numerical methods for pdes introduction, fundamentals of...

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Dr. Noemi Friedman

Numerical methods for PDEs Introduction, fundamentals of functional analysis

Fundamentals| Dr. Noemi Friedman | PDE tutorial | Seite 2

Introduction

Numerical methods for PDEs lectures: Dr. Elmar Zander e-mail: [email protected] tutorials: Dr. Noemi Friedman e-mail: [email protected] assignments (in group of 2 or 3)

mailto:[email protected]






Fundamentals | Dr. Noemi Friedman | PDE2 tutorial | Seite 3

Contents of the course

• Fundamentals of functional analysis • Abstract formulation FEM • Application to conrete formulations • Convergence, regularity • Variational crimes • Implementation • Mixed formulations (e.g. Stokes) • Stabilisation for flow problems • Error indicators/estimation • Adaptivity


Fundamentals of functional analysis

• Vector spaces • Norms • Banach spaces • Inner products • Hilbert spaces • Completeness/Denseness

• Classical funcion spaces • Lebesque spaces • Weak derivatives • Sobolev spaces* • Linear functionals & dual space • Bilinear forms

• Riesz Represenation Theorem • Lax-Milgram Lemma • Céa’s Lemma


Vector space

• functional is a function from a vector space into its underlying scalar field

• Vector spaces (V) 𝑢, 𝑣 ∈ 𝑉

• addition of 𝑢 and 𝑣 (z = 𝑢 + 𝑣) is defined such that 𝑧 ∈ 𝑉 • multiplication by scalars is defined such that α𝑢 ∈ 𝑉

For addition and multiplication the following axioms are satisfied: 𝒖 + (𝒗 + 𝒘) = (𝒖 + 𝒗) + 𝒘 (associativity)

𝒖 + 𝒗 = 𝒗 + 𝒖 (commutativity)

There exists an element 𝟎 ∈ 𝑉, such that 𝒗 + 𝟎 = 𝒗 for all 𝒗 ∈ 𝑉.

For every 𝒗 ∈ 𝑉, there exists an element −𝒗 ∈ 𝑉 such that 𝒗 + (−𝒗) = 𝟎

𝑎(𝑏𝒗) = (𝑎𝑏)𝒗

1𝒗 = 𝒗, where 1 denotes the multiplicative identity in F.

𝑎(𝒖 + 𝒗) = 𝑎𝒖 + 𝑎𝒗 (distributivity)

(𝑎 + 𝑏)𝒗 = 𝑎𝒗 + 𝑏𝒗 (distributivity)

𝑢, 𝑣,𝑤 ∈ 𝑉

𝑎, 𝑏 ∈ ℝ

with:

[On vector space and inner product see Chapter 2.4.1]


Vector spaces, linear subspaces

Important vector spaces, examples: • Euclidian n-space:

• C[a, b] : set of all continuous, real-valued functions defined on the interval [a, b]. • C1 [a, b]: set of all real-valued, continuously differentiable functions defined on

the interval [a, b]. (A function is continuously differentiable if its derivative exists and is continuous.)

• Ck [a, b]: space of real-valued functions defined on [a, b] that have k continuous derivatives

Dirichlet

Subspaces linear subspaces (any linear combination of two elements of the subspace is also en element of the subspace) examples:

Neumann

𝑉0 ∈ 𝑉, 𝑥,𝑦 ∈ 𝑉0 , a, b ∈ R, ax + by ∈ 𝑉0


Vector space - Norm

A vector space may be endowed with additional structures, such as a norm and inner product Norm in Euclidian spaces: any mapping 𝑔:𝑉 → ℝ, that satisfies: 1. The zero vector, 0, has zero length; every other vector has a positive

