Inf-sup Conditions

where u can be interpreted as the velocity field of an incompressible fluid motion, and p is then the associated pressure, the constant μ is the viscosity coefficient of the fluid. For simplicity, we consider homogenous Dirichlet boundary condition for the velocity, i.e. u|∂Ω = 0 and μ = 1. The conditions for the well posedness is known as inf-sup condition or Ladyzhenskaya-Babuška-Breezi (LBB) condition. Multiplying test function v ∈ H0(Ω) to the momentum equation (1) and q ∈ L(Ω) to the mass equation (2), and applying integration by part for the momentum equation, we obtain the weak formulation of the Stokes equations: Find u ∈ H0(Ω) and a pressure p ∈ L(Ω) such that (∇u,∇v)− (p,div v) = (f ,v), for all v ∈H0(Ω) −(divu, q) = 0 for all q ∈ L(Ω).