Multispace and Multilevel BDDC

0.32 MB·English

by  Jan Mandel

Multispace and Multilevel BDDC
Jan Mandel∗† Bedˇrich Soused´ık†‡ Clark R. Dohrmann§
February 2, 2008
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2 Abstract

n BDDC method is the most advanced method from the Balancing family of iterative
a
substructuring methods for the solution of large systems of linear algebraic equations arising
J
from discretization of elliptic boundary value problems. In the case of many substructures,
1
solving the coarse problem exactly becomes a bottleneck. Since the coarse problem in BDDC
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hasthesamestructureastheoriginalproblem,itisstraightforwardtoapplytheBDDCmethod

] recursively to solve the coarse problem only approximately. In this paper, we formulate a new
A
family of abstract Multispace BDDC methods and give condition number bounds from the
N
abstract additive Schwarz preconditioning theory. The Multilevel BDDC is then treated as
. a special case of the Multispace BDDC and abstract multilevel condition number bounds are
h
t given. The abstract bounds yield polylogarithmic condition number bounds for an arbitrary
a
fixed number of levels and scalar elliptic problems discretized by finite elements in two and
m
three spatial dimensions. Numerical experiments confirm the theory.
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AMS Subject Classification: 65N55, 65M55, 65Y05
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Key words: Iterative substructuring, additive Schwarz method, balancing domain
v
decomposition, BDD, BDDC, Multispace BDDC, Multilevel BDDC
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1 Introduction
.
2
1
7 The BDDC (Balancing Domain Decomposition by Constraints) method by Dohrmann [4] is the
0 mostadvancedmethodfromtheBDDfamilyintroducedbyMandel[12]. ItisaNeumann-Neumann
:
v iterative substructuring method of Schwarz type [5] that iterates on the system of primal variables
i
X reduced to the interfaces between the substructures. The BDDC method is closely related to
r the FETI-DP method (Finite Element Tearing and Interconnecting - Dual, Primal) by Farhat et
a
al. [6, 7]. FETI-DP is a dual method that iterates on a system for Lagrange multipliers that
enforce continuity on the interfaces, with some “coarse” variables treated as primal, and it is
a further development of the FETI method by Farhat and Roux [8]. Polylogarithmic condition
number estimates for BDDC were obtained in [13, 14] and a proof that the eigenvalues of BDDC
and FETI-DP are actually the same except for eigenvalue equal to one was given in Mandel et
∗Corresponding author. [email protected].
†DepartmentofMathematicalSciences,UniversityofColorado Denver,Denver,CO80217-3364, USA.Supported
in part by theNational Science Foundation undergrants CNS-0325314, DMS-071387, and CNS-0719641.
‡Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Th´akurova 7, 166 29
Prague 6, Czech Republic. Supported by the program of the Information society of the Academy of Sciences of the
Czech Republic1ET400760509 and bythe Grant Agency of theCzech RepublicGA CˇR 106/08/0403.
§Structural Dynamics Research Department, Sandia National Laboratories, Albuquerque NM 87185-0847, USA.
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United
States Department of Energy’s National Nuclear Security Administration underContract DE-AC04-94-AL85000.
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al. [14]. Simpler proofs of the equality of eigenvalues were obtained by Li and Widlund [10], and
also by Brenner and Sung [3], who also gave an example when BDDC has an eigenvalue equal
to one but FETI-DP does not. In the case of many substructures, solving the coarse problem
exactly becomes a bottleneck. However, since the coarse problem in BDDC has the same form as
the original problem, the BDDC method can be applied recursively to solve the coarse problem
only approximately. This leads to a multilevel form of BDDC in a straightforward manner, see
Dohrmann [4]. Polylogarithmic condition number bounds for three-level BDDC (BDDC with two
coarse levels) were proved in two and three spatial dimensions by Tu [20, 19].
