Yassine Boubendir

A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Stokes-Darcy boundary integral solutions using preconditioners

In a system where a free fluid flow is coupled to flow in a porous medium, different PDEs are solved simultaneously in two subdomains. We consider steady Stokes equations in the free region, coupled across a fixed interface to Darcy equations in the porous substrate. Recently, the numerical solution of this system was obtained using the boundary integral formulation combined with a regularization-correction procedure. The correction process also results in the improvement of the condition number of the linear system. In this paper, an appropriate preconditioner based on the singular part of corrections is introduced to improve the convergence of a Krylov subspace method applied to solve the integral formulation.

An integral preconditioner for solving the two-dimensional scattering transmission problem using integral equations

The solution of the two-dimensional scattering problem for an homogeneous dielectric cylinder of arbitrary shape is considered. The numerical approach is based on a two-field system of integral equations solved by a Krylov subspace method. To accelerate and to improve the convergence of this solver, an efficient and robust preconditioner, based on the Calderón formulae, is developed. Several numerical simulations, for a wide range of physical parameters, validating the choice of this preconditioner are presented.

Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.

Domain Decomposition Methods for Solving Stokes--Darcy Problems with Boundary Integrals

We consider a coupled problem of Stokes and Darcy equations. This involves solving PDEs of different orders simultaneously. To overcome this difficulty, we apply a nonoverlapping domain decomposition method based on a Robin boundary condition obtained by combining the velocity and force interface conditions. The coupled system is then reduced to solving each problem separately by an iterative procedure using a Krylov subspace method. The numerical solution in each subdomain is based on the boundary integral formulation, where the kernels are regularized and correction terms are added to reduce the regularization error.

Regularized Combined Field Integral Equations for Acoustic Transmission Problems

We present a new class of well-conditioned integral equations for the solution of two and three dimensional scattering problems by homogeneous penetrable scatterers. Our novel boundary integral equations result from using regularizing operators which are suitable approximations of the admittance operators that map the transmission boundary conditions to the exterior and, respectively, interior Cauchy data on the interface between the media. We refer to these regularized boundary integral equations as generalized combined source integral equations (GCSIE). The admittance operators can be expressed in terms of Dirichlet-to-Neumann operators and their inverses. We construct our regularizing operators in terms of simple boundary layer operators with complex wavenumbers. The choice of complex wavenumbers in the definition of the regularizing operators ensures the unique solvability of the GCSIE. The GCSIE are shown to be integral equations of the second kind for regular enough interfaces of material discontinu...

Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

Abstract We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined magnetic and electric dipole layer potentials that involves a composition of the latter type of boundary layers with regularizing operators. The regularizing operators are of two types: (1) modified versions of electric field integral operators with complex wavenumbers, and (2) principal symbols of those operators in the sense of pseudodifferential operators. We show that the boundary integral operators that enter these CFIER formulations are Fredholm of the second kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering problems. We present a spectral analysis of CFIER operators with regularizing operators that have purely imaginary wavenumbers for spherical geometries—we refer to these operators as Calderon–Ikawa CFIER. Under certain assumptions on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing operators, we show that the ensuing Calderon–Ikawa CFIER operators are coercive for spherical geometries. These properties allow us to derive wavenumber explicit bounds on the condition numbers of Calderon–Ikawa CFIER operators. When regularizing operators with complex wavenumbers with non-zero real parts are used—we refer to these operators as Calderon-Complex CFIER, we show numerical evidence that those complex wavenumbers can be selected in a manner that leads to CFIER formulations whose condition numbers can be bounded independently of frequency for spherical geometries. In addition, the Calderon-Complex CFIER operators possess excellent spectral properties in the high-frequency regime for both convex and non-convex scatterers. We provide numerical evidence that our solvers based on fast, high-order Nystrom discretization of these equations converge in very small numbers of GMRES iterations, and the iteration counts are virtually independent of frequency for several smooth scatterers with slowly varying curvatures.

Analysis of Open Waveguides Using the Finite-Element Method and Boundary Integral Equations

To analyze dielectric waveguides, an iterative procedure coupling the finite-element method in the interior to an integral equation of the exterior domain is developed. The robustness of this method is confirmed by the numerical results presented in this paper and corresponding the computation of the propagation constant.

A Non-overlapping Quasi-optimal Optimized Schwarz Domain Decomposition Algorithm for the Helmholtz Equation

In this paper, we present a new non-overlapping domain decomposition algorithm for the Helmholtz equation. We are particularly interested in the method introduced by P.-L. Lions [6] for the Laplace equation and extended to the Helmholtz equation by B. Despres [3]. However, this latest approach provides slow convergence of the iterative method due to the choice of the transmission conditions. Thus, in order to improve the convergence, several methods were developed [4, 5, 9, 10].