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

10 1 Function SpacesFirst of all, we recall how normal derivatives are treated in the finite element method.To derive the weak formulation of a boundary value problem we will use the first Green’sidentity.Theorem 1.5 (First Green’s Identity). Let Ω ⊂ R d be a bounded C 1 domain and let ndenote the unit exterior normal vector to ∂Ω. Then for u ∈ C 2 (Ω), v ∈ C 1 (Ω) the firstGreen’s identityis satisfied.Ω∆u(x)v(x) dx =∂Ω∂u∂nΩ(x)v(x) ds − ∇u(x)∇v(x) dx (1.7)Remark 1.6. The normal derivatives in the preceding theorem should be understood as∂u(x) := lim ⟨∇u(x − hn(x)), n(x)⟩∂n h→0 +for x ∈ ∂Ω.Corollary 1.7 (Second Green’s Identity). Let Ω ⊂ R d be a bounded C 1 domain and let ndenote the unit exterior normal vector to ∂Ω. Then for u, v ∈ C 2 (Ω) the second Green’sidentityΩ∆u(x)v(x) dx −is satisfied.Ωu(x)∆v(x) dx =Consider the boundary value problem∂Ω∂u∂n∂Ω(x)v(x) ds − u(x) ∂v (x) ds (1.8)∂n⎧−(∆u + κ⎪⎨2 u) = 0 in Ω,u = g D on Γ D ,(1.9)⎪⎩ ∂u∂n = g N on Γ Nwith a bounded C 1 domain Ω, non-overlapping sets Γ D , Γ N ⊂ ∂Ω such that Γ D ∪ Γ N = ∂Ωand g D , g N ∈ C(∂Ω). Considering a classical solution u ∈ C 2 (Ω), we can multiply theequation by a test function v ∈ V := {v ∈ C 2 (Ω): v| ∂Ω = 0 on Γ D } to obtain− ∆u(x)v(x) dx − κ 2 u(x)v(x) dx = 0 for all v ∈ V.ΩΩApplying the first Greens’s identity (1.7) we obtain∇u(x)∇v(x) dx − κ 2 u(x)v(x) dx = g N (x)v(x) ds for all v ∈ V.ΩΩΓ NThis formulation, however, is also valid for more general settings given in the followingdefinition.