In this paper, we present a new abstract Multispace BDDC method. The method extends a
simple variational setting of BDDC from Mandel and Soused´ık [15], which could be understood as
an abstract version of BDDC by partial subassembly in Li and Widlund [11]. However, we do not
adopt their change of variables, which does not seem to be suitable in our abstract setting. We
provide a condition number estimate for the abstract Multispace BDDC method, which generalizes
the estimate for a single space from [15]. The proof is based on the abstract additive Schwarz
theory by Dryja and Widlund [5]. Many BDDC formulations (with an explicit treatment of
substructure interiors, after reduction to substructure interfaces, with two levels, and multilevel)
are then viewed as abstract Multispace BDDC with a suitable choice of spaces and operators, and
abstract condition number estimates for those BDDC methods follow. This result in turn gives
a polylogarithmic condition number bound for Multilevel BDDC applied to a second-order scalar
elliptic modelproblems,withanarbitrarynumberoflevels. Abriefpresentation ofthemainresults
of the paper, without proofs and with a simplified formulation of the Multispace BDDC estimate,
is contained in the conference paper [16].
Thepaperis organized asfollows. InSec. 2weintroducetheabstractproblemsetting. InSec. 3
we formulate an abstract Multispace BDDC as an additive Schwarz preconditioner. In Sec. 4 we
introduce the settings of a model problem using finite element discretization. In Sec. 5 we recall
the algorithm of the original (two-level) BDDC method and formulate it as a Multispace BDDC.
In Sec. 6 we generalize the algorithm to obtain Multilevel BDDC and we also give an abstract
condition number bound. In Sec. 7 we derive the condition number bound for the model problem.
Finally, in Sec. 8, we report on numerical results.
2 Abstract Problem Setting
We wish to solve an abstract linear problem
u ∈X : a(u,v) = hf ,vi, ∀v ∈ X, (1)
X
where X is a finite dimensional linear space, a(·,·) is a symmetric positive definite bilinear form
defined on X, f ∈ X′ is the right-hand side with X′ denoting the dual space of X, and h·,·i is the
X
duality pairing. The form a(·,·) is also called the energy inner product, the value of the quadratic
form a(u,u) is called the energy of u, and the norm kuk = a(u,u)1/2 is called the energy norm.
a
The operator A : X 7→ X′ associated with a is defined by
X
a(u,v) = hA u,vi, ∀u,v ∈X.
X
A preconditioner is a mapping B : X′ → X and we will look for preconditioners such that
hr,Bri is also symmetric and positive definite on X′. It is well known that then BA : X → X has
X
only real positive eigenvalues, and convergence of the preconditioned conjugate gradients method
is bounded using the condition number
λ (BA )
max X
κ = ,
λ (BA )
min X
2
which we wish to bound above.
All abstract spaces in this paperare finite dimensionallinear spaces and we make nodistinction
between a linear operator and its matrix.
3 Abstract Multispace BDDC
TointroduceabstractMultispace BDDC preconditioner, supposethat thebilinear formais defined
and symmetric positive semidefinite on some larger space W ⊃ X. The preconditioner is derived
from the abstract additive Schwarz theory, however we decompose some space between X and
W rather than X as it would be done in the additive Schwarz method: In the design of the
preconditioner, we choose spaces V , k = 1,...,M, such that
k
M
X ⊂ V ⊂ W. (2)
k
k=1
X
Assumption 1 The form a(·,·) is positive definite on each V separately.
k
Algorithm 2 (Abstract Multispace BDDC) Given spaces V and linear operators Q , k =
k k
1,...,M such that a(·,·) is positive definite on each space V , and
k
M
X ⊂ V , Q : V → X,
k k k
k=1
X
define the preconditioner B : r ∈ X′ 7−→ u ∈X by
M
B : r 7→ u= Q v , v ∈ V : a(v ,z )= hr,Q z i, ∀z ∈ V . (3)
k k k k k k k k k k
k=1
X
We formulate the condition number bound first in the full strength allowed by the proof. The
bound used in the rest of this paper will be a corollary.
