General useful results and Elliptic Problems

Numerical methods for differential equations

General useful results (spaces, functional analysis, weak/strong form, FEM)

Elliptic problems

We want to define spaces in which points are functions: how could we define them?

Properties of a space of vectors (ℝⁿ, ℝ³, ...)

_αu₁ + _βu₂ is still a vector, α, β ∈ ℝ; u₁, u₂ ∈ X

1. u + v = v + u (linear combination, commutativity)
2. 0: y + 0 = u (identity element of addition)
3. -y: y + (-y) = 0 (inverse element of addition)
4. _λ(β)y = α(βF) v ∈ X, α, β ∈ ℝ (compatibility)
5. (d + β) v = α u + β
6. 1 · y = y 1∈ℝ; v ∈ X (identity element of scalar product)

Vector space X

All properties are general counterparts of those in classical Euclidean space, which is, in fact, a particular case of vector space.

Subspace

Y is said to be a subspace of X when Y is a subset of vector space X.

e.g.: in ℝ³ a subspace can be any plane.

Consider 2 subspaces Y, W:

• Y + W = {u ∈ X : u = y + w, y ∈ Y, w ∈ W} -> SUM
• e.g. X ∈ ℝ³, Y = { x∈ℝ³ : x = (α, 0, 0), α∈ℝ } -> Y+W = ℝ³
• W = { x∈ℝ³ : x = (α, β, 0), α, β∈ℝ }

Consider 2 subspaces Y, W such that

• {Y+W = X Y∩W = {0} -> DIRECT SUM ⊕}
• e.g. X ⊂ ℝ³ U = { x∈ℝ³ : x = (α, β, 0) } -> X = U + V
• V = { x∈ℝ³ : x = (α, β, 0) } -> X = U ∪ W
• W = { x∈ℝ³ : x = (0, 0, 0) }

Scalar product

u · v = |u| |v| cos θ, u, v ∈ ℝ³ u · v ∈ ℝ

We want to generalize it to a vector space (pencil).

From _sp we derive:

• Length of a vector: ||u|| = √(u · u)
• Distance: d(u, v) = ||u - v||

Properties

1. (u, v) ∈ ℝ, u, v ∈ X (SP4)
2. (u, v) = (v, u) (SP2)
3. (dU + βv, w) = α(u, w) + β(v, w) u, v, w ∈ X, α, β ∈ ℝ (SP3)
4. (u, u) ≥ 0 , (u, u) = 0 iff u = 0 (SP4)

(u, v) = 0 u and v are orthogonal

General useful results (spaces, functional analysis, weak/strong form, FEM)

Elliptic problems

We want to define spaces in which points are functions: how could we define them?

Properties of a space of vectors (R², R³, ...)

1. αu + βv is still a vector, α, β∈R; u, v∈X (linear combination)
2. u + v = v + u (symmetry)
3. 0: y + 0 = y (identity element of addition)
4. -y: y + (-y) = 0 (inverse element of addition)
5. α(β)v = (αβ)(v), v∈X, α, β∈R (compatibility)
6. (α + β)u = αu + βu (distributivity)
7. 1⋅u = u, 1∈R, u∈X (identity element of scalar product)

Vector space X

All properties are general counterparts of those in classical Euclidean space, which is, in fact, a particular case of vector space.

Subspace

Y is said to be a subspace of X when Y is a subset of vector space X.

e.g.: in R³ a subspace can be any plane.

Consider 2 subspaces Y, W:

• Y + W = {u ∈ X : u = y + w, y ∈ Y, w ∈ W} ➔ SUM
• e.g. X ∈ R³ Y = {x ∈ R³: x = (α,0,0), α∈R}
• W = {x ∈ R³: x = (α,β,0), α,β∈R} ➔ Y⊥W={x∈R³: x=(α,β,0), α,β∈R}

Consider 2 subspaces Y, W such that {Y⊥W = X} {Y∩W = {0}} ➔ DIRECT SUM (⊕)

e.g. X ∈ R³

• U = {x ∈ R³: x = (α,β,0)}
• V = {x ∈ R³: x = (x,β,y)}
• W = {x ∈ R³: x = (0,0,0)}

X = U⊥V

X = U⊕W

Scalar Product

y⋅v = |u||v| cos θ, u,v ∈ R³, u⋅v ∈ R

(We want to generalize it to a vector space: pencil.)

From S.P. we derive:

• Length of a vector: ||u|| = √(u⋅u)
• Distance: d(u,v) = ||u-v||

Properties

1. (u,v) ∈ R, u,v∈X (SP1)
2. (u,v) = (v,u) (SP2)
3. (u + v, w) = α(u,w) + β(v,w), u,v,w∈X,α,β∈R (SP3)
4. (u,u) = 0, (u,u) = 0 iff u = 0 (SP4)

(u,v) = 0 ➔ u and v are orthogonal