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

11Definition 1.8 (Weak Solution). Consider the boundary value problem (1.9) with abounded Lipschitz domain Ω, non-overlapping measurable sets Γ D , Γ N ⊂ ∂Ω such thatΓ D ∪ Γ N = ∂Ω, g D ∈ H 1/2 (Γ D ) and g N ∈ L 2 (Γ N ). Then u ∈ H 1 (Ω) is a weak solution to(1.9) if it satisfies⎧ ⎪⎨ ∇u(x)∇v(x) dx − κ 2 u(x)v(x) dx = g N (x)γ 0 v(x) ds for all v ∈ V,ΩΩΓ N⎪⎩γ 0 u = g D on Γ DwithV := {v ∈ H 1 (Ω): γ 0 v = 0 on Γ D }The advantage of the weak formulation is that the Neumann boundary condition istransformed into the term Γ Ng N (x)γ 0 v(x) dsand thus the normal derivatives of the solution do not appear in the weak formulation.However, for the purposes of the boundary element method the concept of normalderivatives has to be generalized. Using the definition of distributional partial derivatives(1.1) we get for u ∈ L 1 loc (Ω)d ∂ 2 ud ∂u⟨∆u, ϕ⟩ =∂x 2 , ϕ = − , ∂ϕ =k=1 k∂x k ∂x kk=1= u(x)∆ϕ(x) dx for all ϕ ∈ C0 ∞ (Ω).dk=1For u ∈ H 1 (Ω, ∆ + κ 2 ) we may use the middle term from (1.10)⟨∆u, ϕ⟩ = −dk=1 ∂u, ∂ϕ = −∂x k ∂x kand rewrite (1.10) as∆u(x)ϕ(x) dx = −ΩΩdk=1Ω∂u(x) ∂ϕ (x) dx∂x k ∂x k u, ∂2 ϕ∂x 2 = ⟨u, ∆ϕ⟩kfor all ϕ ∈ C ∞ 0 (Ω)(1.10)∇u(x)∇ϕ(x) dx for all ϕ ∈ C ∞ 0 (Ω). (1.11)Let u ∈ H 1 (Ω, ∆ + κ 2 ) be an arbitrary but fixed function. We define a functional˜L u : H 1 (Ω) → C as˜L u (v) := ∇u(x)∇v(x) dx + ∆u(x) + κ 2 u(x) v(x) dx − κ 2 u(x)v(x) dxΩΩΩApparently, ˜L u is linear and due to the Hölder inequality we have|˜L u (v)| ≤ ∥∇u∥ L 2 (Ω)∥∇v∥ L 2 (Ω) + ∥∆u + κ 2 u∥ L 2 (Ω)∥v∥ L 2 (Ω) + κ 2 ∥u∥ L 2 (Ω)∥v∥ L 2 (Ω)≤ (2 + κ 2 )∥u∥ H 1 (Ω,∆+κ 2 )∥v∥ H 1 (Ω),