Stokes Problems

Numerical Methods for Differential Eqns

Stokes Problems

• Stokes equation
  • \( u : \Omega \rightarrow \mathbb{R}^d \) (We focus on 2D problems)
  • \( u(x,y) = \begin{pmatrix} u_x \\ u_y \end{pmatrix} \)
  • \(\nabla u = \begin{pmatrix} \frac{\partial u_x}{\partial x} & \frac{\partial u_x}{\partial y} \\ \frac{\partial u_y}{\partial x} & \frac{\partial u_y}{\partial y} \end{pmatrix}\)
  • \(\text{div} \, u = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = \text{tr}(\nabla u)\)
  • \(A : B = \sum_{i,j=1}^{d} a_{ij} b_{ij} = a_{11} b_{11} + a_{12} b_{12} + b_{21} a_{21} + a_{22} b_{22}\)
  • \(\text{div} \, u = \text{tr} (\nabla u) = \nabla \cdot u = f\)
  • \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

• Stokes System
  • \(\text{div} \, \sigma(u,p) + g = 0\) (Conservation of momentum)
  • We focus on fluids, Incompressible Newtonian fluids
    • \(\sigma = 2\mu \mathcal{E}(u) - pI\)
    • \(\text{div} \, \sigma = \text{div}(2\mu \mathcal{E}(u) + pI) = 0\)
    • \(\text{div} \, u = 0 \rightarrow \nabla \cdot u = 0\)
  • We'll study a simplified problem: -div (σ(u) - pI) + f = g
    • Strong form:
      • \(-\Delta u + \nabla p = f \quad \text{in } \Omega \)
      • \(\text{div} \, u = 0 \quad \text{in } \Omega\)
      • \(u = 0 \quad \text{on } \partial \Omega\)
• Weak Formulation
  • \(\int_{\Omega} \nabla u : \nabla v - p \text{div} \, v + f v \, dx = 0 \quad \forall v \in H^1_0(\Omega)\)
  • \(\int_{\Omega} \text{div} \, q \, dx = 0 \quad \forall q \in L^2(\Omega)\)
• Introducing notation: \(V = H^1_0(\Omega, \mathbb{R}^d), \, Q = L^2_0(\Omega)\)

2: ∀v x V_α → ℝ; u,v → a(u,v) = ∫_Ω ∇u : ∇v dx

b: V x Q → ℝ ; u,p → b(u,p) = - ∫_Ω div u p dx

Can rewrite the problem as: find (u,p) ∈ V x Q such that

• a(u,v) + b(v,p) = F(u) ∀v ∈ V
• + b(v,q) = 0 ∀q ∈ Q

F: x ∈ V → F(x) = ∫_Ω f x dx

|F(x)| ≤ ∫_Ω |f x| dx ≤ ||f||_L²(Ω) ||x||_L²(Ω) ≤ ||f||_L²(Ω) ||x||_V, ∀x ∈ V

|F(x)| is bounded, so it is a continuous functional

a(u,v) := ∫_Ω ∇u : ∇v dx

Continuity: |a(u,v)| ≤ ||u||_V ||v||_V

Coercivity: a(u,v) ≥ α ||u||², α>0

b(u,q) := -∫_Ω div u q dx

|b(u,q)| = |∫_Ω div u q dx| ≤ ∫_Ω |div u q| dx ≤ ||div u||_L²(Ω) ||q||_L²(Ω)

∫_Ω |∂u_k/∂x + ∂u_x/∂y| dx = ∫_Ω | ∂u_k/∂x | + | ∂u_y/∂y | + 2 ∂u_x ∂u_y / x q dy

≤ || ∂u_x

|| div u ||²_L²(Ω) ≤ ||u||²_L²(Ω)

|b(u,q)| ≤ 2 ||div u|| ||q|| bounded

b(u,q) is continuous

Stability

• u=x, p=q
• a(u,q) + b(u,p) = F(u)
• b(u,p)=0

⇒ a(u,q) = F(u)

α ||u||² ≤ a(u,q) = F(u) ≤ ||f||_L² ||u||_L²(Ω)

⇒ ||u||/√α ≤ ||f||_L²(Ω) ⇒ y bounded

Stability of P?

INF – SUP condition

Property:

sup b(x,q) / ||x||_V ≥ β ||q||_L ∀q ∈ Q