Parabolic Problems

NUMERICAL METHODS FOR DIFFERENTIAL EQNS

PARABOLIC PROBLEMS

_∂u^∂t - ∆u = f

∫_Ω ∂u/∂t v dx + ∫_Ω ∇u ∇v dx = ∫_Ω f v dx ∀ t

i.c. u(x, t_o) = u₀(x)

b.c. generally could be u(x, t) = g(x, t) on ∂Ω

time and space treated in n ways, we separate the 2 coordinates

a(u,v) = ∫_Ω ∇u ∇v dx

We assume existence and uniqueness of solution, no proofs will be given

Stability?

∂u/∂t + ∆u = f → can provide an estimate of stability

∂u/∂t ≠ f → not possible to provide stability → great issue!

for Laplace problem u ∈ H¹₀(Ω) = {u ∈ L² : ∇u ∈ L²}

here ∀ t u(., t) ∈ H¹₀(Ω)

Continuity ⇒ ||u(α t_m) - u(t)| ‖_H¹→ 0

lim_h→0 (u(t₀+h) - u(t₀))/h = ∂u/∂t

NUMERICAL METHODS FOR DIFFERENTIAL EQNS

PARABOLIC PROBLEMS

∂u/∂t - Δu = f

∫_Ω ∂u/∂t v - ∫_Ω ∇⋅∇v dx = ∫_Ω f v dx  ∀ t

i.c.  u (x, t₀) = u₀ (x)

• We assume homogeneous Dirichlet conditions on the boundary:
• time and space treated in t ways, we separate the 2 coordinates
• sort of time fixed epn.
• a(u,v) = ∫_Ω ∇u ∇v dx  ⟶ a(u,u) ⩾ α ‖u‖²  α>0
• a(u,u) ⩽ G ‖u‖_H¹ ‖u‖_H¹
• We want to use these results to estimate solution

Well-posedness

(existence and uniqueness of solution, stability)

• unknown u ; data f, g, u₀, ...
• We assume existence and uniqueness of solution, no proofs will be given
• Stability?  ⟶ ∂u/∂t + Δu = f
• ⟹ can provide an estimate of stability
• ∂u/∂t - ∆u = f ⟹ not possible to provide stability ⟶ great issue
• for Laplace problem  u ∈ H¹₀(Ω) = {u ∈ L²: ∇u ∈ L² }
• here  ∀ t  u (., t) ∈ H¹₀(Ω)
• Bochner space  u : (t₀,t_f) ⟶ H¹₀(Ω)
• Continuity ⇒ ‖ u(., tn) - u(E) ‖_H¹ ⟶ 0
• Assuming that u ∈ C¹((t₀,t_f),H¹(Ω)) is like assuming ∃ and uniqueness of the solution

∫_Ω ∂_tu(t) ∇v(t) ∇u(t) dx = ∫_Ω fu(t) dx

We expect V to be L², in order to perform ∫_Ω∇fv dx

Inequality ‖f g‖_X ≤ ‖f‖_P ‖g‖_q ¹/_p + ¹/_q = 1

ƒ ∈ L²(Ω), ‖f‖_L^p(Ω) = (∫_Ω|f|^P dx)^1/p c ∞, L² is a particular case (removed space if p=2)

At least must ask v ∈ H_0¹(Ω), here or v = u(t)

- ∂_tu(t) + ¹/₂ ∇u(t) = - ∫_Ω u(t) ∇u(t) dx = ⟨u ⋅ ∇u(u[1])⟩ = α(u,u)

¹/₂ ∫_Ω |∇u(t)|² dx + α(u(t),u(t)) = ⟨fu(t) dx ⟩

≤ ‖∫_Ω ρ_u (t)‖ dx ‖u(t)‖_L²(Ω)‖∇u(t)‖_H¹0(Ω)

¹/₂ ∂_t ‖u(t)‖²_L² + α‖u(t)‖²_H¹0(Ω) = ∫_Ω fu(t)) dx = ⟨f ⋅ ∇u⟩ + α(u(t),u(t)) = ∫_Ω fρ_u(t) dx = e dtilde L f ⋅‖u(t)‖_L²

Young Inequality: (a|b| ≤ ¹/₂ (α² + b²) or (a|b| ≤ ^ε/₂ (εa² + ¹/_ε |b|²)

¹/₂ ∂_t ‖u(t)‖²_L² + α‖u(t)‖²_H¹0(Ω)

≤ ¹/_2α β‖u(t)⁄_L²

∫_t₀^tp ‖u(t)⁄_L² ‖² dt = ‖u(t)⁄_L²‖² − ‖u(t)⁄_L²‖

⇒ Stability Estimate: ∫ ‖u(tε)⁄_L²‖² + α ∫_t₀^tp ‖u(t)⁄_H¹0(Ω)‖² dt ¹ α Wim’Im Ballyt ¹/_α + |u(tε)‖²

◻ Problem Discretization

• Weak form: ∫_Ω ∂_tv dx + α(u,v) = ∫_Ω f ⋅ dx ∀ (v ϵ H¹⁰(Ω)) (∀ ∈ (t₀,tp))

• v ≠ ϕ_i β(x,t) = ∑ ε_i υ_i(t) + ∋

(ε_iυ_j(ϕ_j,ϕ_i) + ε ∫_Ω ∂(ε_j;υ_iϕ_j) ⋅ dx = ∑ ε_jι∂ϕ_i = ∫_Ω ░ϕ_i dx bzärer b₂ = &exists; other row, associated to adjoint things rows, associated tobasis functions

Stiffness matrix = A_ij = α(ϕ_j,ϕ_i);

Mass matrix = M_ι = ∫_Ω ι (ϕ_i) dx)