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dominated convergence theorem Let {f } If V is finite dimensional, then dim(V ) =
n
1
be a sequence in L such that f → f almost dim(V ). See also dual basis.
∗
n
everywhere and there exists a nonnegative g ∈
1
L such that |f |≤ g almost everywhere for all
n
1
n. Then f ∈ L and f = lim f .
n
n→∞ duality techniques (Aubin-Nitsche trick)
These are used to establish a priori estimates
donor A compound which breaks a chem- for the discretization error of finite element
ical bond, yielding a substituent group which schemes in norms weaker than the natural norms
forms a new bond in a bimolecular chemical or associated with the continuous variational
biochemical reaction. See acceptor. problem. For example, let H, V be Hilbert
spaces, V continuously embedded in H, and
a : V × V → C a continuous and V-elliptic
dot product See angle between vectors.
sesqui-linear form. Write u ∈ V, u ∈ V for
h
h
the solutions of
double helix The structure of DNA in all
biological species is in the form of a double helix
a(u, v) = f(v) ∀v ∈ V,
made of two chain molecules. Each chain is a
polymer made of four types of nucleotide: A
(adenine), G(guanine), T(thymine), andC(cyto- a(u ,v ) = f(v ) ∀v ∈ V ,
h
h
h
h
h
sine). The structure that was first proposed by
J.D. Watson and F.H.C. Crick immediately leads
where V is some closed subspace of V (a con-
h
to a possible mechanism of biological heredity.
forming finite element space) and f ∈ V .For

This was later confirmed by experiments and
φ ∈ H denote by g(φ) ∈ V the solution of
hence provides a molecular basis for genetics.
a(v, g(φ)) = (φ, v) H ∀v ∈ V.
drawing See rendering.
Then, choosing φ := u − u we obtain via
h
dual See dual vector space, complement. Galerkin orthogonality
2
dual basis Let E be a finite dimensional vec- u − u = a(u − u ,g(u − u )) ≤
h
h H
h
tor space with basis (e ,...,e ) . The dual basis
1 n
1
n
(α ,...,α ) of the dual space E is defined by
∗
j
j
j
h V
α (e ) = δ , where δ = 1if j = i and 0 other- a u − u inf v h ∈V h g(u − u ) − v
h
h V
i
i
i
wise.
If g(φ) possesses extra regularity beyond merely
belonging to V , the second term on the right-
dual vector space Let V be a vector space
hand side will become very small compared to
(over the field K); the set V of all linear func-
∗
u − u , if the resolution of the finite ele-
h H
tionals α : V → K can be endowed with a struc-
ment space is increased. Thus, when considering
ture of vector space by defining (λα + µβ)(v) =
families of finite element spaces, the H-norm of
λ α(v) + µ β(v) (where λ, µ ∈ K; v ∈ V ). The
the discretization error may converge asymptot-
vector space V so obtained is called the (alge-
∗
ically faster than the V -norm.
braic) dual vector space of V .
If V is a topological vector space the set of
all linear and continuous functionals α : V →
K with the standard linear structure introduced Duffing equation The Duffing equation
above is called the topological dual space of V ,
3
and it is still denoted by V . ¨ x + x + "x = 0
∗
© 2003 by CRC Press LLC