Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

Multitrace/singletrace formulations and Domain Decomposition Methods for the solution of Helmholtz transmission problems for bounded composite scatterers

“…Remark 4.6 While the new blended operators P S(T j ), j = 0, 1, 2 give rise to well-posed Helmholtz equations in each subdomain, the issue of the invertibility of the operator (P S(T 1 ) + P S(T 2 ))| Γ 12and hence the equivalence between conditions expressions for the integrals of products of periodic singular and weakly singular kernels and Fourier harmonics. These discretizations were introduced in [12] where this methodology is presented in full detail. The main idea of our Nyström discretization is to incorporate sigmoid transforms [23] in the parametrization of a closed Lipschitz curve Γ and then split the kernels of the Helmholtz BIO into smooth and singular components.…”

“…The main idea of our Nyström discretization is to incorporate sigmoid transforms [23] in the parametrization of a closed Lipschitz curve Γ and then split the kernels of the Helmholtz BIO into smooth and singular components. Using graded meshes that avoid corner points and classical singular quadratures of Kusmaul and Martensen [24,25], we employ the Nyström discretization presented in [12] to produce high-order approximations of the BIO that enter the Calderón projectors. We note that the role of weighted traces is to increase the regularity of the densities to be integrated, regularity that can be tuned by simply increasing the polynomial order in the sigmoid transforms.…”

“…Nyström discretization of the MTF equation (3.8). Following the prescriptions in [12] for the discretization of transmission boundary integral equations for one subdomain, i.e. no triple/multiple junctions, we define weighted Neumann traces on the boundary of each subdomain Ω j ∂ w n j u j := |x j |∂ n j u j where x j is a [0, 2π] parametrization of the boundary ∂Ω j that incorporates sigmoid transforms on each of the smooth arcs of ∂Ω j .…”

“…A major advantage of MTF is the ease with which they can be incorporated into existing boundary integral equation solvers. We present in this work a straightforward extension of the Helmholtz transmission Nyström solvers introduced in [12] to MTF. We investigate two types of simple preconditioners for the MTF of the first kind: the Calderón preconditioner proposed in [19] and a new preconditioner that combines a Schur complement approach with a Calderón preconditioner.…”

“…The second approach we pursue is to incorporate square root approximations-using complex wavenumbers-of the DtN operators as generalized Robin operators in DDM. The operators were used successfully for regularizing boundary integral equations at high frequencies [2,1,6,5,12,7] and also incorporated successfully in DDM [4,34]. Since for composite scatterers, adjacent domain decomposition subdomains can have different wavenumbers, we blend the approximations of DtN operators corresponding to different subdomains-with different wavenumbers as well-via smooth cutoff functions.…”