Theorem 3 Define for all k = 1,...,M the spaces VM by
k
VM = v ∈ V :∀z ∈ V :Q v = Q z =⇒ kv k2 ≤ kz k2 .
k k k k k k k k k k a k a
n o
If there exist constants C , ω, and a symmetric matrix E = (e )M , such that
0 ij i,j=1
M M
∀u∈ X ∃v ∈ V , k = 1,...,M :u = Q v , kv k2 ≤ C kuk2 (4)
k k k k k a 0 a
k=1 k=1
X X
∀k = 1,...,M ∀v ∈ VM :kQ v k2 ≤ ωkv k2 (5)
k k k k a k a
∀z ∈ Q V , k = 1,...,M : a(z ,z ) ≤ e kz k kz k , (6)
k k k i j ij i a j a
then the preconditioner from Algorithm 2 satisfies
λ (BA )
max X
κ= ≤ C ωρ(E).
0
λ (BA )
min X
3
Proof. WeinterprettheMultispaceBDDCpreconditionerasanabstractadditiveSchwarzmethod.
AnabstractadditiveSchwarzmethodisspecifiedbyadecompositionofthespaceX intosubspaces,
X = X +...+X , (7)
1 M
and by symmetric positive definite bilinear forms b on X . The preconditioner is a linear operator
i i
B :X′ → X, B : r 7→ u,
defined by solving the following variational problems on the subspaces and adding the results,
M
B :r 7→ u = u , u ∈ X : b (u ,y ) =hr,y i, ∀y ∈ X . (8)
k k k k k k k k k
k=1
X
Dryja and Widlund [5] proved that if there exist constants C , ω, and a symmetric matrix
0
E = (e )M , such that
ij i,j=1
M M
∀u∈ X ∃u ∈ X , k = 1,...,M :u = u , ku k2 ≤ C kuk2 (9)
k k k k bk 0 a
k=1 k=1
X X
∀k = 1,...,M ∀u ∈ X : ku k2 ≤ ωku k2 (10)
k k k a k bk
∀u ∈ X , k = 1,...,M :a(u ,u )≤ e ku k ku k (11)
k k i j ij i a j a
then
λ (BA )
max X
κ= ≤ C ωρ(E),
0
λ (BA )
min X
where ρ is the spectral radius.
Now theidea oftheproofis essentially to maptheassumptionsof theabstractadditive Schwarz
estimate from the decomposition (7) of the space X to the decomposition (2). Define the spaces
X = Q V .
k k k
We will show that the preconditioner (3) satisfies (8), where b is defined by
k
b (u ,y ) = a(G x,G z), x,z ∈ X, u = Q G x, y = Q G z. (12)
k k k k k k k k k k k
with the operators G :X → VM defined by
k k
M
1
G : u7→ v , a(v ,v ) → min, s.t. v ∈ VM, u= Q v , (13)
k k 2 k k k k k k
k=1
X
First, from the definition of operators G , spaces X , and because a is positive definite on V by
k k k
Assumption (1), it follows that G x and G z in (12) exist and are unique, so b is defined correctly.
k k k
To prove (8), let v be as in (3) and note that v is the solution of
k k
1
a(v ,v )−hr,Q v i→ min, v ∈ V .
k k k k k k
2
Consequently, the preconditioner (3) is an abstract additive Schwarz method and we only need to
verify the inequalities (9)–(11). To prove (9), let u ∈ X. Then, with v from the assumption (4)
k
and with u = Q G v as in (12), it follows that
k k k k
M M M
u= u , ku k2 = kv k2 ≤ C kuk2.
k k bk k a 0 a
k=1 k=1 k=1
X X X
4
Next, let u ∈ X . From the definitions of X and VM, it follows that there exist unique v ∈ VM
k k k k k k
such that u = Q v . Using the assumption (5) and the definition of b in (12), we get
k k k k
ku k2 = kQ v k2 ≤ ωkv k2 = ωku k2 ,
k a k k a k a k bk
which gives (10). Finally, (6) is the same as (11).
The next Corollary was given without proof in [16, Lemma 1]. This is the special case of
Theorem 3 that will be actually used. In the case when M =1, this result was proved in [15].
