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n
First, consider the unit simplex K in R . The
local space V agrees with the space of multi-
K
variate polynomials of total degree ≤ k, k ∈ N,
L V
K

(
α 1
α n
:= x → a x ··· x ,a ∈ C .
α 1 a α
n
label A unique identifier for a node or edge α∈N ,|α|≤k
0
of a graph, network, or a subnetwork (subgraph).
The local degrees of freedom are based on point
Comment: Thus, compound names are labels
evaluations
for those nodes, and an edge is labeled by giving
the tuple of nodes which it joins, so (v ,v ). X
i
j
k
(
laboratory sample The sample or subsam- := φ : C(K) → C, φ(u) = u(p), p ∈ P ,
p
ple(s) sent to or received by the laboratory.
When the laboratory sample is further prepared
where
(reduced) by subdividing, mixing, grinding, or
by combinations of these operations, the result P
)
is the test sample. When no preparation of the 1 T
n
laboratory sample is required, the laboratory := (l , ··· ,l ) ,l ∈ N , l ≤ k ⊂ K.
1
i
0
i
n
k
sample is the test sample. A test portion is i=1
removed from the test sample for the perfor-
Obviously, we have
mance of the test or for analysis. The laboratory
sample is the final sample from the point of view
n + k
of sample collection but it is the initial sample dimV = .
K
k
from the point of view of the laboratory. Several
laboratory samples may be prepared and sent to On the unit hyper-cube K ⊂ R , the geomet-
N
different laboratories or to the same laboratory
ric reference element for quadrilateral and hexa-
for different purposes. When sent to the same
hedral meshes, the local spaces V are given by
K
laboratory, the set is generally considered as a
polynomials with degree ≤ k in each indepen-
single laboratory sample and is documented as a
dent variable
single sample.
V K
Ladyshenskaja-Babuˇska-Brezzi (LBB) condi- (
α 1 α n
tion A sufficient condition for the stability := x → α∈N ,α i ≤k a x ··· x a ,a ∈ C .
α
n
α 1
0
of finite element schemes for saddle point prob-
lems. It amounts to requiring uniform inf-sup Local degrees of freedom are given by point
conditions for the pair of finite element spaces evaluation in the points
used in a Galerkin discretization of the saddle
(
point problem. 1 T
P := (l , ··· ,l ) ,l ∈ N ,l ≤ k ⊂ K,
n
0
i
i
1
k
Lagrange finite elements A parametric
equivalent family of finite elements giving rise to which means
1
H (;)-conforming finite element spaces. Para-
n
metricequivalenceisbasedonthefollowingpull- dimV = (k + 1) .
K
back mapping for functions
Lagrangian The Lagrangian formulation
F (u)(x) := u()(x)) x ∈ K.
)
of mechanics is based on the variational principle
Thus, it is sufficient to specify the finite elements of Hamilton. To describe a mechanical system
for reference elements. Lagrangian finite ele- one chooses a configuration space Q with coor-
i
ments can be distinguished by their polynomial dinates q ,i = 1,...,n. Then one introduces
degree k ∈ N. the Lagrangian function L = K − V , where K
© 2003 by CRC Press LLC