Parabolic Problems

Numerical Methods for Differential Eqns

Parabolic Problems

_du/_dt - Δu = f

_∂(c_pu)/_∂t - div(k∇u) = g

example of parabolic problem, we assume temperature could change in time

Weak Formulation:

∫_Ω ^∂u/_∂t v dx + ∫_Ω ∇u ∇v dx = ∫_Ω fv dx

∀t we assume homogeneous Dirichlet conditions on the boundary:

• u=0 on ∂Ω
• v=0 on ∂Ω

i.c. u(x,t_o) = u₀(x)

_∂ ∈ [t_o, T]

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

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

● sort of time fixed eqn:

a(u,v) = ∫_Ω ∇u ∇v dx

• a(u,u) ≤ G(||u||H1+||u||H1)2
• a(u,u) ≥ α ||u||H12, α>0

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 - Δu = f → can provide an estimate of stability

_∂u + Δu = f → not possible to provide stability → great issue!

for Laplace problem u ∈ H^1₀(Ω) = {u∈L²: ∇u ∈L²}

• here ∀t u(.,t) ∈ H^1₀(Ω)

• ∀x u(x,.) ∈ C¹((t₀,t_f)) = {u∈C^.: _∂u/_∂t ∈ C^,}

→ u ∈ C¹((t₀,t_f), H^1₀(xL))

Bachner space

u: (t₀,t_f) → H^1₀(Ω)

Continuity ⇒ || u((t_n) - u(t)||^H1 → 0

^lim_h→0 (u(t_n+h) - u(t))/h = _∂u/_∂t