The analysis of FETI-DP preconditioner for full DG discretization of elliptic problems

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by  Maksymilian Dryja

Noname manuscript No.
(will be inserted by the editor)
The analysis of FETI-DP preconditioner for
full DG discretization of elliptic problems
Maksymilian Dryja · Juan Galvis ·
Marcus Sarkis
Received:date/Accepted:date
4
1
0
Abstract InthispaperadiscretizationbasedondiscontinuousGalerkin(DG)
2
methodforanelliptictwo-dimensionalproblemwithdiscontinuouscoefficients
n
is considered. The problem is posed on a polygonal region Ω which is a union
a
J ofN disjointpolygonalsubdomainsΩi ofdiameterO(Hi).Thediscontinuities
of the coefficients, possibly very large, are assumed to occur only across the
6
subdomaininterfaces∂Ωi.IneachΩi aconformingquasiuniformtriangulation
] withparametersh isconstructed.Weassumethattheresultingtriangulation
A i
in Ω is also conforming, i.e., the meshes are assumed to match across the
N
subdomain interfaces. On the fine triangulation the problem is discretized by
. a DG method. For solving the resulting discrete system, a FETI-DP type
h
t method is proposed and analyzed. It is established that the condition number
a
of the preconditioned linear system is estimated by C(1+max logH /h )2
m i i i
with a constant C independent of h , H and the jumps of coefficients. The
i i
[
methodiswellsuitedforparallelcomputationsanditcanbeextendedtothree-

1 dimensional problems. This result is an extension, to the case of full fine-grid
v DG discretization, of the previous result [SIAM J. Numer. Anal., 51 (2013),
1 pp. 400–422] where it was considered a conforming finite element method
6
9
0 M.Dryja
1. Department of Mathematics, Warsaw University, Banacha 2, 00-097 Warsaw, Poland.
This research was supported in part by the Polish Sciences Foundation under grant
0
2011/01/B/ST1/01179.
4
E-mail: [email protected]
1
: J.Galvis
v
DepartamentodeMatem´aticas,UniversidadNacionaldeColombia,Bogot´a,Colombia.
i
X E-mail:[email protected]
r M.Sarkis
a Department of Mathematical Sciences at Worcester Polytechnic Institute, 100 Institute
Road, Worcester, MA 01609, USA, and Instituto Nacional de Matem´atica Pura e Apli-
cada(IMPA),EstradaDonaCastorina110,CEP22460-320,RiodeJaneiro,Brazil.
E-mail:[email protected]
2 Dryja,Galvis&Sarkis
inside the subdomains and a discontinuous Galerkin method only across the
subdomain interfaces. Numerical results are presented to validate the theory.
Keywords Interior penalty discretization · discontinuous Galerkin · elliptic
problems with discontinuous coefficients · finite element method · FETI-DP
algorithms · preconditioners · AMS: 65F10, 65N20, 65N30
1 Introduction
In this paper we consider a boundary value problem for elliptic second order
partial differential equations with highly discontinuous coefficients and homo-
geneous Dirichlet boundary condition. The problem is posed on a polygonal
region Ω which is a union of disjoint two-dimensional polygonal subdomains
Ω ofdiameter O(H ).Weassume thatthispartition {Ω }N isgeometrically
i i i i=1
conforming, i.e., for all i and j with i(cid:54)=j, the intersection ∂Ω ∩∂Ω is either
i j
empty, a common corner or a common edge of Ω and Ω . We consider the
i j
case where the discontinuities of the coefficients are assumed to occur only
across ∂Ω . The problem is approximated by the symmetric interior penalty
i
discontinuousGalerkin(SIPDG)methodinsideeachΩ ,withh asameshpa-
i i
rameter.Themeshesareassumedtomatchacrosstheinterfaces∂Ω ∩∂Ω .In
i j
order to deal with the nonconformity of the DG spaces across ∂Ω , a discrete
i
problemisfurtherformulatedusingtheSIPDGmethodonthe∂Ω ;see[1,2,3,
i
14,18].ThemaingoalofthispaperistodeveloptheFETI-DPmethodologyfor
thisDGdiscretization.WeshowthatthedesignedFETI-DPmethodisalmost
optimal with rate of convergence independent of the coefficients jumps. The
analysis of the discussed precondicioner is based on the analysis used in [10].
