The Boundary Element Method for the Helmholtz Equation ... - FEI VÅ B

293 Boundary Integral EquationsAt the beginning of this section we introduce some operator properties, which will be usedto prove existence and uniqueness of solutions to boundary integral equations derived laterin Section 3.5. In the following definitions we assume X and Y to be some Hilbert spaces.Definition 3.1. A linear operator A: X → X ∗ is X-elliptic if there exists a constantc A 1 ∈ R + such that the inequalityholds for all u ∈ X.Re ⟨Au, u⟩ ≥ c A 1 ∥u∥ 2 XDefinition 3.2. A linear operator A: X → Y is bounded if there exists a constant c A 2 ∈ R +such that the inequality∥Au∥ Y ≤ c A 2 ∥u∥ Xholds for all u ∈ X.Remark 3.3. Note that for linear operators boundedness is equivalent to continuity.Definition 3.4. A linear operator A: X → X ∗ is coercive if there exists a compactoperator C : X → X ∗ and a constant c A 1 ∈ R + such that the Gårdings inequalityholds for all u ∈ X.∂ΩRe ⟨(A + C)u, u⟩ ≥ c A 1 ∥u∥ 2 XIn the previous section we showed that in the smooth case the solution to the Helmholtzequation on a bounded domain can be represented as∂uu(x) =∂n (y)v κ(x, y) ds y − u(y) ∂v κ(x, y) ds y for x ∈ Ω. (3.1)∂n yFor the solution on an unbounded domain we have similarly∂uu(x) = −∂n (y)v κ(x, y) ds y + u(y) ∂v κ(x, y) ds y for x ∈ Ω ext . (3.2)∂n y∂Ω∂ΩIn both cases the solution u is determined by the Cauchy data, i.e., by the values of thesolution on the boundary and its normal derivatives. However, these data are not givenbeforehand as the boundary value problem would be overdetermined. When considering amixed problem, we are only given Dirichlet data on a part of the boundary and Neumanndata on the remaining part. In this section we show how the Cauchy couple can be obtainedfrom boundary integral equations derived from the representation formulae (3.1) and (3.2).Afterwards, these formulae can be used to compute the solution in any point x ∈ Ω orx ∈ Ω ext , respectively.∂Ω