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minimum-energy reaction path The path This is a symmetric saddle point problem,
corresponding to the steepest descent from the which can be discretized by means of v ∈
col of a potential-energy surface into the two H(div;;)-conforming mixed finite elements for
valleys. The reaction coordinate corresponds the flux unknown. Solving the second-order
to this minimal path. Some workers refer to elliptic boundary value problem is equivalent
the minimum-energy reaction path as simply the to minimizing a convex energy functional on
reaction path but this is not recommended as it H (;). Convex analysis teaches that this gives
1
0
leads to confusion. rise to primal and dual variational problems. The
latter leads to the mixed variational formulation.
Minkowski space The metric vector space
4
(R ,η) with the signature (1, 3).
module (over a ring R) A set M endowed
mixed finite elements A rather general term with an inner binary operation, called the sum
for finite element spaces meant to approximate + : M × M → M and a binary operation called
vector fields. Prominent representatives are the product (by a scalar) · : R × M × M. The
(i.) H(div; ;)-conforming Raviart-Thomas structure (M, +) is a commutative group and the
elements required for the discretization of mixed product by scalars satisfies the following axioms:
variational formulations of second-order elliptic
(i.) ∀λ, µ ∈ R, ∀v ∈ M: λ(µ·v) = (λµ)·v
boundary value problems.
(ii.) H(curl; ;)-conforming N´ed´elec ele- (ii.) λ · (v + w) = λ · v + λ · w
ments (edge elements) used to approximate elec- (iii.) (λ + µ) · v = λ · v + µ · v
tromagnetic fields.
(iv.) I · v = v
R
(iii.) a variety of schemes for the approxi-
mation of velocity fields in computational fluid If R is a field then an R-module is nothing but
mechanics. a vector space.

mixed variational formulation Consider a
second-order elliptic boundary value problem mole An Avogadro’s number of molecules
23
(N ≈ 6.023 × 10 ).
n
−div(Agradu) = f on ; ⊂ R ,u |∂; = g .
D
Formally its mixed formulation is obtained by molecular dynamics Based on Newton’s
introducing the flux j := Agradu as new law of motion and treating atoms in a molecule
unknown, which results in a system of first-order as classical particles with interaction, molecu-
differential equations lar dynamics simulate the motion of the entire
molecule on a computer. This approach has been
j − Agradu = 0 , −divj = f.
successful in studying the dynamics of small
molecules but has not yielded a complete pic-
The first equation is tested with v ∈ H(div;;)
ture of the dynamics of biologically important
and integration by parts is carried out. The sec-
2
ond equation is tested with q ∈ L (;).We molecules, e.g., proteins. Thedifficultyismainly
the stiff nature of the many-body system and the
end up with the variational problem: seek j ∈
2
H(div;;), u ∈ L (;) such that limited computational power.

−1
A j · vdx + divvudx = gv · ndS
molecular entity Any constitutionally or
; ; N
isotopically distinct atom, molecule, ion, ion
∀v ∈ H(div;;), pair, radical, radical ion, complex, conformer,

2 etc., identifiable as a separately distinguishable
divjqdx =− fqdx ∀q ∈ L (;).
entity.
; ;

© 2003 by CRC Press LLC