For that we construct refinement of the fine meshes on each Ω and introduce
i
special interpolation operators which allow to switch from spaces of piecewise
linear functions defined on the fine mesh on Ω to spaces of piecewise linear
i
andcontinuousfunctionsdefinedontherefinementofthefinemeshonΩ and
i
vise-versa. The interpolation operators used in this paper are similar to those
introduced [4,19] and also used in [5]. Properties of the interpolations opera-
tors are proved in this paper. In this paper we extend our results published in
[10] to the full DG discretization of the considered problem. In [10] a FETI-
DP preconditioner was designed and analyzed for the problem discretized by
composed finite element and DG methods, i.e., by continuous Finite Element
Method(FEM)insideeachΩ andthesymmetricinteriorpenaltyDGmethod
i
ontheinterfaces∂Ω only,whilehereinthispaperaFETI-DPpreconditioner
i
for the case full DG discretization of the problem is designed and analyzed.
The proposed FETI-DP algorithm is essentially algebraic and does not re-
quire any interpolation operator for the design of the algorithm, therefore, it
can be naturally extended to three-dimentional problems and for high-order
and discretizations elasticity, Stokes and Maxwell. An interpolation operator
isonlyrequiredfortheanalysis,andtothebestofourknowledgeitisthefirst
complete analysis ever developped for any Neumann-Neumann type of dis-
cretizationsuchasFETI,FETI-DP,BDD,BDDCandNN,andforanyofthe
FETI-DPforfullDG 3
classical interior penalty DG discretizations considered in [1,2,3]. We expect
thattheanalysisdeveloppedherecanbeextendedtomoregeneralproblemsas
the one mentioned above. We also remark that a new technique based on two
interpolationoperatorswereintroducedtoavoidaconditiononthenumberof
elements that can touch a corner of a substructure interface,therefore, the al-
gorithmdevelopedherecanappliedalsoforsubdomainsgeneratedfromgraph
partitioners. Furthermore, we believe that such technique can be extended to
other types of full DG discretizations beyond [1,2,3].
For the developing of FETI-DP methods for the continuous FEM, see the
Introduction of [10] and the references therein. See also [12,13,15,16,17].
Inthispaper,thefullDGdiscreteproblemisreducedtotheSchurcomple-
ment problem with respect to unknowns on the interfaces of the subdomains
Ω . For that, discrete harmonic functions defined in a special way, i.e., in the
i
DG sense, are used. We note that there are unknowns on both sides of the
interfaces ∂Ω . This means that the unknowns on both sides of the interfaces
i
shouldbekeptasdegreesoffreedomofthelinearSchurcomplementsystemto
be solved. These issues characterize some of the main difficulties on designing
and analyzing FETI-DP type methods for full DG discretizations. Distinc-
tively from the classical conforming FEM discretizations, here a double layer
of Lagrange multipliers are needed on interfaces rather than a single layer of
Lagrange multipliers as normally is seen in FETI-DP for conforming FEMs.
Despite the fact we follow the FETI-DP abstract approach, which was aimed
to single layer of Lagrange multipliers, see for example [20], in this paper we
successfully overcome this difficulty.
The algorithm we develop in this paper is as follows. Let Γ(cid:48) be the union
i
of all edges E¯ and E¯ which are common to Ω and Ω , where E¯ and E¯
ij ji i j ij ji
refer to the Ω and Ω sides, respectively, see Figure 1. We note that each
i j
Γ(cid:48) hasinterfaceunknowns(degreesoffreedom)correspondingtonodalpoints
i
which are the endpoint of edges of fine triangulation belonging to ∂Ω \∂Ω
i
and E ⊂ ∂Ω . Unknowns corresponding to vertices of fine triangles which
ji j
intersect E¯ ⊂∂Ω and E¯ ⊂∂Ω by only one vertex are treated as interior
ij i ji j
unknowns. We now need to couple Γ(cid:48) with the other side of the interface
i
Γ(cid:48). We first impose continuity at the interface unknowns at the corners of
j
Γ(cid:48) (which are corners of Γ and common endpoints of E¯ ) with the interface
i i ji
unknownsatcornersoftheΓ(cid:48),seeFigure3.Theseunknownsarecalledprimal.