length. 𝑔 𝒙 ≥ 0, 𝑔 𝒙 = 0 only if 𝒙 = 𝟎

with the usual notation 𝒙 ≥ 0, 𝒙 = 0 only if 𝒙 = 𝟎

2. Multiplying a vector by a positive number changes its length without changing its direction.

𝑔 𝑎𝒙 = 𝑎 𝑔 𝒙 3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

𝑔 𝒙 + 𝒚 ≤ 𝑔 𝒙 + 𝑔 𝒚


Vector space - Norm

Examples for norms in the Euclidien space

𝒙 =𝑥1𝑥2𝑥3

𝒙 1 = 𝑥1 + 𝑥2 + 𝑥3

𝒙 2 = 𝑥12 + 𝑥22 + 𝑥32

𝒙 𝑝 = 𝑥1𝑝 + 𝑥2𝑝 + 𝑥3𝑝1𝑝

𝒙 ∞ = max 𝑥1 , 𝑥2 , 𝑥3


Vector space -Norm Norm in function spaces, equivavelently: any mapping 𝑔:𝑉 → ℝ, that satisfies: 1.

𝑔 𝑓(𝑥) ≥ 0, 𝑔 𝑓(𝑥) = 0 only if𝑓(𝑥) = 0 with the usual notation

𝑓(𝑥) ≥ 0, 𝑓(𝑥) = 0 only if𝑓(𝑥) = 0 2.

𝑔 𝑎𝑓(𝑥) = 𝑎 𝑔 𝑓(𝑥) 3.

𝑔 𝑓(𝑥) + ℎ(𝑥) ≤ 𝑔 𝑓(𝑥) + 𝑔 ℎ(𝑥) Examples:

𝑓 2 = � 𝑓 𝑥 2dx1

0 𝑓 𝑝 = � 𝑓 𝑥 𝑝dx

1

0

1𝑝

𝑓 ∞ = max𝑥∈ 0,1

𝑓 𝑥


Vector sapace - Inner product

Inner product: any mapping 𝑔:𝑉 × 𝑉 → ℝ, that satisfies 1. Positivity

𝑔 𝑥, 𝑥 ≥ 0, 𝑔 𝑥, 𝑥 = 0 only if 𝑥 = 0 with the usual notation

𝑥, 𝑥 ≥ 0, 𝑥, 𝑥 = 0 only if 𝑥 = 0 or

𝑥, 𝑥 ≥ 0, 𝑥, 𝑥 = 0 only if 𝑥 = 0

2. Linearity 𝑔 𝑎𝑥, 𝑦 = 𝑎𝑔 𝑥, 𝑦

𝑔 𝑥 + 𝑦, 𝑧 = 𝑔 𝑥, 𝑧 + 𝑔 𝑦, 𝑧 3. Symmetry

𝑔 𝑥, 𝑦 = 𝑔 𝑦, 𝑥

Inner product space: vector space with an inner product


Vector sapace - Inner product

In Euclidian space (ℝ𝑛) • Dot product

• Hermitian form

if A: pos. def. symm.

In function spaces • 𝑓,𝑔 = ∫ 𝑓 𝑥 𝑔 𝑥 𝑑𝑥1

0 𝐶1[0,1] • 𝑓,𝑔 𝐻1 = ∫ 𝑓 𝑥 𝑔 𝑥 + 𝑓′ 𝑥 𝑔′ 𝑥 𝑑𝑥1

0

Examples

𝒙,𝒚 = 𝒙 ∙ 𝒚 =

𝒙,𝒚 𝐴 = 𝒙𝒙𝒚


Vector sapace - Inner product ↔ Norm

A normed space is not necessarily an inner product space (parallelogram equality doesn’t necessarily hold) An inner product space is also normed space with the norm:

