A(i) A(i)

492 since the inﬁmum over a larger set is smaller. It follows that

2 2 (i) (i) 493 u ≤ u . Γ (i) Γ (i) SΓ,E SΓ

494 Next, we prove the connection between the edge velocity seminorms deﬁned by the 495 Schur complements of the elliptic and Stokes problems for the same subdomain. Sim- 496 ilar proof for the conforming discretizations can be found in [2]. Lemma (i) (i) 497 12. For any uΓ ∈ VΓ , we have

2 2 2 2 β (i) (i) (i) 498 C u ≤ u ≤ u , 2 Γ (i) Γ (i) Γ (i) (1 + β) SΓ SΓ,E SΓ

499 where β is the inf-sup stability constant defined in Lemma 6. 500 Proof. The second inequality directly follow from the Remark. 501 We prove the first inequality as follows. Denote the discrete harmonic and Stokes (i) (i) (i) (i) (i) 502 extension of uΓ ∈ VΓ by H uΓ and S uΓ , respectively. Using v = S uΓ − (i) 503 H uΓ as the test function in Definition 11, we have

(i) (i) (i) (i) (i) 504 a S uΓ , S uΓ − H uΓ − b S uΓ − H uΓ , ρ = 0,

505 where ρ is the corresponding pressure extension with zero mean value living in the 506 space Wk−1(Ωi). (i) 507 Since b S uΓ , ρ = 0, it follows that

(i) (i) (i) (i) (i) 508 a S uΓ , S uΓ = a S uΓ , H uΓ + b H uΓ , ρ .

509 By the part (4) in Lemma 6, we have

2 (i) (i) (i) (i) (i) 510 (6.4) S uΓ ≤ S uΓ H uΓ + C H uΓ kρkL2(Ω ) A(i) A(i) A(i) A(i) i

511 By the inf-sup condition (the part (5) in Lemma 6), 15

This manuscript is for review purposes only. 2 2 −2 b (v, ρ) 512 k ρ k 2 ≤ β sup L (Ωi) 2 0 v∈Vk (Ωi) |||v||| 2 a S u(i) , v −2 Γ 513 (6.5) = β sup 2 0 v∈Vk (Ωi) |||v||| 2 2 −2 (i) −2 (i) 514 ≤ β S uΓ = β S uΓ , A(i)

515 where we have used Deﬁnition 11 for the second equality and the parts (2) and (3) in 516 Lemma 6 for the last inequality. 517 Substituting (6.5) into (6.4), we have

2 (i) (i) (i) −1 (i) (i) 518 S uΓ ≤ S uΓ A(i) H uΓ A(i) + Cβ S uΓ H uΓ A(i) A(i) A(i) 1 + β (i) (i) 519 ≤ C S uΓ H uΓ . 520 β A(i) A(i) 521 It follows that 2 2 2 2 β2 (i) β2 (i) (i) (i) 522 C 2 u = C 2 S u ≤ H u = u . (1+β) Γ (i) (1+β) Γ (i) Γ (i) Γ (i) SΓ,S A A SΓ,E 523 In order to prove the condition number bounds for the BDDC preconditioner, 524 we deﬁne an averaging operator for the Stokes problem, denoted by ED, which 525 maps VeΓ × W0, with generally discontinuous interface velocities, to the same space 526 with continuous interface velocities. Speciﬁcally, for any u = (uΓ, p0) ∈ VeΓ × W0, T 527 ED [uΓ, p0] ∈ VeΓ × W0, where

T T ReΓ ReD,Γ ED,Γ 528 ED = ReRe = = , D I I I

T 529 and ED,Γ = ReΓReD,Γ is the interface averaging operator for the velocities across the 530 interface Γ. The operator ED,Γ computes a weighted average for the edge velocity 531 across the subdomain interface Γ, and then distributes the average back to the original 532 degree of freedoms on the interface. 533 To facilitate further analysis, we introduce a useful norm as deﬁned in [12]:  1/2 ∗ X 1 2 534 (6.6) |||λ||| = kλ − m (λ)k , h,D  h K ∂K  K∈Th,K⊆D