j
The remaining interface unknowns on Γ(cid:48) and Γ(cid:48) are called dual and have
i j
jumps, hence, Lagrange multipliers are introduced to eliminate these jumps,
see Figure 2. For the dual system with Lagrange multipliers, a special block
diagonalpreconditionerisdesigned.Itleadstoindependentlocalproblemson
Γ(cid:48) for 1 ≤ i ≤ N. It is proved that the proposed method is almost optimal
i
with a condition number estimate bounded by C(1+max logH /h )2, where
i i i
C does not depend on h , H , the number of subdomains Ω and the jumps in
i i i
the coefficients.
ThemethodcanbeextendedtofullDGdiscretizationsofthree-dimensional
problems.
4 Dryja,Galvis&Sarkis
Thepaperisorganizedasfollows.InSection2thedifferentialproblemand
a full DG discretization are formulated. In Section 3, the Schur complement
problem is derived using discrete harmonic functions defined in a special way
(in the DG sense). In Section 4, the so-called FETI-DP method is introduced,
i.e.,theSchurcomplementproblemisreformulatedbyimposingcontinuityfor
the primal variables and by using Lagrange multipliers at the dual variables,
andaspecialblockdiagonalpreconditionerisdefined.Themainresultsofthe
paperareTheorem1andLemma4.Section5isdevotedtosometechnicaltools
and auxiliary lemmas to analyze the FETI-DP preconditioner. The proofs of
these results are given in the Appendix A and B. In Section 6 numerical tests
are reported which confirm the theoretical results.
2 Differential and discrete problems
In this section we discuss the continuous problem and its DG discretization.
Theresultingdiscreteproblemistakenintoconsiderationforpreconditioning.
2.1 Differential problem
Consider the following problem: Find u∗ ∈H1(Ω) such that
ex 0
a(u∗ ,v)=f(v) for all v ∈H1(Ω), (1)
ex 0
where ρ >0 is a constant, f ∈L2(Ω) and
i
N (cid:90) (cid:90)
(cid:88)
a(u,v):= ρ (x)∇u·∇vdx and f(v):= fvdx.
i
i=1 Ωi Ω
We assume that Ω =∪N Ω and the substructures Ω are disjoint shaped
i=1 i i
regular polygonal subregions of diameter O(H ). We assume that the parti-
i
tion {Ω }N is geometrically conforming, i.e., for all i and j with i (cid:54)= j, the
i i=1
intersection∂Ω ∩∂Ω iseitherempty,acommoncorneroracommonedgeof
i j
Ω andΩ .Forclaritywestressthathereandbelowtheidentifieredge means
i j
a curve of continuous intervals and its two endpoints are called corners. The
collection of these corners on ∂Ω are referred as the set of corners of Ω . Let
i i
us denote E¯ := ∂Ω ∩∂Ω as an edge of ∂Ω and E¯ := ∂Ω ∩∂Ω as an
ij i j i ji j i
edge of ∂Ω . Sometimes we use the notation E and E to refer the sets
j ijh jih
of nodal points of the triangulation on E and E inherit from Ti and Tj,
ij ji h h
respectively, where the triangulations are defined below. Additionally we use
the notation E¯ and E¯ when the endpoints are included. Let us denote
ijh jih
by Ji,0 the set of indices j such that Ω has a common edge E with Ω . To
H j ji i
take into account edges of Ω which belong to the global boundary ∂Ω, let us
i
introduce a set of indices Ji,∂ to refer these edges. The set of indices of all
H
edges of Ω is denoted by Ji =Ji,0∪ Ji,∂.
i H H H
FETI-DPforfullDG 5
2.2 Discrete problem
Let us introduce a shape regular and quasiuniform triangulation (with trian-
gular elements) Ti on Ω and let h represent its mesh size. The resulting
h i i
triangulation on Ω is matching across ∂Ω . Let
i
(cid:89)
X (Ω ):= X
i i τ
τ∈Ti
h
be the product space of finite element (FE) spaces X which consist of lin-
τ
ear functions on the element τ belonging to Ti. We note that a function
h
u ∈X (Ω )allowsdiscontinuitiesacrosselementsofTi.Wealsonotethatwe
i i i h
do not assume that functions in X (Ω ) vanish on ∂Ω.
