General useful results and Elliptic Problems

NUMERICAL METHODS FOR DIFFERENTIAL EQNS

• GENERAL USEFUL RESULTS (SPACES, FUNCTIONAL ANALYSIS, WEAK/STRONG FORM, FEM)
• ELLIPTIC PROBLEMS

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

Properties of a space of vectors

• ^✒ Y is said to be 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 = (0,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 = (0,β,0)}
• 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 S.P. we derive: - length of a vector: ||u|| = sqrt(u · u) - distance: d(u,v) = ||u - v||

Properties (u,v) ∈ R, u,v ∈ X _SP1 (u,v) = (v,u) _SP2 (u + v, u) > 0 ↔ u = 0 _SP4

(u, v) = 0 ↔ u and v are orthogonal

in vectors

can be generalized to any space admitting s.p.

Cauchy-Schwarz inequality:

We introduce a generic norm defining its properties

• (N1)

• (N2)

• (N3)

More than one norm could be defined, with α concepts, meanings:

• (p-norm) classical euclidean norm

• (infinity norm) (can be seen as extension of p norm)

Vector space where u “lives”, is called NORMED vector space

Norm and scalar product properties have much in common:

defining

we have

_(N1)

_(N2) assured by _(SP4)

_(N3) direct consequence of _(SP3)

_(N4)

Cauchy-Schwarz

We can define a norm without s.p, but in practice it’s easier, as just seen

If I have a vector space in which I define a s.p. I can define a norm, so it is also a normed space

What about the converse (norm without s.p. = s.p. in the space)?

theorems of cosine into triangles

Parallelogram law ( ∀ vector space )

A norm does not satisfy parallelogram law there is no scalar product

The “right way to go” is from s.p. to norm

Norm is essential to check if a method is convergent or not

Two norms are said to be EQUIVALENT

e.g. in: