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to pull-back. An important consequence is that
nodal interpolation operators and pull-backs
commute:
P F (I u ) = I (F (u)) ,
h
h |T
|K
)
)
for all sufficient smooth functions/vector fields u
n
p-Laplacian equation Let U ⊂ R be open on T .
and u : U × R → R. The p-Laplacian equation Given an infinite family of meshes {; } ,
h h∈H
H an index set, a related family of finite ele-
for u is
p−2 ment spaces is considered parametric equivalent,
div(|Du| Du) = 0,
if there is a small number of finite elements to
where Du = D u = (u ,...,u ) denotes the
x x 1 x n which all elements of the family are parametric
gradient of u with respect to the spatial variable equivalent. These elements are called reference
x = (x ,...,x ).
1 n elements.
Parametric equivalence is a key tool in most
proofs of local a priori interpolation estimates for
p-version of finite elements The finite families of finite element spaces and their associ-
element discretization of a boundary value prob- ated nodal interpolation operators. If the meshes
lem is built upon a fixed mesh. A family of are shape regular, it is often possible to gauge
finite elements is employed that supplies finite the change of norms under pull-back. First the
element spaces for a wide range of local polyno- interpolation error is estimated on the reference
mial degrees. The idea of the p-version of finite element(s). Then, the relationship of commuta-
elements is to get an acceptable discrete solu- tivity given above is used to conclude an estimate
tion by using a sufficiently high degree polyno- for an arbitrary element.
mial. A local variant of the p-version raises the
polynomial degree only on cells where a better partial differential equation A differential
approximation is really needed to reduce the dis- equation involving a function of more than one
cretization error; cf., adaptive refinement. variable, and hence partial derivatives.

parallel transport A vector field X defined partition A set of subgraphs, {G (V , E ),

along a curve γ in a manifold M with a connec- G (V , E ),...} of a graph G(V, E), such that

tion N such that the subgraphs are disjoint.
∇ X = 0 partition of unity Let M be a paracompact
˙ γ
manifold and {U } a locally finite covering
α α∈I
where ˙γ denotes the tangent vector to the (see open covering). A partition of unity relative
curve γ .
to {U } is a family of (smooth) local functions
α α∈I
f : M → R such that:
α
parametric equivalence Two finite ele-
(i.) ∀x ∈ M, f (x) ≥ 0;
ments (K, V ,X ) and (T, V ,X ) are para- α
T
T
K
K
metric equivalent, if there is a piecewise smooth (ii.) suppf ⊂ U ;
α
α
bijective mapping ) : K → T and a bijective (iii.) ∀x ∈ M, α∈I f (x) = 1;
α
linear mapping F ) : V → V , the pull-back,
T
k
α
such that where suppf denotes the support of the function
f , i.e., the closure of the set {x ∈ M : f (x)
α
α
X ={κ : V → C, κ(u) = φ(F u), φ ∈ X }. = 0}.
T T ) k
Notice that the sum in P3 is finite since the
Sloppily speaking, this means that both the geo- covering {U } is locally finite. The functions
α α∈I
metric elements and the local spaces can be f of a partition of unity can be required to be
α
mapped onto each other, and that the local real-analytic only in trivial situations. In fact,
degrees of freedom are invariant with respect any analytical function which is identically zero
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