→ if orthogonality

Pythagoras’ theorem:

inner product → parallelogram equality


Vector sapace - Inner product ↔ Norm

Show that 𝑥 = 𝑥, 𝑥 is really a norm 1. 𝑥 ≥ 0, 𝑥 = 0 → 𝑥 = 0

𝑥 = 𝑥, 𝑥 ≥ 0 𝑥, 𝑥 = 0 → 𝑥 = 0 (from definition of the inner product) 2. 𝑎𝑥 = 𝑎 𝑥

𝑎𝑥 = 𝑎𝑥, 𝑎𝑥 = 𝑎 𝑥, 𝑎𝑥 = 𝑎2 𝑥, 𝑥 = 𝑎 𝑥, 𝑥 = 𝑎 𝑥 3. 𝑥 + 𝑦 ≤ 𝑥 + 𝑦

𝑥 + 𝑦 = 𝑥 + 𝑦, 𝑥 + 𝑦 = 𝑥, 𝑥 + 2 𝑥, 𝑦 + 𝑦, 𝑦 =

= 𝑥 2 + 2 𝑥, 𝑦 + 𝑦 2 ≤ 𝑥 2 + 2 𝑥, 𝑥 𝑦,𝑦 + 𝑦 2 =

𝑥 2 + 2 𝑥 𝑦 + 𝑦 2 = 𝑥 + 𝑦 2 = 𝑥 + 𝑦 Causchy-Schwarz inequality:


Completeness, denseness

A normed space V is called complete (or a Cauchy space) if every Cauchy sequence of points in V has a limit that is also in V or, alternatively, if every Cauchy sequence in V converges in V

Cauchy sequence

For every positive real number ε, there is a positive integer N such that for all m, n > N

𝑥𝑚 − 𝑥𝑛 < 𝜀

A subset W of a space V is called dense (in V) if every point v in V either belongs to W or arbitrarily "close" to a member of W

Cauchy sequence (source: Wiki)

Cauchy sequence (source: Wiki)

[Chapter 2.4.2]


Important vector spaces

Banach space: complete, normed vector space examples: • Lp spaces • Hilbert space with norm

Hilbert space: complete, inner product space examples • L2 space • Sobolev spaces

Lebesque space (Lp)

with the norm: 𝑓 𝑝 = ∫ 𝑓 𝑥 𝑝dx Ω

1𝑝

L2 space (square-integrable functions): inner product: norm:

𝑓 2 = 𝑓, 𝑓 = �𝑓 𝑥 2dx

Ω

�𝑓 𝑥 2dx

Ω< ∞

𝑓,𝑔 = �𝑓 𝑥 𝑔 𝑥 dx

Ω

[Chapter 2.4.2]


Important vector spaces

Sobolev space (Hp) [On Sobolev space and weak derivatives see Chapter 2.2]: norm: combination of Lp-norms of the function itself and its derivatives up to p order derivatives: weak derivatives→ complete space →Banach space Examples:

𝐻1 ℝ = 𝑢 𝑢, 𝑢′ ∈ 𝐿2 ℝ

𝐻1 ℝ2 = 𝑢 𝑢,𝜕𝑢𝜕𝑥 ,

𝜕𝑢𝜕𝑦 ∈ 𝐿2 ℝ

𝐻2 ℝ = 𝑢 𝑢, 𝑢′,𝑢′′ ∈ 𝐿2 ℝ

𝐻2 ℝ2 = 𝑢 𝑢,𝜕𝑢𝜕𝑥 ,

𝜕𝑢𝜕𝑦 ,

𝜕2𝑢𝜕𝑥2 ,

𝜕2𝑢𝜕𝑦2 ,

𝜕2𝑢𝜕𝑥𝜕𝑦 ∈ 𝐿2 ℝ

𝐻𝑝0 ℝ = 𝑢 𝑢,𝑢′ ∈ 𝐿2 ℝ ,𝑢�Γ𝐷

= 0


Weak derivative

gerneralization for the derivative for functions that are in the Lp space (integrable) but not differentiable (in the strong sense) weak derivative of u(x) is g(x) if g(x) satisfies:

�𝑔𝑏

𝑎

𝑥 𝑣 𝑥 = −�𝑢𝑏

𝑎

𝑥 𝑣′ 𝑥 on ∀ 𝑣 𝑥 ∈ 𝐶∞ 𝑣 𝑎 = 𝑣 𝑏 = 0

(from integration by parts)