Corollary 4 Assume that the subspaces V are energy orthogonal, the operators Q are projections,
k k
a(·,·) is positive definite on each space V , and
k
M M
∀u∈ X : u= v , v ∈ V =⇒u = Q v . (14)
k k k k k
" #
k=1 k=1
X X
Then the abstract Multispace BDDC preconditioner from Algorithm 2 satisfies
λ (BA ) kQ v k2
κ = max X ≤ ω = max sup k k a . (15)
λmin(BAX) k vk∈Vk kvkk2a
Proof. We only need to verify the assumptions of Theorem 3. Let u ∈ X and choose v as
k
the energy orthogonal projections of u on V . First, since the spaces V are energy orthogonal,
k k
u = v , Q are projections, and from (14) u = Q v , we get that kuk2 = kv k2 which
k k k k a k a
proves (4) with C = 1. Next, the assumption (5) becomes the definition of ω in (15). Finally, (6)
0
P P P
with E = I follows from the orthogonality of subspaces V .
k
Remark 5 The assumption (14) can be written as
M
Q P = I,
k k
(cid:12)
Xk=1 (cid:12)X
(cid:12)
(cid:12)
where P is the a-orthogonal projection from M(cid:12) V onto V . Hence, the property (14) is a type
k j=1 j k
of decomposition of unity.
L
In the case when M = 1, (14) means that the projection Q is onto X.
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4 Finite Element Problem Setting
Let Ω be a bounded domain in Rd, d = 2 or 3, decomposed into N nonoverlapping subdomains
Ωs, s= 1,...,N, which form a conforming triangulation of the domain Ω. Subdomains will be also
called substructures. Each substructure is a union of Lagrangian P1 or Q1 finite elements with
characteristic meshsizeh, andthenodesofthefiniteelements between substructurescoincide. The
nodes contained in the intersection of at least two substructures are called boundary nodes. The
union of all boundary nodes is called the interface Γ. The interface Γ is a union of three different
types of open sets: faces, edges, and vertices. The substructure vertices will be also called corners.
For the case of regular substructures,such as cubes or tetrahedrons, we can usestandard geometric
definition of faces, edges, and vertices; cf., e.g., [9] for a more general definition.
In this paper, we findit more convenient to use the notation of abstract linear spaces and linear
operators between them instead of the space Rn and matrices. The results can be easily converted
5
to the matrix language by choosing a finite element basis. The space of the finite element functions
on Ω will be denoted as U. Let Ws be the space of finite element functions on substructure Ωs,
such that all of their degrees of freedom on ∂Ωs∩∂Ω are zero. Let
W = W1×···×WN,
and consider a bilinear form arising from the second-order scalar elliptic problem as
N
a(u,v) = ∇u∇vdx, u,v ∈ W. (16)
s=1ZΩS
X
Now U ⊂W is thesubspaceof allfunctionsfromW thatarecontinuous across thesubstructure
interfaces. We are interested in the solution of the problem (1) with X = U,
u∈ U :a(u,v) = hf,vi, ∀v ∈ U, (17)
where the bilinear form a is associated on the space U with the system operator A, defined by
A :U 7→ U′, a(u,v) = hAu,vi for all u,v ∈U, (18)
and f ∈ U′ is the right-hand side. Hence, (17) is equivalent to
Au =f. (19)
Define U ⊂U as the subspace of functions that are zero on the interface Γ, i.e., the “interior”
I
functions. Denote by P the energy orthogonal projection from W onto U ,
I
P :w ∈ W 7−→ v ∈U :a(v ,z ) = a(w,z ), ∀z ∈ U .
I I I I I I I
Functionsfrom(I −P)W,i.e.,fromthenullspaceofP,arecalleddiscreteharmonic;thesefunctions
are a-orthogonal to U and energy minimal with respect to increments in U . Next, let W be the
I I
space of all discrete harmonic functions that are continuous across substructureboundaries, that is
c
W = (I −P)U. (20)
In particular, c
U = U ⊕W, U ⊥ W. (21)
I I a
A common approach in substructuring is to reduce the problem to the interface. The problem
c c
(17) is equivalent to two independentproblems on the energy orthogonal subspaces U and W, and
I
the solution u satisfies u= u +u, where
I
c
b
u∈ U :a(u ,v ) = hf,v i, ∀v ∈ U , (22)
I I I I I I
u∈ W : a(u,v) = hf,vi, ∀v ∈ W. (23)
The solution of the interior probclem (2b2)bdecompboses inbto incdependent problems, one per each
substructure. The reduced problem (23) is then solved by preconditioned conjugate gradients. The
reduced problem (23) is usually written equivalently as
u∈ W :s(u,v) = hg,vi, ∀v ∈ W,
c b b 6 b b c
where s is the form a restricted on the subspace W, and g is the reduced right hand side, i.e., the
functional f restricted to the space W. The reduced right-hand side g is usually written as
c
hg,vi= hf,vi−a(u ,v), ∀v ∈ W, (24)
c I
because a(u ,v) = 0 by (21). In the implementation, the process of passing to the reduced problem
I b b b b c
becomes the elimination of the internal degrees of freedom of the substructures, also known as
static condensbation. The matrix of the reduced bilinear form s in the basis defined by interface
degrees of freedom becomes the Schur complement, and (24) becomes the reduced right-hand side.
For details on the matrix formulation, see, e.g., [17, Sec. 4.6] or [18, Sec. 4.3].
TheBDDC method is a two-level preconditioner characterized by the selection of certain coarse
degrees of freedom, such as values at the corners and averages over edges or faces of substructures.
DefineW ⊂ W asthesubspaceofallfunctionssuchthatthevaluesofanycoarsedegreesoffreedom
haveacommonvalueforallrelevantsubstructuresandvanishon∂Ω,andW ⊂ W asthesubspace
∆
of all fufnction such that their coarse degrees of freedom vanish. Next, define W as the subspace
Π
of all functions such that their coarse degrees of freedom between adjacenft substructures coincide,
and such that their energy is minimal. Clearly, functions in W are uniquelyfdetermined by the
Π
values of their coarse degrees of freedom, and
f
W ⊥ W , and W = W ⊕W . (25)
∆ a Π ∆ Π
We assume that
f f f f f
a is positive definite on W. (26)
That is the case when a is positive definite on the space U, where the problem (1) is posed, and
f
there are sufficiently many coarse degrees of freedom. We further assume that the coarse degrees
of freedom are zero on all functions from U , that is,
I
U ⊂ W . (27)
I ∆
In other words, the coarse degrees of freedom depend on the values on substructure boundaries
f
only. From (25) and (27), it follows that the functions in W are discrete harmonic, that is,
Π
WΠ = (I −P)WΠf. (28)
Next, let E be a projection from Wf onto U, deffined by taking some weighted average on
substructure interfaces. That is, we assume that
f
E : W → U, EU =U, E2 = E. (29)
Since a projection is the identity on its range, it follows that E does not change the interior degrees
f
of freedom,
EU = U , (30)
I I
since U ⊂ U. Finally, we show that the operator (I −P)E is a projection. From (30) it follows
I
that E does not change interior degrees of freedom, so EP = P. Then, using the fact that I −P
and E are projections, we get
[(I −P)E]2 = (I −P)E(I −P)E
= (I −P)(E−P)E (31)
= (I −P)(I −P)E = (I −P)E.
7
Remark 6 In [14, 15], the whole analysis was done in spaces of discrete harmonic functions after
eliminating U , and the space W was the solution space. In particular, W consisted of discrete
I
harmonic functions only, while the same space here would be (I−P)W. The decomposition of this
space used in [14, 15] would be cin our context written as f
f
(I −P)W = (I −P)W ⊕W , (I −P)W ⊥ W . (32)
∆ Π ∆ a Π
In the next section, the spface X will befeitherfU or W. f f
c
5 Two-level BDDC as Multispace BDDC
We show several different ways the original, two-level, BDDC algorithm can be interpreted as
multispace BDDC. We consider first BDDC applied to the reduced problem (23), that is, (1) with
X = W. This was the formulation considered in [14]. Define the space of discrete harmonic
functions with coarse degrees of freedom continuous across the interface
c
W = (I −P)W.
