Well-Posedness of the Xxxxxx Iterative Method Sample Clauses

Well-Posedness of the Xxxxxx Iterative Method. 0 We first introduce two auxiliary linear problems. Let β ∈ H1 be a given vector field. For the first problem we want to find u and p such that  −∇ · ..2µ + τs Σ DuΣ + ∇p = f |Dβ|ε ∇ · u = 0 in Ω (4.3) with u = 0 on ∂Ω for simplicity. We define the bilinear form ∫ aβ(u, v) = Ω 2µ + τs Du : Dv . Σ |Dβ|ε on V × V. The weak formulation of (4.3) reads: find (u, p) ∈ H1 × L2 such that 0 aβ(u, v) + b(p, v) − b(q, u) = (f, v) for all (v, q) ∈ H1 × L2. (4.4) Proposition 4.1 Given f ∈ L2 and β ∈ H1 there exists a unique solution (u, p) to