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

38 3 Boundary Integral EquationswithB := D −1 A = D −1 (D − C) = I − D −1 C.From X-ellipticity of D and boundedness of A we get that B is bounded. Since we assumethat A is injective, it follows that the homogeneous equationD −1 Au = (I − D −1 C)u = 0only has the trivial solution and thus D −1 C is injective. The Fredholm Alternative 3.19with the compact operator K := D −1 C then guarantees a unique solution to (3.19) satisfyingthe estimate (3.20).Theorem 3.21. Let X denote a Hilbert space. Then it holds(∀x, y ∈ X, x ≠ y)(∃f ∈ X ∗ ): ⟨f, x⟩ ≠ ⟨f, y⟩.Proof. The theorem is a direct consequence of the well-known Hahn-Banach Theorem (see,e.g., [21], Theorem 1.B, Corollary 2 on pages 4–5).In Section 2.3 we introduced representation formulae for the solution to the Helmholtzequation with smooth enough data. The following theorems generalize the previous resultsfor a broader range of functions. For more details see [12], Theorems 6.10 and 7.12.Theorem 3.22. Let Ω be a bounded Lipschitz domain and let u ∈ H 1 (Ω, ∆ + κ 2 ) satisfythe Helmholtz equation in the weak sense. Then we have the representationu(x) = ( V κ γ 1,int u)(x) − (W κ γ 0,int u)(x) for x ∈ Ω. (3.21)Theorem 3.23. Let Ω be a bounded Lipschitz domain and let us denote Ω ext := R 3 \ Ω.Let u ∈ H 1 loc (Ωext , ∆ + κ 2 ) satisfy the Sommerfeld radiation condition and the Helmholtzequation in the weak sense. Then we have the representationu(x) = −( V κ γ 1,ext u)(x) + (W κ γ 0,ext u)(x) for x ∈ Ω ext . (3.22)Before we proceed to individual boundary value problems, we provide theorems regardingexistence and uniqueness of the weak solution to general boundary value problems forthe Helmholtz equation (see [12] and [15]).Theorem 3.24. The interior boundary value problem⎧⎪⎨∆u + κ 2 u = 0 in Ω,γ 0,int u = g D on Γ D ,⎪⎩γ 1,int u = g N on Γ Nwith a bounded Lipschitz domain Ω, non-overlapping sets Γ D , Γ N satisfying Γ D ∪ Γ N = ∂Ωand boundary conditions g D ∈ H 1/2 (Γ D ), g N ∈ H −1/2 (Γ N ) has a unique weak solution