Alternating Schwarz Method Research Articles

Review of the Methods of Reflections

The methods of reflections were invented to obtain approximate solutions of the motion of more than one particle in a given environment, provided that one can represent the solution for one particle rather easily. This motivation is quite similar to the motivation of the Schwarz domain decomposition method, which was invented to prove existence and uniqueness of solutions of the Laplace equation on complicated domains, which are composed of simpler ones, for which existence and uniqueness of solutions was known. Like for Schwarz methods, there is also an alternating and a parallel method of reflections, but interestingly, the parallel method is not always convergent. We carefully trace in this paper the historical development of these methods of reflections, give several precise mathematical formulations, an equivalence result with the alternating Schwarz method for two particles, and also an analysis for a one dimensional model problem with three particles of the alternating, parallel, and a recent averaged parallel method of reflections.

A hybrid, coupled approach for modeling charged fluids from the nano to the mesoscale

We develop and demonstrate a new, hybrid simulation approach for charged fluids, which combines the accuracy of the nonlocal, classical density functional theory (cDFT) with the efficiency of the Poisson–Nernst–Planck (PNP) equations. The approach is motivated by the fact that the more accurate description of the physics in the cDFT model is required only near the charged surfaces, while away from these regions the PNP equations provide an acceptable representation of the ionic system. We formulate the hybrid approach in two stages. The first stage defines a coupled hybrid model in which the PNP and cDFT equations act independently on two overlapping domains, subject to suitable interface coupling conditions. At the second stage we apply the principles of the alternating Schwarz method to the hybrid model by using the interface conditions to define the appropriate boundary conditions and volume constraints exchanged between the PNP and the cDFT subdomains. Numerical examples with two representative examples of ionic systems demonstrate the numerical properties of the method and its potential to reduce the computational cost of a full cDFT calculation, while retaining the accuracy of the latter near the charged surfaces.

Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains

We describe a multidomain spectral-tau method for solving the three-dimensional helically reduced wave equation on the type of two-center domain that arises when modeling compact binary objects in astrophysical applications. A global two-center domain may arise as the union of Cartesian blocks, cylindrical shells, and inner and outer spherical shells. For each such subdomain, our key objective is to realize certain (differential and multiplication) physical-space operators as matrices acting on the corresponding set of modal coefficients. We then achieve sparse realizations through the integration “preconditioning” of Coutsias, Hagstrom, Hesthaven, and Torres. Since ours is the first three-dimensional multidomain implementation of the technique, we focus on the issue of convergence for the global solver, here the alternating Schwarz method accelerated by GMRES. Our methods may prove relevant for numerical solution of other mixed-type or elliptic problems, and in particular for the generation of initial data in general relativity.

A concurrent multiscale method based on the alternating Schwarz scheme for coupling atomic and continuum scales with first-order compatibility

As well known, one of the major challenges in developing a multiscale model is how to ensure a seamless interface between the constituent length/time scales. In order to overcome this challenge, a novel concurrent multiscale numerical method is proposed in this paper, which is based on the alternating Schwarz method, to provide the seamless coupling between the atomic and continuum scales. The novelty in this method is the use of the strong-form meshless Hermite---cloud method in the continuum domain for approximation of both the field variable and corresponding first-order derivative simultaneously. As a result, the coupling between the domains is achieved by ensuring the compatibility of both the field variable and the first-order derivative simultaneously across the overlapping transition region. The proposed scheme is validated numerically through both the static and transient benchmark case studies in 1- and 2-D domains. The numerical results show that the proposed method is simple, efficient, and accurate, and also provides a seamless coupling between the two domains.

