Numerical modeling of differential problems - Appunti completi

Numerical Modelling of Differential Problems

Introduction to Numerical Analysis

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

• F can be any type of eqt: linear, int, partial diff...
• F, d known → direct problem
• F, x known → inverse problem
• x, d known → identification problem

Problem is said to be WELL-POSED when the pb admits a solution, UNIQUE and continuously depending on the data.

If introduced a small perturbation on data is expected to have a small perturbation on the solution F(x+δx, d+δd)=0

∇y∃δ : 3K(y, d) : ||δd|| < ε ⇒ ||δx|| < K(d, y, d) ||δd||

Can be defined the CONDITION NUMBER as the number that quantifies the well-posed property of a problem

Relative Condition Number:

K(d) = sup_δdε ||δx||/||x|| / ||δd||/||d||

"Which is the entity of the relative perturbation when introduced a pert. on the data"

Absolute Condition Number:

K_abs(d) = sup_δdε ||δx|| / ||δd||

For big K (≥1000) the pb is ILL-CONDITIONED and the solution is totally different

small K (≤1-100) the pb is WELL-CONDITIONED

NUMERICAL METHODS

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

A method can be said CONSISTENT if: F_n(x, d) = F_n(x, d) - F(x, d) ⇒0

∇n→∞

If this property is true for every n the method is said STRONGLY CONSISTENT

F_m(x, d) = F∀n

A method is said to be STABLE (or well-posed) if ∇ fixed n, ∃! solution

x_n fort d and x_n depends continuously on the data, similarly to the continuous pb.

F_n(x_n+δx_n, d_n+δd_n)=0

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

Remark. Sobolev spaces are Banach spaces for 1 ≤ p ≤ ∞; for p = 2, W^1,2(Ω) is also a Hilbert space. The case for p = 2 is the most common in FE analysis and we will use the notation H¹(Ω) = W^1,2(Ω). For k = 1 we have

H¹(Ω) = {u ∈ L²(Ω); ∇u ∈ L²(Ω)}

with the following inner product and norm

(u,v)_H¹(Ω) = (u,v)_L²(Ω) + (∇u, ∇v)_L²(Ω)

‖u‖_H¹(Ω) = ‖u‖_L²(Ω) + ‖∇u‖_L²(Ω)

Remark. For d = 1 it can be shown that its functions are continuous for d = 1, may lack values at certain isolated points for d = 2 and be discontinuous along a curve for d = 3.

How to define the boundary values of a function belonging to a Sobolev space?

IDEA: make a smooth approximation from within the domain, and then evaluate this approximation on the boundary. This is called the trace of the function. We can define the trace operator

γ : W^1,p(Ω) → L^p(∂Ω)

and the space

H₀¹(Ω) = {v ∈ H¹(Ω) : (γv)|_∂Ω = 0}

Classification of PDEs

The equations of elasticity (no inertial terms) or the Laplacian are elliptic PDEs.

∇⋅σ + F = 0, ε = ¹/₂ [∇u + (∇u)^T], σ = C : ε, -△u = f

The heat conduction equation is an example of a parabolic PDE.

^∂T/_∂t - ∇⋅(k∇T) = 0

Hyperbolic PDEs describe wave propagation and transport phenomena.

^∂2u/_∂t² - c²△u = 0, ^∂u/_∂t + a ⋅ ∇u = 0

FINITE DIFFERENCES

f^′(x_i) = lim_h→0 [f(x_i + h) - f(x_i)]/h

u_i^FD = [f(x_i+1) - f(x_i)]/h, u_i^BD = [f(x_i) - f(x_i-1)]/h, u_i^CD = [f(x_i+1) - f(x_i-1)]/2h

f(x_i+1) = f(x_i) + h f^′(x_i) + ^h2/₂ f^″(ξ_i), f^′(x_i) = f^′(x_i) - u_i^FD = - ^h/₂ f^″(ξ_i)

f^′(x_i) - u_i^BD = ^h/₂ f^″(ξ_i), f^′(x_i) - u_i^CD = - ^h2/₆ f^‴(ξ_i)

∫₀¹ w''v'dx = ∫₀¹ fvdx - ∫₀¹ r''v'dx

the solution of the pb is a translation of the solution obtained for w

A non-homo Dirichlet problem can be recast into an homo one,

where the pb can be solved for w then recall u = w+r

MINIMIZATION FORMULATION of the pb

A third formulation of the DIFFERENTIAL PROBLEM can be introduced

J(v) = ½ ∫₀¹ v'v'dx - ∫₀¹ fvdx

the solution u to the diff. pb is the one which minimizes the internal

energy of the system F(u)

(H) min J(v) = min ½ ∫₀¹ v'v'dx - ∫₀¹ fvdx

(D) - u'' = f in (0,1)

u(0) = u(1) = 0

(V) find u ∈ H¹₀(0,1) s.t.

∫₀¹ u'v'dx = ∫₀¹ fvdx ∀v ∈ H¹₀(0,1)

(M) find u ∈ H¹₀(0,1) s.t.

min F(v) = min ½ ∫₀¹ v'v'dx - ∫₀¹ fvdx

v ∈ H¹₀(0,1) v ∈ H¹₀(0,1)

⇒ (D) ⇔ (V) ⇔ (M)

⇒ if u'' ∃ and is continuous

(V) ⇒ (M) ∀ε H¹₀(0,1)

if u is solution

of (V) then is

solution of (M)

Now we want to check if our pb is well-posed, so Lax-Milgram lemma's hypothesis must be satisfied.

-Δuⁿ = f in Ω

u = 0 on ∂Ω

a(u,v) = ∫_Ω∇v·∇v dx

1. V is an Hilbert space
2. Continuity of a(u,v),

|a(u,v)| = |∫_Ω∇v·∇v dx| ≤ ∫_Ω||u||_L2||v||_L2 ≤ ||u||_L2||v||_L2

Continuity of F(v):

|F(v)| = |∫_Ωf·v dx| ≤ ||f||_L2||v||_L2 ≤ ||F||_L2||v||_L2

3. Coercivity of a(v,v)

a(v,v) ≥ α||v||_H0¹² >0

∫_Ω||v||^L2

a(v,v) = ∫_Ω∇v·∇v dx = ||∇v||^L2

||v||^L2 = ||v1||^L2 + ||v2||^L2

≤ c∑||v1||^L2 + ||v2||^L2

≤ (r+c2)∑||v||^L2

||v||_H0¹² =

a(v,v) = ∫_Ω∇v·∇v dx = ||∇v||² ≥

||v||^H0¹

Proved these hy it is possible to say that the continuous problem: find u∈Γ a(u,v)=F(v) ∀ v∈ V in well-posed → solution ∃, it's unique and depends continuously on data

Using the considerations made with ii the discrete pb (c) is well-posed too

Strong Form

-Δu = f in Ω u = 0 on ∂Ω

Weak Form: find u∈Γ a(u,v)=F(v) ∀ v∈ V

Algebraic pb

Au=f where A contains the coeff of u_h in V_h

a(∫_φ∫_φ) = ∫₀ ∫₀ ∫_φ∫_φ'