Gradient-prolongation commutativity and graph theory

0.15 MB·English

by  François Musy

Numerical Analysis
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2 Gradient-prolongation commutativity and graph theory

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Francois Musya, Laurent Nicolasb, Ronan Perrussela,b
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8 aInstitut Camille Jordan, Ecole Centrale de Lyon, F-69134 Ecully Cedex
1 bCentre de G´enie E´lectrique de Lyon, Ecole Centrale de Lyon, F-69134 Ecully Cedex

Received*****; accepted afterrevision+++++
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Presentedby
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a Abstract
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Thisnotegivesconditionsthatmustbeimposedtoalgebraicmultileveldiscretizationsinvolvingatthesametime
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nodalandedgeelementssothatagradient-prolongation commutativitycondition willbesatisfied;thiscondition

1 isveryimportant,sinceit characterizes thegradientsof coarsenodal functionsinthecoarse edgefunction space.
v They will beexpressed using graph theory and they providetechniquesto compute approximation bases at each
5 level. To cite this article: A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
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5 R´esum´e
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Commutativit´e entre gradient et prolongement et th´eorie des graphesCettenotedonnedesconditions
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quidoiventˆetreimpos´eesauxdiscr´etisationsmultiniveaualg´ebriquesen´el´ementsfinisnodauxetd’arˆetedefa¸con
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0 `a assurer la commutativit´e entre gradient et prolongement; cette relation importante caract´erise les gradients
/ desfonctions nodales grossi`eres dansl’espace desfonctions d’arˆetegrossi`eres. Ces conditions seront exprim´eesen
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termedegraphesetellespermettentd’introduiredesm´ethodesdecalculdesbasesd’approximationauxdiff´erents
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a niveaux.Pour citer cet article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
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Version franc¸aise abr´eg´ee
L’approximationnum´erique du champ´electrique ou magn´etique utilise fr´equemment les ´el´ements finis
d’arˆete dont la relation avec les ´el´ements finis nodaux traduit des propri´et´es importantes au niveau
discret [1]. Dans ce qui suit, nous consid`ererons les ´el´ements de plus bas degr´e : P en nodal et ordre 1
1
incomplet pour les arˆetes. D`es qu’on traite des probl`emes de grande taille, une strat´egie multiniveau est
Email addresses: [email protected] (FrancoisMusy),[email protected] (LaurentNicolas),
[email protected] (RonanPerrussel).
PreprintsubmittedtoElsevierScience 2f´evrier2008
un choix int´eressant. Pour les syst`emes provenant de discr´etisations par ´el´ements finis d’arˆete, Hiptmair
a introduit des m´ethodes multiniveau pour une hi´erarchiede maillages emboˆıt´es [2].
Cependant, dans des applications r´ealistes,on ne disposeg´en´eralementpas de maillages structur´es.La
strat´egiemultiniveau alg´ebriqueva donc s’imposer: il s’agitde d´efinirdes fonctions grossi`eresnodales et
d’arˆetegrˆaceauxcontributionsdepaquetsdefonctionsfinesnodalesetd’arˆete;lescombinaisonslin´eaires
(1a) et (1b) d´efinissent respectivement ces fonctions grossi`eresnodales et d’arˆete.
Parconstructionlesgradientsdesfonctionsnodalesfinesappartiennent`al’espacedesfonctionsd’arˆete
fines ce que traduit la relation (2). Dans cette relation, Gh est la matrice d’incidence arcs-sommets du
graphe orient´e naturellement associ´e au maillage de travail. Les orientations des arcs sont arbitraires.
Pouradapterauxm´ethodesalg´ebriquesleslisseursdesm´ethodesg´eom´etriques,ReitzingeretScho¨berl[3]
ont introduit une repr´esentationexplicite des gradients des fonctions grossi`eresnodales dans la base des
fonctions grossi`eresd’arˆete, donn´ee par la relation (3) ou` GH est une matrice d’incidence arcs-sommets.
En regroupant les relations (1) `a (3), nous obtenons la relation matricielle (4). La matrice α est
construiteparexempleparlesm´ethodesd´efiniesdans[4]quipermettentd’obtenirlesfonctionsgrossi`eres
nodales comme partition de l’unit´e et de contraindre leurs supports `a ˆetre inclus dans des ensembles
g´eom´etriquesconvenablement choisis.
ConnaissantGh etα,noussouhaitonschoisirGH commematriced’incidencearcs-sommetsd’ungraphe
orient´e SH. Nous donnons dans cette note une condition n´ecessaire et suffisante sur ce graphe,la propo-
sition (2.3), qui assure l’existence d’une solution de (4). Eneffet, nous associerons,par un proc´ed´ed´ecrit
dans la partie en anglais, `a chaque arˆete fine un sous-graphe du graphe grossier, qui doit ˆetre connexe.
