Robin type boundary conditions

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Robin-type boundary conditions April 30, 2016 In the course, only for the sake of time, we have not dealt with Robin-type boundary conditions, sometimes called conditions of the third type. Since they may be relevant and may have a consequence on the stability of transport-advection problem I am writing this note as a complement to the things told at lecture. The notes are rather brief, you need to complement them with what seen at lecture and during the exercise sessions (or in the books that have been suggested). 1 Robin b.c. for Laplace problem Let’s consider    − div(µ∇u) =f in Ω, u = 0 on Γ D⊂∂Ω, µ∂u ∂n +γu =hr on ΓR⊂∂Ω µ∂u ∂n =hn on ΓN⊂∂Ω, (1) where µ satisfies the usual conditions, in particular µ(x≥ µ0 > 0 and f ∈ L2(Ω), hr ∈ H1/2(ΓR). Here, γ is a constant value, but in general may a function. Note that if γ = 0 we recover the standard Neumann condition. We have used homogeneous Dirichlet boundary conditions for simplicity (but the considerations exposed here are valid in general). We exploit the fact that, for any regular function v that takes zero values on ΓD, we have − ∫ Ω div(µ∇u)v = ∫ Ω µ∇u·∇v− ∫ ∂Ω µ∂u ∂nv = ∫ Ω µ∇u·∇v− ∫ ΓN hnv− ∫ ΓR hrv + ∫ ΓR γuv, where we have exploited the boundary conditions. If we defineV ={w∈H1(Ω) : w|ΓD = 0} we immediately recognize that the weak form is ?u∈V such that a(u,v ) =F (v) ∀v∈V, (2) where a(u,v ) = ∫ Ω µ∇u·∇v +γ ∫ ΓR uv, and F (v) = ∫ Ω fv + ∫ ΓN hnv + ∫ ΓR hrv. Continuity and (bi)linearity is standard, with standard hypotheses on the data. Let’s look at coercivity, so we consider a(v,v ) for v∈ V . We first assume that |ΓD| > 0, i.e. we are 1 imposing Dirichlet conditions on a portion of the boundary (of non null measure). In this case we can apply Poincar´ e inequality and ∫ Ω µ∇v·∇v≥ µ0 1 +C2 Ω ∥v∥2 V (3)…