Appunti di Analisi Matematica Avanzata

Numerical Modeling of Differential Problem

Definitions

Ω to denote a domain - subset of ℝ Ω ⊂ ℝ^d, d = 1, 2, 3

Ω is an open domain, &partial; maybe is unbounded, but we want Ω bounded.

Ω is also connected, if you take 2 points inside the domain, you can connect them with a line inside the domain. This is a differential problem on ℝ

Normally we denote the parred domain from time variations.

Q = Ω × I where I = (t₀, T), often t₀ = ∅

Ω is a spatial domain, no time variations.

Differential Problems:

• Elliptic Problem

- ∇ · (μ ∇ u ) + β ∇ u + δ u = f   in   Ω   u : ℝ → k

μ(x) → ℝ

β is a vector   ℝ → ℝ^d can be still a function β(x) ∈ ℝ^d

β represent typically a convective field, like velocity.

&nablan; (μ∇u) is a diffusion term, for example thermal problems.

In every elliptic problem μ(x) > μ₀ > ∅ ∀x ∈ Ω

μ must be everywhere greater than zero, is always bounded away from ∅

-∇⋅(μ∇u) + (β⋅∇)u + σu = f in Ω where β is a velocity field

(β⋅∇)u is a convective field or an advective field, β can be ∅, it doesn't disappear

σ(u) is a reaction term, but σ can be ∅

f is called source term or forcing term, in thermal problem is an external source of heat

Example (membrane) - elasticity

u = displacement

μ = elastic coefficient

σ = ∅

β = ∅

f = external forces

To define a differential problem is not enough to write a differential equation, we must add boundary conditions

In elliptic problem we need to specify conditions on all the boundary of Ω (∂Ω)

Kinds of boundary conditions

1. Dirichlet boundary condition (essential condition)

In Dirichlet boundary condition we impose the value u=g on M_D ⊂ ∂Ω, the condition as knowledge of the value of the boundary

2. Neumann boundary condition (natural condition)

μ ∂u/∂n = h ∂u/∂n = ∂u/∂n, n the flux across ∂Ω

With this condition we doesn't know the value of the boundary

boundary conditions are slightly the same, small portions of the data corrupt to small

corrections in the solution.

The solution of a differential problem depends with continuity on the data,

if ∀ε > 0 ∃ δ |0 < δ < 0.5| such that P(u, g) ≠ ∅ , P(u^δ, g+ δ g) with ||δf|| ≤ δ

Then ||u – u^δ|| ≤ ε

Example:

f(x) = x, g(x) = x²+x³

(no solution)

Assign u^δ by a “lottery”, can have ∅, 1, ∞ solution—no well posedness

Continuity for linear problem

Continuity with respect to data means that exist a constant, ∃ C < ∞: ||u || ≤ C ||f ||

It suffices to prove that the problem depends with continuity from the data

A numerical scheme replaces the original problem with

P_N (u_j, g_j) where u_j an approximation of u.

That depends on a finite

number (N) of coefficients called degrees of freedom, those coefficients are an

unknown.

h is a discretization parameter (mesh size or time step), such that N → ∞

when h → 0

The finer your discretization, the more coefficients we will need to find an approximate

solution.

For an approximation of the data such that g ^_h → g as h → 0

Consistent numerical scheme

A numerical scheme is consistent if (converges to)

P_N (u_j, g_j) = P_N (u_j^δ), P(u^j, g^j) → ^{. . .}

for h, N → ∞

I take my numerical scheme and I plug in the exact solution, converges to know it

instead of the approximation

BANACH SPACE

A normed, complete, and linear is called BANACH SPACE

A Banach space equipped with internal product and induced norm is called

HILBERT SPACE

Let Ω be a domain, we want to consider spaces of function ℝ → ℝ or (ℝ → ℝ^d)

• C^k(Ω) = space of functions continuous up to k-th derivative
• C^k(Ω̅) = space of functions continuous up to the border

EXAMPLE

Ω = (0, π/2)

• tan(x) ∈ C^ω(Ω)
• tan(x) ∉ C⁰(Ω)
• C^k(Ω̅) ∈ (C^ℓ(Ω̅)) k ≤ ℓ

||u|| _C⁰(Ω) = max_{x ∈ Ω} |u(x)|

The gradient (∇) and the laplacian (∇², Δ) in general notation are independent from the frame of reference (Tensorial)

EXAMPLE

^∂2u/_∂x² + ^∂2u/_∂y² = ≤ in Ω (0,1)²

u = Ø on ∂Ω

CLASSICAL SOLUTION

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