535 where 1 Z 536 mK (λ) = λds. |∂K| ∂K ∗ ∗ 537 Denote |||λ|||h = |||λ|||h,Ω. 538 Deﬁne the local lifting operators Q(·) and U(·) for the weak Galerkin (WG) 539 method as below: given λ on ∂K,

d×d 540 (6.7a) (Qλ, r)K + (Uµ, ∇ · r)K = hλ, r · ni∂K for all r ∈ [Pk−1(K)] , −1 d 542541 (6.7b) −(w, ∇ · Qλ)K + hhK (QbUλ − λ),Qbwi∂K = 0 for all w ∈ [Pk(K)] . 16

This manuscript is for review purposes only. 543 Let u = (u0, ub) = (Uλ, λ). We have Qλ = ∇wu and can obtain a reduced norm of 544 λ using the norm from the WG bilinear form as given in (6.3) as

2 2 2 545 (6.8) |||λ||| = P k∇ uk + P h−1 kQ u − u k . D K∈Th,K⊆D w K K∈Th,K⊆D K b 0 b ∂K

546 Denote |||λ||| = |||λ|||Ω 547 We will show the equivalence between the triple-bar norm deﬁned above in (6.6) 548 and (6.8). To denote the triple-bar norm deﬁned over an element K, we add a sub- 549 script K to it. Note that similar strategy was used to prove the equivalence between 550 the norm generated by the bilinear form from a hybridized mixed method and triple- 551 bar norm (6.6) in [12]. Lemma 552 13. The function |||λ|||K is zero on K ∈ Th if and only if λ is constant on 553 ∂K.

554 Proof. Assume that |||λ|||K = 0 on K. It follows that −1 555 0 = (∇wu, ∇wu) + hK hQbUλ − λ, QbUλ − λi∂K ,

556 where u = {Uλ, λ}, and ∇wu = Qλ. This implies that ∇wu = 0 on element K and 557 QbUλ = λ on ∂K. Further, we have from the deﬁnition of the discrete weak gradient n 558 operator or the lifting operator Q given in (6.7b) that for any τ ∈ [Pk−1(K)] ,

559 0 = (∇wu, τ)K

560 = −(Uλ, ∇ · τ)K + hλ, τ · ni∂K

561 = (∇Uλ, τ)K − hUλ − λ, τ · ni∂K

562 = (∇Uλ, τ)K − hQbUλ − λ, τ · ni∂K

564563 = (∇Uλ, τ)K .

565 Let τ = ∇Uλ. Then we have ∇Uλ = 0 on K. It follows that Uλ = const. on K. 566 Thus, QbUλ = const. on ∂K. Since QbUλ = λ on ∂K, we have λ = const. Note that 567 similar argument as above was provided in [38] to prove that (6.3) gives a norm. 568 Conversely, assume λ is a constant on ∂K. Substituting the ordered pair (r, w) 569 in (6.7) with (Qλ, Uλ) and adding up, we obtain

2 −1 570 |||λ|||K = hλ, Qλ · ni∂K − hK hQbUλ − λ, λi∂K .

571 Let w = λ be the test function in (6.7b). Since λ is constant, λ = Qbλ. It follows 572 from (6.7b) that

−1 573 −hλ, Qλ · ni∂K + hK hQbUλ − λ, λi∂K = 0.

574 Therefore, |||λ|||K = 0. Lemma 0 575 14. Let Mh = {vb : v = {v0, vb} ∈ Vk }. For all λ ∈ Mh,

∗,2 2 ∗,2 576 c|||λ|||h ≤ |||λ||| ≤ C|||λ|||h .

577

578 Proof. First, we prove the lower bound. By Lemma 13, |||λ|||K = 0 implies that λ 579 is constant on ∂K. Similarly as in [12], by a scaling argument, it can be shown that

c c ∗ 580 |||λ||| ≥ inf kλ − κk = kλ − mK (λ)k = c|||λ||| , K |∂K|1/2 κ∈R ∂K |∂K|1/2 ∂K h,K 17

This manuscript is for review purposes only. 581 for some constant c independent of λ. 582 Next, we prove the upper bound. Let r = Qλ, and w = Uλ. Plugging the ordered 583 pair (r, w) into (6.7), and adding up, we obtain

2 −1 584 |||λ|||K = hλ, Qλ · ni∂K − hK hQbUλ − λ, λi∂K −1 585 = hλ, Qλ · n − hK (QbUλ − λ)i∂K −1 586 = hλ − mK (λ), Qλ · n − hK (QbUλ − λ)i∂K C 587 ≤ kλ − m (λ)k |||λ||| |∂K|1/2 K ∂K K ∗ 588 = C|||λ|||h,K |||λ|||K ,

590589

591 where we have used (6.7b) for the third equality, the trace inequality (6.1) and inverse 592 inequality (6.2) for the second-to-last inequality. It follows that

∗,2 2 ∗,2 593 c|||λ|||h,K ≤ |||λ|||K ≤ C|||λ|||h,K .

