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finite element space Given a triangulation first integral For a dynamical system
; of ;, consider a finite element (K, V , X ) (M, X) a function F : M → R such that X(F)
h
K
K
for each cell K ∈ ; . A related (global) finite = 0. The definition is equivalent to requiring
h
element space V has to satisfy (for suitable that F ◦ γ is constant for any integral curve γ
h
m ∈ N) of X.
More generally, a function F : R × M → R
∗
V ⊂ V :=
h
h
which is constant along the curves of motion.
M Equivalently, let us define X = ∂ + X; F is a
ˆ
{v : ; → C defined a.e. in ;, t
ˆ
first integral if X(F) = 0.
v ∈ V ∀K ∈ ; }.
|K K h
fixed point A point x ∈ M is called a fixed
0
The global finite element space is said to be point of a map T : M → M if T(x ) = x .
0 0
conforming with respect to a space V of func-
tions/vector fields defined a.e. on ;, if V ⊂ V . flow The flow of a vector field X on a
h
Often, this leads to the definition manifold M is the one-parameter group of
diffeomorphisms F : M → M,(F t+s = F ◦F )
s
t
t
∗
V = V ∩ V. such that
h
h
d
In many cases, the requirement of conformity F (x) = X(F (x)), for all x ∈ M.
t
t
dt
amounts to conditions on the continuity of global
finite element functions at boundaries between In other words, t → F (x) is an integral curve
t
geometric elements. This is due to the fact (trajectory) of X with initial condition x ∈ M.
∗
that functions in V are piecewise smooth. For Flows are also called dynamical systems.
1
instance, conformity in H (;) entails global
2
continuity, and finite element functions in H (;) flow rate (of a quantity) Quantity X (e.g.,
even have to be globally continuously differen- heat, amount, mass, volume) transferred in a time
tiable. interval divided by that time interval. General
˙
Actually, the glue between the (local) finite symbols: q , X.
X
elements is provided by the degrees of free-
F, Z,
F, Z,
dom, because in practice V is introduced as fluence, F, Z, H H H 0 0 0 At a given point in space,
h
∗ the radiant energy incident on a small sphere
V := {v ∈ V , ∀K, T ∈ ; : φ(v ) =
h
|k
h
h
κ(v ) if φ ∈ X , κ ∈ X are of the same type divided by the cross-sectional area of that sphere.
|T K T
and are associated with the same vertex, edge, It is used in photochemistry to specify the energy
face, etc. of the global mesh}. It goes without delivered in a given time interval (for instance,
saying that this construction imposes tight con- by a laser pulse).
straints on the choice of the local finite elements.
flux The flux of a compound x is ∂x /∂t
Local degrees of freedom of the finite elements j j
or x .
belonging to adjacent cells have to match. jt
Comment: Biochemically, flux is frequently
This makes it possible to convert the local
used to describe the consumption of indi-
degreesoffreedomintonodaldegreesoffreedom.
vidual molecules by a series of reactions in
These are obtained by collecting all local degrees
dynamic equilibrium. Experimentally, flux can
of freedom into one set X and weeding out func-
h be determined by giving a fixed amount of a par-
tionals that agree on V . By definition of V , the ticular compound with a radioactive or heavy
h
h
remaining functionals in X :={φ , ··· ,φ }, isotope to a population of cells, then measur-
1
h
N
N = dimV , form a basis of the dual space.
h
ing the amount and rate of movement of the
isotope into compound(s) derived from the first.
finite set A set whose cardinality is either 0 Since the cells are usually in a metabolic steady
or a natural number. See also cardinality, count- state, the assumption is that all the reactions
able set, denumerably infinite set, infinite set, and are in dynamic equilibrium. Compound con-
uncountably infinite set. centrations do not change over time (˙x = 0).
i
© 2003 by CRC Press LLC