Forward backward domain decomposition method for finite element solution of electromagnetic boundary value problems

AbstractWe introduce the forward–backward domain decomposition method (FB‐DDM), which is basically an improved version of the classical alternating Schwarz method with overlapping subdomains, for electromagnetic boundary value problems. The proposed method is noniterative in some cases involving smooth geometries, or it usually converges in a few iterations in other cases involving challenging geometries, via the utilization of the locally‐conformal PML method. We report some numerical results for two‐dimensional electromagnetic scattering problems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2582–2590, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22757

Convergent iterative schemes for time parallelization

Parallel methods are usually not applied to the time domain because of the inherit sequentialness of time evolution. But for many evolutionary problems, computer simulation can benefit substantially from time parallelization methods. In this paper, we present several such algorithms that actually exploit the sequential nature of time evolution through a predictor-corrector procedure. This sequentialness ensures convergence of a parallel predictor-corrector scheme within a fixed number of iterations. The performance of these novel algorithms, which are derived from the classical alternating Schwarz method, are illustrated through several numerical examples using the reservoir simulator Athena.

On the Schwarz algorithms for the Elliptic Exterior Boundary Value Problems

Tuning the alternating Schwarz method to the exterior problems is the subject of this paper. We present the original algorithm and we propose a modification of it, so that the solution of the subproblem involving the condition at infinity has an explicit integral representation formulas while the solution of the other subproblem, set in a bounded domain, is approximated by classical variational methods. We investigate many of the advantages of the new Schwarz approach: a geometrical convergence rate, an easy implementation, a substantial economy in computational costs and a satisfactory accuracy in the numerical results as well as their agreement with the theoretical statements.

Flow simulation on moving boundary-fitted grids and application to fluid-structure interaction problems

We present a method for the parallel numerical simulation of transient three-dimensional fluid-structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non-overlapping subdomains, the subproblems are solved altematingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time-dependent domains. To this end, we present a technique to solve the incompressible Navier-Stokes equation in three-dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time-dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian-Eulerian formulation of the Navier-Stokes equations. Here the grid velocity is treated in such a way that the so-called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well-known MAC-method to a staggered mesh in moving boundary-fitted coordinates which uses grid-dependent velocity components as the primary variables

Solving the Stress Problem for a Finite Cylinder by the Schwarz Method with Hybrid Approximations

An axisymmetric elastic problem is solved by a heterogeneous numerical scheme of the alternating Schwarz method based on the finite- and boundary-element methods. The Schwarz method and the modified scheme are compared. A relaxation parameter is used to accelerate the iterative process

A multilevel Schwarz shooting method for the solution of the Poisson equation in two dimensional incompressible flow simulations

This work presents a Multilevel Schwarz Shooting method for the numerical solution of the Poisson equation, using a five point finite difference molecule, and subject to Dirichlet boundary conditions, which arises in two dimensional incompressible flow simulations. The iterative simple shooting algorithm utilized has optimal complexity, but becomes unstable when the problem size is increased. In this work, this limitations is overcome using a domain decomposition technique, more specifically, the alternating Schwarz method, which is used to generalize the iterative multiple shooting approach in the partial differential equation (PDE) context. In order to accelerate the convergence of the algorithm a multilevel approach is used. The resulting iterative algorithm has optimal complexity and is scalable in terms of problem size. Results of computational experiments are presented for the Poisson equation and for the two dimensional incompressible Navier–Stokes equation using the stream function-vorticity formulation.

Sur le traitement des conditions aux limites à l'infini pour quelques problèmes extérieurs par la méthode de Schwarz alternée

We propose an iterative algorithm to solve the exterior Helmholtz and Poisson problems. It can be written as an alternating Schwarz method which allows us to state a geometrical convergence result in the elliptic case. To cite this article: F. Ben Belgacem et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

The alternating schwarz method applied to some biharmonic variational inequalities

Recently there has been a renewed interest in the alternating Schwarz method and its numerical applications; this interest has also regarded developments in parallel computing (see [19] for an ample bibligraphy). In this paper we deal with the problems known as the free and clamped plate with two obstacles problems. We study the theoretical and numerical aspects of such problems with response to the Schwarz method, applying the mixed finite element method. Finally, we construct a simple algorithm permitting an estimate of the m.th step Schwarz error in the continuous and discrete case.