594 Summing up over all elements in Th, we obtain

∗,2 2 ∗,2 595 c|||λ|||h ≤ |||λ||| ≤ C|||λ|||h .

596 Based on the equivalence of norms in Lemma 14, similar to the proof of [35, 597 Lemma 5], we can obtain that the interface averaging operator ED,Γ satisﬁes the 598 following bound:

599 Lemma 15. For any wΓ ∈ VeΓ, 2 2 H 2 600 |ED,ΓwΓ| ≤ C 1 + log |wΓ| , SeΓ,E h SeΓ,E

601 where C is a positive constant independent of the domain size H, and mesh size h.

602 Now, we are in a position to prove the bound of the averaging operator ED for 603 the Stokes problem.

604 Lemma 16. There exists a positive constant C, which is independent of H and h, 605 such that 2 2 2 1 + β H 2 606 |EDw| ≤ C 1 + log |w| ∀w = (wΓ, q0) ∈ VeΓ,B × W0, Se β h Se

607 where β is the inf-sup stability constant. T 608 Proof. For any vector w = (wΓ, q0) ∈ VeΓ,B ×W0, by Lemma 9, ReDw ∈ VbΓ,B ×W0. T 609 Thus, EDw = ReReDw ∈ VeΓ,B × W0. 2 2 610 From the deﬁnition of the Se-seminorm, we have |EDw| = kED,ΓwΓk = Se SeΓ 611 |R¯ (E w ) |2 . Γ D,Γ Γ SΓ (i) 612 Noting that SΓ = diag(SΓ ), and applying Lemma 12 to each subdomain, we 613 have 2 1+β 614 |R¯ (E w ) |2 ≤ C |R¯ (E w ) |2 Γ D,Γ Γ SΓ β Γ D,Γ Γ SΓ,E 18

619 Combining these inequalities, we have 2 2 2 1+β H 2 1+β H 2 2 620 |EDw| ≤ C 1 + log |wΓ| = C 1 + log |w| . Se β h SeΓ β h Se 621 7. Condition number estimate for the BDDC preconditioner. We are 622 now ready to formulate and prove our main results. It follows by proving the lower 623 and upper bound for uT M −1Sub . See similar proof in [18]. 624 Theorem 17. Assume the divergence free constraint holds for the interface veloc- 625 ities. The preconditioned operator M −1Sb is symmetric, positive deﬁnite with respect 626 to the bilinear form h·, ·i on the space V × W . Its eigenvalues are bounded from Sb bΓ,B 0 (1+β)2 H 2 627 below by 1 and from above by C β2 1 + log h , where C is a constant which is 628 independent of the domain size H, and the mesh size h, and β is the inf-sup stability 629 constant.

630 Proof. It is suﬃcient to prove that for any u = (uΓ, p0) ∈ VbΓ,B ×W0, with uΓ 6= 0, 2 D E 1+β 2 631 hu, ui ≤ u, M −1Su ≤ C 1 + log H hu, ui . Sb b β h Sb Sb D E 632 In what follows, we prove the lower and upper bound for u, M −1Sub respec- Sb 633 tively. −1 634 Let ue = Se ReDSub . Obviously, ue ∈ VeΓ,B × W0. T T 635 Note that Re ReD = ReDRe = I. The details for the proof of the lower bound go as 636 follows: D E 637 hu, ui = uT SRT Ru = uT SRT S−1SRu = u, Ru Sb b eD e b eD e e e e e Se 1/2 1/2 D E 1/2 1/2 638 ≤ hu, ui Ru,e Rue = hu, ui hu, ui . 639 e e Se Se e e Se Sb

