Appendix A
Table of Notations
Here mathematical expressions and corresponding FreeFem++
commands are noted.
A.1 Generalities
-
δij
- Kronecker delta (0 if i ⇔ j, 1 if i = j for integers i,j)
-
∀
- for all
-
∃
- there exist
-
i.e.
- that is
-
PDE
- partial differential equation (with boundary conditions)
-
∅
- the empty set
-
ℕ
- the set of integers (a
ℕ ⇔ int a); “int” means long integer inside FreeFem++
-
- the set of real numbers (a
⇔ real a) ;double inside FreeFem++
-
ℂ
- the set of complex numbers (a
ℂ ⇔ complex a); complex¡double¿
-
d - d-dimensional Euclidean space
A.2 Sets, Mappings, Matrices, Vectors
Let E,F,G be three sets and A subset of E.
-
{x
E|P} - the subset of E consisting of the elements possessing the property P
-
E ∪F
- the set of elements belonging to E or F
-
E ∩F
- the set of elements belonging to E and F
-
E \A
- the set {x
E|x ℜ A}
-
E + F
- E ∪F with E ∩F = ∅
-
E ×F
- the cartesian product of E and F
-
En
- the n-th power of E (E2 = E ×E, En = E ×En-1)
-
f : E → F
- the mapping form E into F, i.e., E ∋ x
f(x)
F
-
IE or I
- the identity mapping in E,i.e., I(x) = x ∀x
E
-
f ∘g
- for f : F → G and g : E → F, E ∋ x
(f ∘g)(x) = f(g(x))
G (see Section 4.6)
-
f|A
- the restriction of f : E → F to the subset A of E
-
{ak}
- column vector with components ak
-
(ak)
- row vector with components ak
-
(ak)T
- denotes the transpose of a matrix (a
k), and is {ak}
-
{aij}
- matrix with components aij, and (aij)T = (a
ji)
A.3 Numbers
For two real numbers a,b
-
- [a,b] is the interval {x
|a ≤ x ≤ b}
-
- ]a,b] is the interval {x
|a < x ≤ b}
-
- [a,b[ is the interval {x
|a ≤ x < b}
-
- ]a,b[ is the interval {x
|a < x < b}
A.4 Differential Calculus
-
∂f∕∂x
- the partial derivative of f :
d →
with respect to x ( dx(f))
-
∇f
- the gradient of f : Ω →
,i.e., ∇f = (∂f∕∂x,∂f∕∂y)
-
div
or ∇.
- the divergence of
: Ω →
d, i.e., div
= ∂f
1∕∂x + ∂f2∕∂y
-
Δf
- the Laplacian of f : Ω →
, i.e., Δf = ∂2f∕∂x2 + ∂2f∕∂y2
A.5 Meshes
-
Ω
- usually denotes a domain on which PDE is defined
-
Γ
- denotes the boundary of Ω,i.e., Γ = ∂Ω (keyword border, see Section 5.1.2)
-
h - the triangulation of Ω, i.e., the set of triangles Tk, where h stands for mesh size (keyword
mesh, buildmesh, see Section 5)
-
nt
- the number of triangles in
h (get by Th.nt, see “mesh.edp”)
-
Ωh
- denotes the approximated domain Ωh = ∪k=1ntT
k of Ω. If Ω is polygonal domain, then it will
be Ω = Ωh
-
Γh
- the boundary of Ωh
-
nv
- the number of vertices in
h (get by Th.nv)
-
[qiqj
- ] the segment connecting qi and qj
-
qk1,qk2,qk3
- the vertices of a triangle T
k with anti-clock direction (get the coordinate of qkj by
(Th[k-1][j-1].x, Th[k-1][j-1].y))
-
IΩ
- the set {i
ℕ|qi ℜ Γ
h}
A.6 Finite Element Spaces
-
L2(Ω)
- the set
-
H1(Ω)
- the set
-
Hm(Ω)
- the set
-
H01(Ω)
- the set
-
L2(Ω)2
- denotes L2(Ω) ×L2(Ω), and also H1(Ω)2 = H1(Ω) ×H1(Ω)
-
Vh
- denotes the finite element space created by “ fespace Vh(Th,*)” in FreeFem++ (see Section 6 for
“*”)
-
Πhf
- the projection of the function f into Vh (“ func f=x^2⋆y^3; Vh v = f;” means v = Πh
f)
-
{v}
- for FE-function v in Vh means the column vector (v1,
,vM)T if v = v
1ϕ1 +
+ vMϕM, which is
shown by “ fespace Vh(Th,P2); Vh v; cout << v[] << endl;”