Numerical modeling of differential problems - Appunti completi

Name: Numerical modeling of differential problems - Appunti completi
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Appunti completi di Numerical modeling of differential problems contenenti i seguenti argomenti (corso in lingua inglese):- Problemi ellittici- Proprietà dei metodi- FEM- Problemi …

Esame Numerical Modeling of Differential Problem

Facoltà Ingegneria industriale

Dal corso del Prof. Miglio Edie

Università Politecnico di Milano

A.A. 2022-2023

63 pagine

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Estratto del documento

Numerical modelling of differential problems

Introduction to numerical analysis

Formulation of a generic problem: F(x,d) = 0

F, d known ⇒ direct problem
F, x known ⇒ inverse problem
x, d known ⇒ identification problem

A problem is said to be well-posed when the problem admits a solution, is unique and continuously depends on the data. If introduced a small perturbation on data, it is expected to have a small perturbation on the solution:

F(x+δx, d+δd)=0 ∀y>0 ∃K(y,d): ‖δd‖

If K is large (>1000) the problem is ill-conditioned and the solution is totally different.

If K is small (≤7-100), the problem is well-conditioned.

Numerical methods

Discrete numerical counterpart F_n(x_n,d_n)=0 n>1 of the continuous problem:

A method can be said consistent if:

F_n(x,d) = F_n(x,d) - F(x,d) ⇒ 0

If this property is true for every n, the method is said to be strongly consistent:

F_n(x,d) = ∀n

A method is said to be stable (or well-posed) if ∀ fixed n, there exists a unique solution x_n for d and x_n depends continuously on the data, similarly to the continuous problem:

F_n(x_n+δx_n,d_n+δd_n)=0

If the continuous problem is well-posed, passing to the numerical method we want to maintain this property, unless it’s useless.

A numerical method is said to be convergent to the true solution if:

∥x-x_n∥ → 0 for n → ∞

Measure of convergence:

ϵ(x_n) = |x-x_n|
ϵ_rel(x_n) = |x-x_n| / |x| if x≠0

Relation between consistency, stability and convergence: Lax-Richtmeyer equivalence method ⇒ for a consistent numerical method, stability is equivalent to convergence.

Ex: -u''(x) = f(x)

Anteprima

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