640 Thus, we obtain hu, ui ≤ hu, ui by canceling a common factor and squaring on Sb e e Se 641 both sides. 642 Since D E D E 643 hu, ui = uT SRT S−1SS−1R Su = u, RT S−1R Su = u, M −1Su , e e Se b eD e ee eD b eD e eD b b D E Sb Sb 644 we have hu, ui ≤ u, M −1Su . Sb b Sb 645 Next, we prove the upper bound. −1 T −1 T −1 646 Since M = ReDSe ReD, we have ReDue = M Sub . 647 By using Lemma 16 and the fact that Sb = ReT SeRe, we obtain D E D E D E 648 M −1Su, M −1Su = RT u, RT u = RRT u, RRT u = |E u|2 b b eD eD e eD e eD D S Sb e e Sb e e Se e e 1 + β 2 H 2 649 ≤ C 1 + log |u|2 β h e Se 1 + β 2 H 2 D E 650 ≤ C 1 + log u, M −1Sub 651 β h Sb 19

This manuscript is for review purposes only. Table 1 Condition number estimates and iteration counts for the BDDC preconditioned operator with H changing subdomains numbers. h = 8, and k = 1.

Number of Subdomains Iterations Condition number 4×4 11 4.12 8×8 13 5.01 16×16 13 4.90 24×24 13 5.05 32×32 12 4.94

Table 2 Condition number estimates and iteration counts for the BDDC preconditioned operator with H changing subdomains numbers. h = 8, and k = 2.

Number of Subdomains Iterations Condition number 4×4 13 7.37 8×8 17 9.24 16×16 20 9.89 24×24 20 10.29 32×32 19 10.26

652 Using the Cauchy-Schwarz inequality, we have

1/2 D E 1/2 D E 653 u, M −1Sub ≤ hu, ui M −1Su,b M −1Sub Sb Sb Sb 1/2 1 + β H 1/2 D E 654 ≤ C 1 + log hu, ui u, M −1Su . S b 655 β h b Sb 2 D E 1+β 2 656 This gives u, M −1Su ≤ C 1 + log H hu, ui . The upper bound of b β h Sb Sb 657 the eigenvalues thus follows.

658 8. Numerical Experiments. In this section, we will report some numerical 659 results for the BDDC algorithm proposed for the weak Galerkin discretization of the 660 Stokes problem. We used the BDDC algorithm to solve the model problem (3.1) on 2 661 the square domain Ω = [0, 1] with zero Dirichlet boundary condition. The analytical 662 solution of the test problem is given by

sin3 (πx) sin2 (πy) cos (πy) 663 u = and p = x2 − y2. − sin2 (πx) sin3 (πy) cos (πx)

664 We decompose the unit square into N ×N subdomains with side length H = 1/N. 665 Each subdomain has a characteristic mesh size h. Both the first order (k = 1) 666 and second order (k = 2) weak Galerkin methods are used to discretize the model 667 equations. The BDDC preconditioned conjugate gradient iterations are stopped when 6 668 the l2−norm of the residual has been reduced by a factor of 10 . 669 In the first set of experiments, we fix the size of the subdomain problem to be H 670 h = 8. Table 1 and 2 show the iteration counts and the estimates of the condition 671 numbers for the BDDC preconditioned operator with changing subdomain numbers 672 for k = 1 and k = 2, respectively. The condition numbers are found to be independent 673 of the number of subdomains. As the second set of experiment, instead of fixing the 20

This manuscript is for review purposes only. Table 3 Condition number estimates and iteration counts for the BDDC preconditioned operator with changing subdomain problem size. 8 × 8 subdomains, and k = 1.

H h Iterations Condition number 4 9 2.49 8 13 5.01 16 15 7.48 24 18 9.12 32 19 10.37

Table 4 Condition number estimates and iteration counts for the BDDC preconditioned operator with changing subdomain problem size. 8 × 8 subdomains, and k = 2.

H h Iterations Condition number 4 14 5.87 8 17 9.24 16 21 12.47 24 23 15.33 32 23 16.09

674 size of the subdomain problems, we ﬁx the subdomain partition to be 8 × 8, and 675 allow the subdomain problem size to vary. The condition number is found to increase 676 logarithmically with the subdomain problem size. Table 3 and 4 demonstrate results 677 for the second set of experiments for k = 1 and k = 2, respectively. 678 To conclude, we have carried out a series of experiments to obtain iteration counts 679 and condition number estimates. The experimental results prove to be consistent 680 with the theory. That is the condition number bound of the BDDC preconditioned (1+β)2 H 2 681 system is of the form C β2 1 + log h , where H and h are the diameters of the 682 subdomains and elements, respectively. Possible future work will be to explore the 683 order of the basis functions eﬀects on C.

684 Acknowledgments. The authors would like to thank the anonymous referees 685 for their comments and suggestions that helped improve the quality of the manuscript.

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