La connaissance de ces sous-graphes donne les degr´es de libert´e disponibles pour d´eterminer des fonc-
tionsd’arˆetegrossi`erescompatiblesaveclesfonctionsnodalesgrossi`eres;enr´esolvantunprobl`emedeflot
sur ces sous-graphes,voir (14), nous pouvons alors construire la matrice β (Section 4).
1. Introduction
Numericalapproximationofelectricormagneticfieldusesoftenedgefiniteelementswhoserelationwith
nodal finite elements contains important properties at discrete level [1]. In this note we restrict ourselves
to lowestorderapproximation:P for nodalelements andincomplete order1 for edge elements.Inorder
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to solve large problems, multilevel methods are an attractive choice. While, for systems coming from
edge element discretisation, Hiptmair [2] proposed multilevel methods using nested meshes, engineering
applications do not usually provide structured meshes. Therefore, algebraic multilevel methods are an
interesting option: we have to build coarse nodal and edge functions by using aggregates of fine nodal
and edge functions. If (φh) and (λh) respectively denote fine nodal and edge bases, the
p p=1,...,Nh i i=1,...,Eh
following linear combinations define coarse nodal and edge functions:
Nh
φH = α φh, ∀n∈{1,...NH}, (1a)
n pn p
Xp=1
Eh
λH = β λh, ∀e∈{1,...,EH}. (1b)
e ie i
Xi=1
By construction, the gradients of fine nodal functions belong to the space of fine edge functions:
Eh
∀p∈{1,...,Nh}, grad(φh)= Ghλh, (2)
p ip i
Xi=1
2
where Gh is the edge-node incidence matrix of the digraph naturally associated with the initial mesh.
The orientation of the edges can be arbitrarily chosen.
In [3], Reitzinger and Scho¨berl deduced their smoother from the matrix GH involved in the relation:
EH
∀n∈{1,...,NH}, grad(φH)= GHλH, (3)
n en e
Xe=1
whichstatesthatthe gradientsofthe coarsenodalfunctions mustalsobelongto the spaceofcoarseedge
functions. The matrix GH is an edge-node incidence matrix as in the structured case. Relation (3) does
not guarantee the efficacy of the algebraic multilevel method but it leads to relevant strategies.
Gathering Equations (1), (2) and (3), we obtain the matrix relation:
Ghα=βGH. (4)
The matrix α is constructed following for instance the methods defined in [4], which provides a family of
coarse nodal functions, making up a partition of unity, whose supports satisfy appropriate conditions.
Knowingtheleft-handsideof (4),wewanttochooseGH asanedge-nodeincidencematrixofadigraph
SH,andwewillgiveconditionsonthecoarsegraphSH,whichensuretheexistenceofamatrixβsatisfying
(4). Moreover,the proof of the propositionindicates how to choose the degreesof freedomwhich enables
us to define the coarse edge functions. It also helps us to construct β.
2. Notation and statement of the problem
Let (L ) be sets of indices in {1,...,Nh} such that:
n n=1,...,NH
NH
L ={1,...,Nh}. (5)
n
n[=1
The matrix α describes the coarsenodal basis;we assume thatit is has been previouslycomputed and
it has the following properties:
– the coarse nodal functions make up a partition of unity, which can be algebraically stated as:
NH
∀p∈{1,...,Nh}, α =1, (6)
pn
nX=1
– inordertorestrictthesupportofeachcoarsebasisfunctionφH,theindicesofthenon-zerocomponents
n
of φH are included in the set L , i.e.:
n n
p∈{1,...,Nh}\L =⇒ α =0. (7)
n pn
The fine nodal function φh contributes to the coarse nodal function φH if p belongs to L .
p n n
We haveareciprocalset-valuedfunction L:the setL isthe setofcoarsenodalfunction indicesto which
p
the fine nodal function φh contributes. For the fine graph in Figure 1(a), we set L = {1,2,3,4,5,6,7},
p e e 1
L ={5,6,8,9,13,14}and L ={7,8,10,11,12}.One obtains, for instance, the set L ={1,3}.
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We define twofamilies ofsets offine edgefunctionindices.Wewill denotea directedfine edgeiby pqh
e
where p and q are respectively the starting andending nodes of the edge i. A similar notation is used for
a directed coarse edge e=mnH.