i i
The global DG finite element space we consider is defined by
N
(cid:89)
X(Ω)= X (Ω )≡X (Ω )×X (Ω )×···×X (Ω ). (2)
i i 1 1 2 2 N N
i=1
We define Ei,0 as the set of edges of the triangulation Ti which are inside
h h
Ω , and by Ei,j, for j ∈Ji, the set of edges of the triangulation Ti which are
i h H h
on E . An edge e ∈ Ei,0 is shared by two elements denoted by τ and τ of
ij h + −
Ti with outward unit normal vectors n+ and n−, respectively, and denote
h
1
{ρ∇u}= (ρ ∇u +ρ ∇u ) and [u]=u n++u n−.
2 τ+ τ+ τ− τ− τ+ τ−
An edge e ∈ Ei,∂ is shared by one element denoted by τ with outward unit
h
normal vectors n, and denote
{ρ∇u}=ρ ∇u and [u]=u n.
τ τ τ
The discrete problem we consider by the DG method is of the form: Find
u∗ ={u∗}N ∈X(Ω) where u∗ ∈X (Ω ), such that
i i=1 i i i
a (u∗,v)=f(v) for all v ={v }N ∈X(Ω), (3)
h i i=1
where the global bilinear from a and the right hand side f are assembled as
h
N N (cid:90)
(cid:88) (cid:88)
a (u,v):= a(cid:48)(u,v) and f(v):= fv dx. (4)
h i i
i=1 i=1 Ωi
Here, the local bilinear forms a(cid:48), i=1,...,N, are defined as
i
a(cid:48)(u,v):=a (u ,v )+s (u ,v )+p (u,v)+s (u,v)+p (u,v) (5)
i i i i i,0 i i i,0 i,∂ i,∂
where a is the local energy bilinear form,
i
(cid:90)
(cid:88)
a (u ,v ):= ρ ∇u ·∇v dx. (6)
i i i i i i
τ∈Ti τ
h
6 Dryja,Galvis&Sarkis
The (interior edges) symmetrized bilinear form s is defined by
i,0
(cid:90)
(cid:88)
s (u ,v ):=− {ρ ∇u }·[v ]+{ρ ∇v }·[u ]ds, (7)
i,0 i i i i i i i i
e∈Ei,0 e
h
and the (interior edges) penalty bilinear form is given by
p (u,v):= (cid:88) (cid:90) δρi[u ].[v ]ds. (8)
i,0 h i i
e∈Ei,0 e e
h
The corresponding symmetric and penalty form over the local interface edges
are given by
s (u,v):= (cid:88) (cid:88) (cid:90) 1 (cid:18)ρ ∂ui(v −v )+ρ ∂vi(u −u )(cid:19) ds (9)
i,∂ l i ∂n j i i∂n j i
j∈JHi e∈Ehi,j e ij
and
p (u,v):= (cid:88) (cid:88) (cid:90) δ ρi(u −u )(v −v )ds (10)
i,∂ l h i j i j
j∈JHi e∈Ehi,j e ij e
respectively. Here and above, h denotes the length of the edge e. When
e
j ∈ Ji,0 we set l = 2, while when j ∈ Ji,∂ we denote the boundary edges
H ij H
E ⊂ ∂Ω by E and set l = 1, and on the artificial edge E ≡ E we
ij i i∂ i∂ ji ∂i
set u =0 and v =0. The partial derivative ∂ denotes the outward normal
∂ ∂ ∂n
derivative on ∂Ω and δ is the penalty positive parameter.
i
Remark 1 Thediscreteformulationusedhereisusefulfortheadequateformu-
lationofourFETI-DPmethod.Wenotethat(9)and(10)canalsobewritten
in the same form as in (7) and (8) without the term (cid:96) (since now the edges
ij
arecountedonce).WenotealsothatothersDGformulationsfordiscontinuous
coefficients can also be considered [3,6]. We note that the design of FETI-DP
methodsforthoseformulationsarethesame,andtheanalysisaresimplemod-
ifications of the proofs we present here in this paper. See for instance [7,9,8]
where a formulation based on harmonic averages of the coefficients is studied.
Some details for these case as well as numerical experiments will be presented
elsewhere.Wenotethatthree-dimensionalversionscanalsobeformulatedand
analyzed by extending naturaly some ideas from this paper and from [11].
For u = {u }N ,v = {v }N ∈ X(Ω), let us introduce the local positive
i i=1 i i=1
bilinear forms
d (u,v):=a (u,v)+p (u,v)+p (u,v) (11)
i i i,0 i,∂
and the global positive bilinear form assembled as
N
(cid:88)
d (u,v):= d (u,v). (12)
h i
i=1
FETI-DPforfullDG 7
Note that the norm defined by d (·,·) is a broken norm in X(Ω) with weights
h
given by ρ , δρi and δ ρi. For u = {u }N ∈ X(Ω), this discrete norm is
i he lij he i i=1
defined by (cid:107)u(cid:107)2=d (u,u).
h h
It is known that there exists a δ = O(1) > 0 and a positive constant c,
0
which does not depend on ρ , H , h and u , such that for every δ ≥ δ we
i i i i 0
obtain |s (u,u)|≤cd (u,u) and therefore, the following lemma is valid.
i i
Lemma 1 There exists δ >0 such that for δ ≥δ and for all u∈X(Ω) we
0 0
have, in each subdomain,
γ d (u,u)≤a(cid:48)(u,u)≤γ d (u,u), 1≤i≤N, (13)
0 i i 1 i
and also we have the following global bilinear forms inequality
γ d (u,u)≤a (u,u)≤γ d (u,u). (14)
0 h h 1 h
Here, γ and γ are positive constants independent of the ρ ,h , H and u.
0 1 i i i
The proof of this lemma is a modification of the proof of [7, Lemma 2.1], [9]
or [6, Theorem 4.1], therefore it is omitted.
Lemma 1 implies that the discrete problem (3) is elliptic and continuous,
therefore, the solution exists and it is unique and stable. An optimal a priori
errorestimateofthismethodwasestablishedin[1,2]forthecontinuouscoeffi-
cientcase.WementionherethatLemma1togetherwithLemma7,seebelow,
are going to be fundamental for establishing condition number estimates for
the FETI-DP preconditioned system developed in the remaining sections.
3 Schur complement matrices and discrete harmonic extensions
The first step of many iterative substructuring solvers, such as the FETI-DP
method that we consider in this paper, requires the elimination of unknowns
interior to the subdomains. In this section, we describe this step for DG dis-
cretizations.
We introduce some notation and then formulate (3) as a variational prob-
lem with constraints. Associated to a subdomain Ω , we define the extended
i
subdomain Ω(cid:48) by
i
Ω(cid:48) :=Ω (cid:91){∪ E¯ }
i i j∈Ji,0 ji
H
i.e., it is the union of Ω and the E¯ ⊂∂Ω such that j ∈Ji,0, and the local
i ji j H
interfaces Γ by
i
Γ :=∂Ω \∂Ω and Γ(cid:48) :=Γ (cid:91){∪ E¯ }.
i i i i j∈Ji,0 ji
H
8 Dryja,Galvis&Sarkis
Fig. 1 IllustrationoftheclassificationofdegreesoffreedomrelatedtosubdomainΩi.
We also introduce the sets
N N N
(cid:91) (cid:89) (cid:89)
Γ := Γ , Γ(cid:48) := Γ(cid:48), I :=Ω(cid:48)\Γ(cid:48) and I := I . (15)
i i i i i i
i=1 i=1 i=1
Associated to these sets, we classify degrees of freedom (DG nodal values)
on Ω(cid:48). We illustrate this classification (along with further classifications to be
i
introduced later) in Figure 1:
– Γ -degresoffreedom:ThenodalpointswhichareendpointsofedgesofEi,j
i h
for j ∈Ji,0.
