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

54 4 Discretization and Numerical Realizationthe well studied Galerkin scheme based on the corresponding variational formulations. Toapproximate the Dirichlet and Neumann boundary data we use continuous piecewise affinefunctions and piecewise constant functions, respectively.4.3.1 Interior Dirichlet Boundary Value ProblemThe solution to the interior Dirichlet boundary value problem (3.24) is given by the representationformula (3.25) with unknown Neumann data g N := γ 1,int u. To compute themissing values we use the Fredholm integral equation of the first kind(V κ g N )(x) = 1 2 g D(x) + (K κ g D )(x) for x ∈ ∂Ωor equivalentlyg N (y)v κ (x, y) ds y = 1 2 g D(x) +∂Ωwith the fundamental solution v κ and the normal derivative∂Ωg D (y) ∂v κ∂n y(x, y) ds y for x ∈ ∂Ω (4.3)∂v κ(x, y) = 1 e iκ∥x−y∥(1 − iκ∥x − y∥)⟨x − y, n(y)⟩.∂n y 4π ∥x − y∥3 The corresponding variational formulation reads 1⟨V κ g N , s⟩ ∂Ω =2 I + K κ g D , sor∂Ω∂Ωfor all s ∈ H −1/2 (∂Ω)s(x) g N (y)v κ (x, y) ds y ds x∂Ω= 1 s(x)g D (x) ds x + s(x) g D (y) ∂v κ(x, y) ds y ds x .2 ∂Ω∂Ω ∂Ω ∂n yWe seek the solution in the approximate formg N ≈ g N,h :=Egl N ψ l ∈ T ψ (∂Ω) (4.4)l=1which can be identified with a vector g N ∈ C E . Furthermore, we use the L 2 projection toapproximate the given Dirichlet datag D ≈ g D,h :=Ngj D ϕ j ∈ T ϕ (∂Ω), (4.5)j=1