Alexandre Vion

Double sweep preconditioner for optimized Schwarz methods applied to the Helmholtz problem

This paper presents a preconditioner for non-overlapping Schwarz methods applied to the Helmholtz problem. Starting from a simple analytic example, we show how such a preconditioner can be designed by approximating the inverse of the iteration operator for a layered partitioning of the domain. The preconditioner works by propagating information globally by concurrently sweeping in both directions over the subdomains, and can be interpreted as a coarse grid for the domain decomposition method. The resulting algorithm is shown to converge very fast, independently of the number of subdomains and frequency. The preconditioner has the advantage that, like the original Schwarz algorithm, it can be implemented as a matrix-free routine, with no additional preprocessing.

A model reduction algorithm for solving multiple scattering problems using iterative methods

In this paper we propose a new method for solving high-frequency scattering problems by multiple objects. A model reduction algorithm based on the macro basis function (MBF) method is used to find an approximate solution within a subspace spanned by the solutions of several single scattering subproblems. Different iterative methods for generating the MBFs are compared. The whole process relies on a finite element approach and is applied to convex obstacle scattering.

Parallel Double Sweep Preconditioner for the Optimized Schwarz Algorithm Applied to High Frequency Helmholtz and Maxwell Equations

The principle of sweeping to accelerate the solution of wave propagation problems has recently retained much interest, yet with different approaches (Engquist and Ying, Multiscale Model Simul 9(2):686–710, 2011; Stolk, J Comput Phys 241:240–252, 2013). We recently proposed a preconditioner for the optimized Schwarz algorithm, based on a propagation of information using a double sequence of subproblems solves, or sweeps (Vion et al., A DDM double sweep preconditioner for the Helmholtz equation with matrix probing of the DtN map, Mathematical and Numerical Aspects of Wave Propagation WAVES 2013, June 2013; Vion and Geuzaine, J Comput Phys, 2014, Preprint, submitted). Although this procedure significantly reduces the number of iterations when many subproblems are involved, the sequential nature of the process hinders the scalability of the algorithm on parallel computer architectures. Here we propose a modified version of the algorithm that concurrently runs partial sweeps on smaller groups of domains, which efficiently reduces the preconditioner application time on parallel machines. We show that the algorithm is applicable to both Helmholtz and Maxwell equations.

GetDDM: An open framework for testing optimized Schwarz methods for time-harmonic wave problems☆

We present an open finite element framework, called GetDDM, for testing optimized Schwarz domain decomposition techniques for time-harmonic wave problems. After a review of Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples for Helmholtz and Maxwell’s equations, is freely available online for further testing.

An open source domain decomposition solver for time-harmonic electromagnetic wave problems

We present a flexible finite element solver for testing optimized Schwarz domain decomposition techniques for the time-harmonic Maxwell equations. After a review of non-overlapping Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples, is available online for further testing.

An Amplitude Finite Element Formulation for Multiple-Scattering by a Collection of Convex Obstacles

We present a multiple-scattering solver for nonconvex geometries obtained as the union of a finite number of convex obstacles. The algorithm is a finite element reformulation of a high-frequency integral equation technique proposed previously. It is based on an iterative solution of the scattering problem, where each iteration leads to the resolution of a single scattering problem in terms of a slowly oscillatory amplitude.