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Galilei group The closed subgroup of
GL(5, R) consisting of Galilei transformations,
i.e., of matrices of the following block structure
G  R v a 

 0 1 t 
0 0 1
G G G-graded algebra An algebra A together 3
where R ∈ SO(3), v, a ∈ R ,r ∈ R.
with a map # : g → A called the degree map
g
which associates a vector subspace A to any
g
element g ∈ G. The subspaces A are not neces-
g
sarily subalgebras. However, the product of A gamma function The complex function
given by
is compatible with the degree map, in the sense
that if a ∈ A and b ∈ A then a · b ∈ A . ∞
gh
g
h
e dt
The elements a ∈ A are called homogeneous of N(z) = t z−1 −t
g
degree g. 0
for complex z with positive real part.
Galerkin method An approach to the dis-
cretization of a variational problem G˙ arding inequality The inequality satisfied
by a coercive sesquilinear form on some Sobolev
u ∈ V :< A(u), u >= 0 ∀v ∈ V, space.
V a Banach space, A : V → V continu- Gateaux derivative Let V, W be Banach
ous. It relies on two finite dimensional subspaces spaces, U ⊂ V open. The Gateaux derivative or
V ,W ⊂ V to obtain the discrete variational directional derivative of a map F : U ⊂ V →
h
h
problem W at the point x ∈ U in direction h ∈ V is
defined by
u ∈ V : < A(u ), v >= 0 ∀v ∈ W . (4)
h h h h h h
1
DF(x, h) = lim [F(x + th) − F(x)].
The space V is called the trial space, the space t→0 t
h
W is known as the test space.If V = W we
h
h
h
confront a Ritz-Galerkin method, the more gen- F is called Gateaux differentiable at x ∈ U if
eral case is often referred to as a Petrov-Galerkin DF(x, h) exists for all h ∈ V .
method. Necessary and sufficient conditions for
existence and uniqueness of solutions of (4) are Gateaux-Levi derivative Let V, W be
supplied inf-sup conditions. Banach spaces, U ⊂ V open. A map F : U →
W is called Gateaux-Levi differentiable at the
point x ∈ U if it is Gateaux differentiable at x
Galerkin orthogonality Let a be a sesqui-
and the map h ∈ V → DF(x, h) from V to W is
linear form on a Banach space V and V be a linear and bounded. If F is Gateaux-Levi differ-
h
subspace of V that may represent a finite element
entiable, we can set DF(x)h = DF(x, h) and

space.For f ∈ V the functions u ∈ V and
get
u ∈ V are to satisfy
h
h
F(x + h) = F(x) + DF(x)h + RF(x, h)
a(u, v) = f(v) ∀v ∈ V,
where
a(u ,v ) = f(v ) ∀v ∈ V .
h
h
h
h
h
RF(x, th)
The Galerkin orthogonality refers to the straight- lim = 0, f or each h ∈ V.
t→0 t
forward relationship
gauge group The group of gauge transfor-
a(u − u ,v ) = 0 ∀v ∈ V . mations.
h
h
h
h
© 2003 by CRC Press LLC