The set C is the set of indices of fine edges which have an extremity in L :
n n
C = i∈{1,...,Eh}:i=pqh, p∈L or q ∈L . (8)
n n n
(cid:8) (cid:9)
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The fine edge function λh contributes to the gradient of the coarse nodal function φH if i belongs to
i n
C . Indeed, for the directed fine edge i=pqh, Gh is equal to −1 if r =p and +1 if r =q. Moreover,if p
n ir
and q are not in L , the components α and α vanish according to (7); therefore:
n pn qn
i∈{1,...,Eh}\Cn =⇒ (Ghα•n)i =0, (9)
where α•n denotes the n-th column of α. The reciprocal set-valued function C is such that Ci is the set
of coarse nodal function indices to whose gradientthe fine edge function λh contributes. On Figure 1(b),
i e e
the fine edges are numbered, set C is highlighted and we can note, for instance, the set C ={1,3}.
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Let e=mnH be an edge of the coarse graph SH; we define:
e
I =C ∩C . (10)
e n m
By analogy with the structured case and for restricting the support of λH, we enforce:
e
i∈{1,...,Eh}\I =⇒ β =0. (11)
e ie
The fine edge function λh contributes to the coarse edge function λH if i belongs to I . The set-valued
i e e
function I is such that I is the set of coarse edge function indices to which the fine edge function λh
i i
contributes.The coarsegraphinFigure 1(c) is relatedto the fine in Figure1(a). SetI is representedin
e e e3
Figure 1(d).
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14 16
5 5 2
1 9 13 13 15
2 4 6 8 12 4 3 6 7 1224 17 1918 1 e1 e2
1 2 8 10 23 20 21 e
3 7 10 11 9 11 22 3 3
(a) Fine graph with node in- (b) Finegraphwithedgeindices. (c) Coarsegraph. (d) SetIe3isrepresentedbybold
dices. Sets (Ln)n=1,...,3 are sur- Set C3 is represented by bold edges.
rounded. edges.
Figure1.Representationofthefineandcoarsegraphes,sets(Ln)n=1,...,3,C3 andIe3.
The following statement can be easily deduced from (8) and the definition of Gh:
Lemma 2.1 If i denotes the edge pqh, C =L ∪L .
i p q
In order to simplify notations, we introduce the set F = i ∈ {1,...,Eh} : I 6= ∅ , since some fine
e e e i
edge functions might not contribute to any coarse edge funct(cid:8)ions. (cid:9)
e e
Foranyfineedgei,letSH,i betheinducedsubgraphdefinedbyC :theverticesofSH,i arethevertices
i
of SH, which are indexed by the elements of C and the edges of SH,i are those edges of SH whose
i
e
extremities are vertices of SH,i.
e
The following lemma is a direct consequence of definition (10):
Lemma 2.2 For any edge i∈F, the edges of SH,i are those edges of SH which are indexed by I .
i
We may now state precisely our main result, which gives a necessary and sufficient condition on the
e e
coarse graph SH permitting the resolution of (4):
Proposition 2.3 For all matrices α satisfying conditions (6) and (7), there exists a matrix β satisfying
(11) and solving (4) iff for all i, the induced subgraph SH,i is connected.
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3. The essential steps of the proof
First step. Many relations in (4) reduce to 0=0: this is the case for n∈/ C .
i
Indeed, according to (9) the (i,n) coefficient of the right-hand side of (4) vanishes.
e
Conversely, if e does not belong to I , according to (11) and the definition of I , β vanishes and:
i i ie
e EH e
β GH = β GH. (12)
ie en ie en
Xe=1 eX∈Ii
H
On the other hand if the directed coarse edge e denotedeby lm belongs to Ii, Lemma 2.2 implies that l
and m belongs to C . However, for GH not to vanish for all e, m or l must be equal to n, which means
i en e
that n belong to C , and this contradicts the assumption n∈/ C .
ie i
Secondstep.Welookatalltheotherequations,i.e.thoseforwhichn∈C .Wenotethat(12)remains
i
e e
and that the edges indexed by I are precisely those of the graph SH,i according to Lemma 2.2.
i e
We assume now i ∈ F and we define GH,i as the edge-node incidence matrix of SH,i and the (i,n)
e
equation of (4) is rewritten:
e
β GH,i =Θ where Θ = Ghα . (13)
ie en i,n i,n ir rn
eX∈Ii rX∈Ln
This could be satisfied for all couples (i,n) such that n ∈ {1,...,NH} and i ∈ C or equivalently
e n
i∈{1,...,Eh} and n∈Ci. For a fixed i, we may write that βi•, the i-th row of β satisfies the system:
e β GH,i =Θ , ∀n∈C . (14)
ie en i,n i
X
e∈Ii e
Thus, we solve line by line for β and we see that (14) is a flow problem whose solution is of the form:
e
′ ′′
βi• =βi•+βi•. (15)
with (β′′)t ∈ker(GH,i)t and β′ a particular solution.