Γ
Because we work in the space of discrete harmonic functions and the output of the averaging
f f
operator E is not discrete harmonic, denote
E = (I −P)E. (33)
Γ
In an implementation, discrete harmonic functions are represented by the values of their degrees of
freedom on substructure interfaces, cf., e.g. [18]; hence, the definition (33) serves formal purposes
only, so that everything can be written in terms of discrete harmonic functions without passing to
the matrix formulation.
Algorithm 7 ([15], BDDC on the reduced problem) Define the preconditioner r ∈ W′ 7−→
u∈ W by
u= E w , w ∈ W :a(w ,z ) = hr,E z i, ∀z ∈ W . c (34)
Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ
c
Proposition 8 ([15]) The BDDC prfeconditioner on the reduced problemfin Algorithm 7 is the
abstract Multispace BDDC from Algorithm 2 with M = 1 and the space and operator given by
X = W, V = W , Q = E . (35)
1 Γ 1 Γ
Also, the assumptions of Corollary 4 are satisfied.
c f
Proof. We only need to note that the bilinear form a(·,·) is positive definite on W ⊂ W by (26),
Γ
and the operator E defined by (33) is a projection by (31). The projection E is onto W because
Γ Γ
E is onto U by (29), and I −P maps U onto W by the definition of W in (20). f f
Using the decomposition (32), we can split the solution in the space W into the incdependent
Γ
solution of two subproblems: mutually indepecndent problems on subcstructures as the solution in
the space W = (I−P)W , and a solution of global coarse problem in tfhe space W . The space
Γ∆ ∆ Π
W has a decomposition
Γ
f f f
W = W ⊕W , and W ⊥ W , (36)
f Γ Γ∆ Π Γ∆ a Π
the same as the decomposition (32), and Algorithm 7 can be rewritten as follows.
f f f f f
8
Algorithm 9 ([14], BDDC on the reduced problem) Define the preconditioner r ∈ W′ 7−→
u∈ W by u= E (w +w ), where
Γ Γ∆ Π
c
c wΓ∆ ∈ WΓ∆ : a(wΓ∆,zΓ∆) = hr,EΓzΓ∆i, ∀zΓ∆ ∈ WΓ∆, (37)
w ∈ W : a(w ,z ) = hr,E z i, ∀z ∈ W . (38)
Π Π Π ΓΠ Γ ΓΠ ΓΠ Π
f f
Proposition 10 The BDDCfpreconditioner on the reduced problem ifn Algorithm 9 is the abstract
Multispace BDDC from Algorithm 2 with M = 2 and the spaces and operators given by
X = W, V = W , V = W , Q = Q = E . (39)
1 Γ∆ 2 Π 1 2 Γ
Also, the assumptions of Corcollary 4 arfe satisfied. f
Proof. Let r ∈ W′. Define the vectors v , i = 1,2 in Multispace BDDC by (3) with V and Q
i i i
given by (39). Let u, w , w be the quantities in Algorithm 9, defined by (37)-(38). Using the
Γ∆ Π
decomposition (36c), any w ∈ W can be written uniquely as w = w +w for some w and
Γ Γ Γ Γ∆ Π Γ∆
w corresponding to (3) as v = w and v = w , and u= E (w +w ).
Π 1 Γ∆ 2 Π Γ Γ∆ Π
To verify the assumptions off Corollary 4, note that the decomposition (36) is a−orthogonal,
a(·,·) is positive definite on both W and W as subspaces of W by (26), and E is a projection
Γ∆ Π Γ Γ
by (31).
Next, we present a BDDC fofrmulationfon the space U wfith explicit treatment of interior
functionsinthespaceU asin[4,13],i.e.,inthewaytheBDDCalgorithmwasoriginallyformulated.