Convergency in a normed space

A sequence 𝑥1, 𝑥2, 𝑥3, .. in a normed space is convergent, if there exists a point x∈ X such that, for every real number 𝜖 > 0, there exists a natural number 𝑁 such that

𝑥 − 𝑥𝑛 < 𝜀 for all 𝑛 > 𝑁 The point x, if it exists, is unique, and is called the limit point or limit of the sequence. One can also say that the sequence 𝑥1, 𝑥2, 𝑥3, .. converges to 𝑥. If a sequence does not converge, it is divergent. Note: Pointwise convergency:

lim𝑛→∞

𝑓𝑛 𝑥 − 𝑓 𝑥 = 0 ∀𝑥 Convergence in a normed space:

lim𝑛→∞

𝑓𝑛 𝑥 − 𝑓(𝑥) = 0 Convergence in the infinity norm = uniform convergence

not equivalent!



Example: Show that the sequance:

𝑓𝑛 𝑥 = 𝑥𝑛 on [0,1] converges to 𝑓 𝑥 = 0 in the 𝐿1 norm Proof:

lim𝑛→∞

𝑓𝑛 𝑥 − 𝑓(𝑥) 1 = lim𝑛→∞

� 𝑥𝑛 − 0 𝑑𝑥1

0= lim

𝑛→∞

𝑥𝑛+1

𝑛 + 1 0

1

=

= lim𝑛→∞

1𝑛+1

𝑛 + 1 =1∞ = 0



Example: The sequence:

𝑓𝑛 𝑥 = 𝑥𝑛 on [0,1] • converges pointwise to

lim𝑛→∞

𝑓𝑛 𝑥 = lim𝑛→∞

𝑥𝑛 = � 0 𝑥 ∈ [0,1)1 𝑥 = 1

• in the 𝐿2 norm? • in the 𝐿∞ norm?


Dual space

V* is the dual space of V : 𝑊 = 𝑉∗ = 𝜑:𝑉 → ℝ 𝜑 bounded and linear

with the norm:

𝜑 𝑉∗ = sup𝑢∈𝑉𝑢 ≠0

𝜑(𝑢 )𝑢

Linear functional A functional ℓ, mapping from a vector space V into a scalar ℝ, is linear if

ℓ(𝛼𝑢 + 𝛽𝑣) = 𝛼ℓ(𝑢) + 𝛽ℓ(𝑣) Bounded functional A linear functional ℓ, mapping from a vector space V into a scalar ℝ, is bounded if there exists some 𝑀 > 0 such that for all 𝑢 in V

ℓ(𝑢 ) ≤ 𝑀 𝑢 Continious functional (every linear functional is bounded if and only if it is continious) A linear functional ℓ, mapping from a vector space V into a scalar ℝ, is continious, if for any sequence 𝑣𝑛 which converges to 𝑢, the corresponding sequence converges to ℓ(𝑢) :

lim𝑛→∞

𝑣𝑛 − 𝑢 = 0 → lim𝑛→∞

ℓ(𝑣𝑛) − ℓ(𝑢) = 0

[Chapter 2.4.3]


Riesz representation theorem

𝑉∗ = 𝜑:𝑉 → ℝ 𝜑 bounded and linear Let V be a Hilbert space, and let V* be its dual space, consisting of all bounded, linear functionals. According to the Riesz theorem for any element of V* (𝜑 ∈ V∗) there exists a unique 𝑢 ∈ 𝑉 such that

𝜑𝑢 𝑣 = 𝑢, 𝑣 ∀𝑣 ∈ 𝑉 and moreover

𝜑 𝑉∗ = 𝑢 𝑉 where:

𝜑 𝑉∗ = sup𝑢∈𝑉𝑢 ≠0

𝜑𝑢𝑢

and 𝑢 𝑉 = 𝑢,𝑢

where 𝑢 is an element of V, and ∙,∙ denotes the inner product of the Hilbert space. In other words all the bounded linear functionals mapping from the Hilbert space to a scalar is the same as integration against a measure.

[Chapter 2.4.]