H
– Degrees V : The nodal points which are both an endpoint of an edges of
i
Ei,j for j ∈Ji,0 and a corner of Ω .
h H i
– Γ(cid:48)-degreesoffreedom:ItistheunionofthedegreesΓ andthenodalpoints
i i
which are endpoints of edges of Ej,i for j ∈Ji,0.
h H
– V(cid:48)-degrees of freedom: It is the union of the degrees V , and the nodal
i i
points which are both an endpoint of an edge of Ej,i for j ∈ Ji,0 and a
h H
corner of Ω .
j
– I -degreesoffreedom:Thenodalpointswhichareverticesofelementsτ of
i
Ti and are not endpoints of an edge of τ in Ei,j for j ∈Ji,0.
h h H
– Ω -degrees of freedom: The union of Γ and I degrees of freedom.
i i i
– Ω(cid:48)-degrees of freedom: The union of Γ(cid:48) and I degrees of freedom.
i i i
FETI-DPforfullDG 9
Remark 2 Nodal points of elements in Ti which intersect E¯ by only one
h ij
vertexaretreatedasI degreesoffreedom.ThetraceofthebasisDGfunctions
i
associated to these nodal points are zero almost everywhere on Γ , hence, the
i
analysis shows that it is convenient to treat these nodes as as I -degrees of
i
freedomasdefinedabove.Forthesamereason,nodalpointsofelementsofTj
h
whichintersectE¯ byvertexonlyforj ∈Ji,0,arenotincludedasΩ(cid:48)-degrees
ji H i
offreedom.Wepointout,however,thatwecouldhaveconsideredthesenodes
pointsasΓ andΩ(cid:48)-degreesoffreedom,respectivelyaswell,andbyintroducing
i i
alsoLagrangemultiplierstoforcecontinuityatthesetypeofnodalvalues,and
theneliminatingtheseLagrangemultipliersandalsothesedegreesoffreedom.
This approach would be equivalent to the approach we consider and analyze
here in this paper.
We mention that if we refer to the classical FETI-DP design, i.e., for con-
tinuous FEM, the set I corresponds to the set of interior degrees of freedom
(thatareblock-uncoupled)andcanbeeliminatedinafirststep.Moreover,the
Γ corresponds to the global interface with the original coupling and Γ(cid:48) corre-
spondstothe(block-wise)torninterface.WerecallthattheclassicalFETI-DP
design extends the original problem to a problem in the torn interface space
and then construct and implement suitable restrictions to interconnect back
some of the broken coupling between degrees of freedom. We now describe
these steps for DG discretization in detail. First we introduce the correspond-
ing functions spaces.
Let W (Ω(cid:48)) be the FE space of functions defined by values on Ω(cid:48)
i i i
W (Ω(cid:48))=W (Ω )× (cid:89) W (E¯ ), (16)
i i i i i ji
j∈Ji,0
H
where W (Ω ):=X (Ω ) and W (E¯ ) is the trace of the DG space X (Ω ) on
i i i i i ji j j
E¯ ⊂∂Ω forallj ∈Ji,0.Afunctionu(cid:48) ∈W (Ω(cid:48))isdefinedbytheΩ(cid:48)-degrees
ji j H i i i i
of freedom. Below, we denote u(cid:48) by u if it is not confused with functions of
i i
X (Ω ). A function u ∈W (Ω(cid:48)) is represented as
i i i i i
u ={(u ) ,{(u ) } },
i i i i j j∈Ji,0
H
where (ui)i := ui|Ωi (ui restricted to Ωi) and (ui)j := ui|E¯ji (ui restricted
to E¯ ). Here and below we use the same notation to identify both DG func-
ji
tions and their vector representations. Note that a(cid:48)(·,·), see (5), is defined on
i
W (Ω(cid:48))×W (Ω(cid:48)) with corresponding stiffness matrix A(cid:48) defined by
i i i i i
a(cid:48)(u ,v )=(cid:104)A(cid:48)u ,v (cid:105) u ,v ∈W (Ω(cid:48)), (17)
i i i i i i i i i i
where (cid:104)u ,v (cid:105) denotes the (cid:96) inner product of nodal values associated to
i i 2
the vector space in consideration. We also represent u ∈ W (Ω(cid:48)) as u =
i i i i
(u ,u ) where u represents values of u at nodal points on Γ(cid:48) and u
i,I i,Γ(cid:48) i,Γ(cid:48) i i i,I
at the interior nodal points in I , see (15). Hence, let us represent W (Ω(cid:48)) as
i i i
10 Dryja,Galvis&Sarkis
the vector spaces W (I )×W (Γ(cid:48)). Using the representation u =(u ,u ),
i i i i i i,I i,Γ(cid:48)
the matrix A(cid:48) can be represented as
i
(cid:18) A(cid:48) A(cid:48) (cid:19)
A(cid:48) = i,II i,IΓ(cid:48) , (18)
i A(cid:48) A(cid:48)
i,Γ(cid:48)I i,Γ(cid:48)Γ(cid:48)
where the block rows and columns correspond to the nodal points of I and
i
Γ(cid:48), respectively.
