Finite Element Methods for Stokes Equations
نویسنده
چکیده
1. STOKES EQUATIONS In this section, we shall study the well posedness of the weak formulation of the steadystate Stokes equations −μ∆u +∇p = f , (1) −divu = 0, (2) 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. 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) (3) −(divu, q) = 0 for all q ∈ L(Ω). (4) The conditions for the well posedness of a saddle point system is known as inf-sup conditions or Ladyzhenskaya-Babuška-Breezi (LBB) condition; see Chapter: Inf-sup conditions for operator equations for details. The setting for the Stokes equations: • Spaces: V = H0(Ω), with norm |v|1 = ‖∇v‖, P = L0(Ω) = {q ∈ L(Ω), ∫
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