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

30 3 Boundary Integral EquationsIntroducing the integral operatorsV κ : H −1/2 (∂Ω) → Hloc 1 (Ω), ( V κ s)(x) :=W κ : H 1/2 (∂Ω) → Hloc 1 (Ω), (W κt)(x) :=∂Ω∂Ωv κ (x, y)s(y) ds y , (3.3)∂v κ∂n y(x, y)t(y) ds y (3.4)with density functions s, t: ∂Ω → R allows us to rewrite the representation formulae (3.1)and (3.2) asu = V κ γ 1,int u − W κ γ 0,int u in Ω,u = − V κ γ 1,ext u + W κ γ 0,ext u in Ω ext .Remark 3.5. The operators V κ and W κ are usually called the single layer potential operatorand the double layer potential operator, respectively. However, there is a naming conflict.In the following sections we also use these terms for composite operators γ 0 Vκ and γ 0 W κ ,where γ 0 denotes the interior or exterior Dirichlet trace operator.Theorem 3.6. The operator V κ : H −1/2 (∂Ω) → Hloc 1 (Ω) is linear and continuous. Hence,for bounded domains there exists a constant c ∈ R + such that∥ V κ s∥ H 1 (Ω) ≤ c∥s∥ H −1/2 (∂Ω)for all s ∈ H −1/2 (∂Ω).Moreover, for all s ∈ H −1/2 (∂Ω) the function V κ s satisfies the Helmholtz equation in theweak sense (including the Sommerfeld radiation condition in the case of an unboundeddomain).Theorem 3.7. The operator W κ : H 1/2 (∂Ω) → Hloc 1 (Ω) is linear and continuous. Hence,for bounded domains there exists a constant c ∈ R + such that∥W κ t∥ H 1 (Ω) ≤ c∥t∥ H 1/2 (∂Ω)for all t ∈ H 1/2 (∂Ω).Moreover, for all t ∈ H 1/2 (∂Ω) the function W κ t satisfies the Helmholtz equation in theweak sense (including the Sommerfeld radiation condition in the case of an unboundeddomain).The preceding theorems allow us to seek the solution to the Helmholtz equation in theform u = V κ s or u = W κ t with unknown density functions s, t. This approach gives rise toindirect boundary element methods, which will be mentioned later.Properties of the above given potential operators, which will be discussed in the followingsections, can be found in [12] (see also [9] and [18]). These properties will be crucialfor the derivation of boundary integral equations (both direct and indirect), which will beused for the computation of the missing Cauchy data.