Riesz representation theorem

𝑉∗ = 𝜑:𝑉 → ℝ 𝜑 bounded and linear

𝜑𝑢 𝑣 = 𝑢, 𝑣 ∀𝑣 ∈ 𝑉

The mapping Φ:𝑉 → 𝑉∗ defined by Φ(𝑢) = 𝜑𝑢 is an isometric isomorphism, meaning that: Φ is bijective (surjective + all 𝑢 element of V is paired with exactly one element of 𝑉∗+ injective) The norms of 𝑢 and Φ(𝑢) agree:

Φ(𝑢) 𝑉∗ = 𝑢 𝑉 Φ is additive:

Φ 𝑢1 + 𝑢2 = Φ 𝑢1 + Φ 𝑢2 For all real numbers 𝜆:

Φ 𝜆𝑢 = 𝜆Φ 𝑢 ∀𝜆 ∈ ℝ


Riesz representation theorem - example

All the bounded linear functionals mapping from the L2 space to a scalar can be written in the form:

𝜑𝑢 𝑣 = 𝑢, 𝑣 𝐿2 = �𝑢 𝑥 𝑣 𝑥 dx

Ω ∀𝑣 ∈ 𝐿𝐿

where 𝑢 ∈ 𝐿𝐿, 𝜑𝑢∈ (𝐿𝐿)∗

All the bounded linear functionals mapping from the H1 space to a scalar can be written in the form:

𝜑𝑢 𝑣 = 𝑢, 𝑣 𝐻1 = �𝑢 𝑥 𝑣 𝑥 + 𝑢′ 𝑥 𝑣′ 𝑥 dx

Ω ∀𝑣 ∈ 𝐻1

where 𝑢 ∈ 𝐻1, 𝜑𝑢∈ (𝐻1)∗

A functional 𝑎, mapping from 𝑈𝑥𝑉 into a scalar ℝ, is bilinear if A bilinear functional 𝑎, is symmetric if A bilinear functional 𝑎, is positive definite if A bilinear functional 𝑎, is bounded if there exists an 𝑀 > 0 such that ∀ 𝑢 ∈ 𝑈 and 𝑣 ∈ 𝑉 A bilinear functional 𝑎, is V-elliptic if there exists a 𝛿 > 0 such that ∀ 𝑢 ∈ 𝑈


Bilinear functionals, and its properties

𝑎 𝑢,𝑢 ≥ 0, 𝑎 𝑢,𝑢 = 0 only if 𝑢 = 0

𝑎 𝑢, 𝑣 ≤ 𝑀 𝑢 𝑣

𝑎 𝑢,𝑢 ≥ 𝛿 𝑢 2

Let 𝑎 ∙,∙ be a bounded, V-elliptic bilinear functional and V a Hilbert space Then for any 𝑓 ∈ 𝑉∗ (that is, for any linear, bounded functionals mapping from V to ℝ) there is a unique solution 𝑢 ∈ 𝑉 to the equation:

𝑎 𝑢, 𝑣 = 𝑓(𝑣) and moreover, this unique solution 𝑢 depends continiously on 𝑓:

𝑢 𝑉 ≤1𝛿 f(𝑢) 𝑉∗

Fundamentals| Dr. Noemi Friedman | PDE2 tutorial | Seite 26

Lax-Milgram Lemma [Chapter 2.6]

• If 𝑎 ∙,∙ is a bilinear functional that is • bounded • V-elliptic • symmetric

then it is an inner product, called the energy inner product:

𝑢, 𝑣 𝐸 = 𝑎(𝑢, 𝑣) The corresponding induced norm is called the energy norm:

𝑢 𝐸 = 𝑎(𝑢, 𝑢) In this case (as 𝑎 ∙,∙ is an inner product) it directly follows from the Riesz theorem, that if 𝑓 is a linear, bounded functional, then the equation

𝑎 𝑢, 𝑣𝑢,𝑣 𝐸

= 𝑓(𝑣)

has a unique solution.


Energy inner product, energy norm

positive definite

numerical methods for pdes introduction, fundamentals of...

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