Appunti di Analisi Matematica Avanzata

Numerical modeling of differential problem

Definitions

Ω to denote a domain → subset of ℝ_d Ω d = 1, 2, 3. Ω is an open domain, ∞ people is exhausted, but we set Ω bounded. Ω is also connected; if you take 2 points inside the domain, you can connect them with a line inside of the domain. This is a differential problem on Ω. Normally we divide the forced domain from time variations Q = Ω x I where I = (t₀, T], often t₀ = ∅. True variation is a cylinder. Ω is a spatial domain, no time variations.

Differential problems

Elliptic Problem - ∇ · (μ ∇u) + β∇u + δu = f. u : Ω → ℝ. ∇ denotes divergence, ∇u denotes gradient, and u is a scalar function. μ(x) = ℝ β is a vector Ω → ℝ^d and can still be a function β(x) ∈ ℝ^d. β represents T, ∇, only a convective field, like velocity. d(β ∇u) = ∑ β_i i=1 δu / ∂x_i.

Let's get some physical meaning: -∇(μ∇u) is a diffusion term, for example, a thermal problem where u = Temperature and μ = Conductivity.

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; ∂ purple is unbounded, but we set Ω bounded. Ω is connected; if you take 2 points inside the domain, you can connect them with a line inside of the domain. This is a differential problem on Ω. Normally we divide the forced domain from true variations Q = Ω × I     where   I = (τ₀, τ) , often τ₀ = ∅. Ω is a spatial domain, no time variations.

Differential problems

Elliptic Problem - ∇⋅(μ∇u) + β⋅∇u + δu = f   in   Ω.     u : Ω → k. ∇ denotes divergence, ∇u denotes gradient, and u is a scalar function. μ(x) ∈ ℝ. β is a vector   Ω → ℝ^d and can still be a function   β(x) ∈ ℝ^d. β represents V_i, often a convective field, like velocity. (β⋅∇)u = Σ_i=1^d β_i 𝔡⁄d_xi. -∇⋅(μ∇u) is a diffusion term, for example, a thermal problem where u = temperature and μ = conductivity.

In every elliptic problem, μ(x) > μ₀ > 0 ∀x ∈ Ω. μ must be everywhere greater than zero; it is always bounded away from 0. -∇ · (μ∇u) + (β∇)u + cδu = f in Ω where β is a velocity field. (β∇)u is a convective field or an advective field. cδu is a reaction term, but it can be 0. f is called a source term or forcing term.

Example (membrane) - linearity u = displacement, μ = elastic coefficient, c = 0, β = 0, f = external forces.

To define a differential problem, it is not enough to write a differential equation; we must add boundary conditions. In an elliptic problem, we need to specify conditions on all the boundary of Ω (∂Ω).

Kinds of boundary conditions

• Dirichlet Boundary Condition (essential condition): In Dirichlet Boundary Condition, we impose the value u = g on M₀ ⊂ ∂Ω. The condition is the knowledge of the value of the boundary.
• Neumann Boundary Condition (natural condition): μ ∂u/_n = h, ∂u/_n = t ∂u/_n = in the flux across M_N. With this condition, we don't know the value of the boundary.
• Robin Boundary Condition: μ_f ^∂u/_∂n + αu = h on Γ_f. The important thing is to fill the whole boundary Γ_u ∪ Γ_f ∪ Γ_n = ∂Ω.

Let's write our elliptic problem in the form L u = f where L is our differential operator L = -∇・(μ∇) + (β∇) + σ.

Classification of PDE's

The differential equations can be classified based on their mathematical formulation in 3 different families: elliptic, parabolic, and hyperbolic. L u = A_∂²u/_∂x1² + B_∂²u/_∂x2² + C_∂²u/_∂x1∂x2 + D_∂u/_∂x1 + ε_∂u/_∂x2 + F u = G. Coeffs A, B, C, D, E, F, ∈ R, the classification is made on the region of the △ = B²-4AC. If △ < 0.