i
The Schur complement of A(cid:48) with respect to u is of the form
i i,Γ(cid:48)
S(cid:48) :=A(cid:48) −A(cid:48) (A(cid:48) )−1A(cid:48) (19)
i i,Γ(cid:48)Γ(cid:48) i,Γ(cid:48)I i,II i,IΓ(cid:48)
and introduce the block diagonal matrix S(cid:48) = diag{S(cid:48)}N . Note that S(cid:48) sat-
i i=1 i
isfies
(cid:104)S(cid:48)u ,u (cid:105) = min a(cid:48)(w ,w ), (20)
i i,Γ(cid:48) i,Γ(cid:48) i i i
wheretheminimumistakenoverw =(w ,w )∈W (Ω(cid:48))suchthatw =
i i,I i,Γ(cid:48) i i i,Γ(cid:48)
u onΓ(cid:48).Thebilinearforma(cid:48)(·,·)issymmetricandnonnegative,seeLemma
i,Γ(cid:48) i i
1. The minimizing function satisfying (20) is called discrete harmonic in the
senseofa(cid:48)(·,·)orinthesenseofH(cid:48).Anequivalentdefinitionoftheminimizing
i i
function H(cid:48)u ∈W (Ω(cid:48)) is given by the solution of
i i,Γ(cid:48) i i
o
a(cid:48)i(Hi(cid:48)ui,Γ(cid:48),vi,Γ(cid:48))=0 vi,Γ(cid:48) ∈Wi(Ωi(cid:48)), (21)
H(cid:48)u =u on Γ(cid:48), (22)
i i,Γ(cid:48) i,Γ(cid:48) i
o
where Wi(Ωi(cid:48)) is the subspace of Wi(Ωi(cid:48)) of functions which vanish on Γi(cid:48). We
note that for substructures Ω which intersect ∂Ω by edges, the nodal values
i
of W (Ω(cid:48)) on ∂Ω \Γ ⊂∂Ω are treated as unknowns and belong to I .
i i i i i
Let us introduce the product space
N
(cid:89)
W(Ω(cid:48)):= W (Ω(cid:48)), (23)
i i
i=1
i.e., u ∈ W(Ω(cid:48)) means that u = {u }N where u ∈ W (Ω(cid:48)); see (16) for
i i=1 i i i
the definition of W (Ω(cid:48)). Recall that we write (u ) = u (u restricted
i i i i i|Ωi i
to Ωi) and (ui)j = ui|E¯ji (ui restricted to E¯ji). Using the representation
u = (u ,u ) where u ∈ W (I ) and u ∈ W (Γ(cid:48)), see (18), let us
i i,I i,Γ(cid:48) i,I i i i,Γ(cid:48) i i
introduce the product space
N
(cid:89)
W(Γ(cid:48)):= W (Γ(cid:48)), (24)
i i
i=1
i.e., u ∈ W(Γ(cid:48)) means that u = {u }N where u ∈ W (Γ(cid:48)). The
Γ(cid:48) Γ(cid:48) i,Γ(cid:48) i=1 i,Γ(cid:48) i i
space W(Γ(cid:48)) which was defined on Γ(cid:48) only, is also interpreted below as the
subspaceofW(Ω(cid:48))offunctionswhicharediscreteharmonicinthesenseofH(cid:48)
i