I
Algorithm 11 ([4, 13], original BDDC) Define the preconditioner r ∈ U′ 7−→ u ∈ U as
follows. Compute the interior pre-correction
u ∈ U : a(u ,z ) = hr,z i, ∀z ∈ U . (40)
I I I I I I I
Set up the updated residual
r ∈ U′, hr ,vi = hr,vi−a(u ,v), ∀v ∈ U. (41)
B B I
Compute the substructure correction
u = Ew , w ∈ W : a(w ,z )= hr ,Ez i, ∀z ∈ W . (42)
∆ ∆ ∆ ∆ ∆ ∆ B ∆ ∆ ∆
Compute the coarse correction f f
u = Ew , w ∈ W : a(w ,z )= hr ,Ez i, ∀z ∈ W . (43)
Π Π Π Π Π Π B Π Π Π
Add the corrections f f
u = u +u .
B ∆ Π
Compute the interior post-correction
v ∈U :a(v ,z ) = a(u ,z ), ∀z ∈U . (44)
I I I I B I I I
Apply the combined corrections
u = u −v +u . (45)
B I I
9
The interior corrections (40) and (44) decompose into independent Dirichlet problems, one
for each substructure. The substructure correction (42) decomposes into independent constrained
Neumann problems, one for each substructure. Thus, the evaluation of the preconditioner requires
three problems to be solved in each substructure, plus solution of the coarse problem (43). In
addition, the substructure corrections can be solved in parallel with the coarse problem.
Remark 12 As it is well known [4], the first interior correction (40) can be omitted in the
implementation by starting the iterations from an initial solution such that the residual in the
interior of the substructures is zero,
a(u,z )−hf ,z i= 0, ∀z ∈ U ,
I X I I I
i.e., such that the error is discrete harmonic. Then the output of the preconditioner is discrete
harmonic and thus the errors in all the CG iterations (which are linear combinations of the original
error and outputs from the preconditioner) are also discrete harmonic by induction.
The following proposition will be the starting point for the multilevel case.
Proposition 13 The original BDDC preconditioner in Algorithm 11 is the abstract Multispace
BDDC from Algorithm 2 with M = 3 and the spaces and operators given by
X = U, V = U , V = (I −P)W , V = W , (46)
1 I 2 ∆ 3 Π
Q = I, Q = Q = (I −P)E, (47)
1 2 3
f f
and the assumptions of Corollary 4 are satisfied.
Proof. Let r ∈ U′. Define the vectors v , i = 1,2,3, in Multispace BDDC by (3) with the spaces
i
V given by (46) and with the operators Q given by (47). Let u , r , w , w , u , v , and u be
i i I B ∆ Π B I
the quantities in Algorithm 11, defined by (40)-(45).
First, with V = U , the definition of v in (3) with k = 1 is identical to the definition of u in
1 I 1 I
(40), so u = v .
I 1
Next, consider w ∈ W defined in (42). We show that w satisfies (3) with k = 2, i.e.,
∆ ∆ ∆
v = w . So, let z ∈ W be arbitrary. From (42) and (41),
2 ∆ ∆ ∆
f
fa(w∆,z∆)= hrB,Ez∆i = hr,Ez∆i−a(uI,Ez∆). (48)
Now from the definition of u by (40) and the fact that PEz ∈ U , we get
I ∆ I
hr,PEz i−a(u ,PEz ) = 0, (49)
∆ I ∆
and subtracting (49) from (48) gives
a(w ,z )= hr,(I −P)Ez i−a(u ,(I −P)Ez )
∆ ∆ ∆ I ∆
= hr,(I −P)Ez i,
∆
becausea(u ,(I −P)Ez ) =0byorthogonality. Toverify(3),itisenoughtoshowthatPw = 0;
I ∆ ∆
then w ∈ (I −P)W = V . Since P is an a-orthogonal projection, it holds that
∆ ∆ 2
f a(Pw∆,Pw∆)= a(w∆,Pw∆) = hrB,EPw∆i= 0, (50)
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