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nodal interpolation operator Given a finite
element space V with set X of nodal degrees
h h
of freedom, we can define a projection
N I : S ∪ V → V ,φ(I u) = φ(u) ∀φ ∈ X ,
h
h
h
h
h
where S is a space of sufficiently smooth func-
tions/vector fields for which the evaluation of
n
Navier-Stokes equations Let U ⊂ R be degrees of freedom is well defined. Note that for
n
open and u : U × R → R . The Navier-Stokes many finite element spaces that are intended as
equations for incompressible, viscous flows are conforming approximating spaces for a function
space V , S will be strictly contained in V . Nodal

u + u · Du = −Dp interpolation operators are local in the sense that
t
div u = 0 for each cell K of the underlying triangulation
the restriction of I u onto K depends on u ¯ .
|K
h
where Du = D u = (u ,...,u ) denotes the
x
x n
x 1
gradient of u with respect to the spatial variable node A node, v (i indexing the nodes of the
i
graph), is a member of the set of nodes (V)ofa
x = (x ,..., x ).
1
n
mathematical graph or network.
Comment: Each node is commonly rendered
neighborhood (of a point p in a topological by a geometric point, but in fact it is simply an
space) A set containing an open set contain- element of any set; so in a set of integers, each
ing p. integer would be a node of a graph. A com-
mon synonym is vertex. When reactions and
compounds are both nodes in the representation
nematic phase See liquid-crystal transi- of a biochemical network, because there are two
tions. types of nodes, the network is said to be bipartite.
nonconforming finite elements A finite
network A mathematical graph N(V, E,
element space V is called nonconforming with
h
P, L) consisting of a set of nodes (or vertices)
respect to a function space V if V ⊂ V . Using
h
V, a set of edges between the nodes E, a set of
V to discretize a variational problem over V
h
parameters P describing properties of subgraphs
amounts to a variational crime. However, it
of the network, and a set of labels L.
might be feasible, if V satisfies certain consis-
h
Comment: If P =∅ the network reduces to its
tency conditions expressed in Strang’s second

corresponding graph [denoted either N (V, E, L) lemma.
or G(V, E, L) according to one’s taste]. In gen-
eral, all graph algorithms apply to networks, but nondeterministic computation Let the sets
there are some specialized algorithms, relying on of presented inputs and produced outputs be K
I
the parameters, which do not apply to graphs. and K , respectively; the cardinalities of sets be

O
denoted by the double overset bars; and σ and
i,I
σ be the particular symbols presented to or
nodal basis The basis {b , ··· ,b } of a i,O
1 N produced by the mapping c .
i
finite element space V dual to the set X of Then a nondeterministic computation C n
h
h
global degrees of freedom, that is, φ (b ) = specifies a computation C : K → K , such
k
i

δ ,i,k = 1, ··· ,N, is called the nodal basis I O
ik thatK ≥ 1; K ≥ 1; atleastonemappingc ∈
I O i
of V . Thanks to the localization of the degrees C transforming presented inputs to presented
h
n
of freedom, the nodal basis functions are locally outputs is many-to-many; and ∀σ ,σ ∈
i,I i,I
supported. More precisely, supp b is contained ,σ ∈ K , the probabilities that
i K , ∀σ i,O i,O O
I
in the closure of the union of all those cells of ; h each exists, P (σ i,I ) and P (σ i,O ), are unity.
e
e
that share the vertex, face, edge, etc. to which the Comment: This computation is not stochastic,
related global degree of freedom is associated. but it is nondeterministic. There is at least one
c
© 2003 by CRC Press LLC
© 2003 by CRC Press LLC