i• i•
More precisely, let Ti be a spanning tree for SH,i; call Γi the edge-node incidence matrix associated
with Ti;weknow thatΓi has|C |−1 rowsand|C |columns,anditis ofrank|C |−1.We choosea vertex
i i i
m in Γi and we solve the system:
e e e
β′ Γi =Θ , ∀n∈C \{m}, (16)
ie en i,n i
e∈XE(Ti)
e
where E(Ti) denotes the set of indices of the edges of Ti. The system (16) is a regular system of |C |−1
i
equations with |C |−1 unknowns, and we put β′ equal to 0 if e is in I \E(Ti).
i ie i e
Itremainsto showthat the forgottenequationofindexm in(16)is automaticallysatisfied.Indeed,by
e e
denoting i by pqh, we sum the right-hand side of (14) with respect to n∈C :
i
Θ = α −α =0, e (17)
i,n qn pn
X X
n∈Ci n∈Lp∪Lq
since in view of (6) and (7), αepn = αepne=1 and αqn = αqn =1.
X X X X
n∈Lp∪Lq n∈Lp n∈Lp∪Lq n∈Lq
On the other hand, if we sum the left-hand side of (14) with respect to n∈C , we obtain:
i
e e e e e e
β GH,i = β GH,i =0, e (18)
ie en ie en
X X X X
n∈Cie∈Ii e∈Ii n∈Ci
e e 5 e e
since each line of GH,i contains only two non-zero coefficients +1 and −1.
EH
If i∈/ F, |C |=|L |=|L |=1 and the relation β GH =Θ is satisfied from (12) and (17).
i p q ie en i,n
Xe=1
e e e e
Now we assume that SH,i is not connected and we denote by C the nodes of a connected component.
i
EH
For the same reasons as in (18), if β satisfies (11) one gets b β GH =0.
ie en
nX∈CiXe=1
Nh
However we can construct a matrix α satisfying (6) and (7)bsuch that Ghα 6=0. In fact, in
ir rn
nX∈CiXr=1
view of (7), for i=pqh we can write:
b
Nh
Ghα = α −α = α − α . (19)
ir rn qn pn qn pn
nX∈CiXr=1 nX∈Ci CXi∩Lq CXi∩Lp
Since Ci is strictly includbed in Lp∪Lq, webwill have Ci∩Lbp 6=e Lp or Cbi ∩eLq 6= Lq. Depending on the
situation, we can construct a suitable matrix α such that:
b e e b e e b e e
α =1 and α =0 or α =0 and α =1 .
qn pn qn pn
(cid:18) (cid:19) (cid:18) (cid:19)
X X X X
Ci∩Lq Ci∩Lp Ci∩Lq Ci∩Lp
For these matrices α, the condition defined by (4) cannot be ensured.
b e b e b e b e
4. Construction of the coarse edge functions
ForacoarsegraphsatisfyingtheconditionofProposition2.3andbyusingthedecomposition(15),any
compatible matrice can be written β = β′ +β′′, where the complete matrices are defined by gathering
the lines of index i βi•, βi′• and βi′′•. The computation of each βi′• can be done by solving system (16).
As concerns β′′, a basis of the kernel of (GH,i)t is given by a set of k independent cycles of SH,i. Then,
i• i
k degrees of freedom should be determined by minimising an appropriate energy functional; such
i∈F i
Pa problem is introduced in [5] and can be related to explanations in [4].
Weethank Michelle Schatzman for many fertile discussions.
References
[1] R.Hiptmair,Finite elements incomputational electromagnetism,ActaNumer.,11:237–339, 2002.
[2] R.Hiptmair,Multigrid method forMaxwell’s equations,SIAMJ.Numer.Anal.,36:204–225, 1999.
[3] S.ReitzingerandJ.Scho¨berl,Analgebraic multigridmethod forfinite element discretizations withedge elements,
Numer.LinearAlgebraAppl.,9:223–238, 2002.
[4] J.Mandel,M.Brezina andP. Vanˇek,Energy optimization of algebraic multigrid bases,Computing,62(3):205–228,
1999.
[5] F.Musy,L.Nicolas,R.PerrusselandM.Schatzman,Compatible coarse nodal and edge elementsthrough energy
functionals,UMRMAPLY,internalreport394,2004.http://maply.univ-lyon1.fr/∼perrussel/report.pdf.
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