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

36 3 Boundary Integral Equationswith the Calderon projection matrix 1C int := 2 I − K κV κD κ12 I + K∗ κFor the solution in an unbounded domain we have the formula.u = W κ γ 0,ext u − V κ γ 1,ext u in Ω ext .Applying the Dirichlet and Neumann trace operators, respectively, and using properties(3.12), (3.10) and the definitions of the boundary integral operators V κ and D κ we get 1γ 0,ext u =2 I + K κ γ 0,ext u − V κ γ 1,ext u on ∂Ω ext , (3.17)γ 1,ext u = −D κ γ 0,ext u + 12 I − K∗ κγ 1,ext u on ∂Ω ext . (3.18)Again, the boundary integral equations (3.17), (3.18) can be rewritten as γ 0,ext uγ 1,ext = C ext γ 0,ext uu γ 1,ext uwith the Calderon projection matrix 1C ext := 2 I + K κ−V κ1−D κ 2 I − K∗ κTheorem 3.16. The Calderon operators C int and C ext are projection operators, i.e.,.(C int ) 2 = C int (C ext ) 2 = C ext .Proof. The proof is analogous to the proof of Lemma 6.18 in [18].Corollary 3.17. For the boundary integral operators we have the following identities 1 1V κ D κ = κ2 I + K 2 I − K κ ,D κ V κ = 12 I + K∗ κ 12 I − K∗ κD κ K κ = K ∗ κD κ ,K κ V κ = V κ K ∗ κ.Proof. The relations follow directly from the comparison of the matrices (C int ) 2 and C int .,