Non-overlapping Schwarz Methods Research Articles

On the convergence of overlapping and non-overlapping Schwarz methods for the Cahn–Hilliard equation

In this study, we develop and investigate the Schwarz method with or without overlap for the Cahn–Hilliard (CH) problem. The CH equation has a wide range of applications, hence it is crucial to develop effective numerical techniques. In this article, we provide the formulation of the classical Schwarz methods (CSM) and optimized Schwarz methods (OSM) for the CH equation and present the convergence behaviour for two as well as multiple subdomain settings. We also formulate the nonlinear variant of the Schwarz and quasi-optimized Schwarz techniques for the same. We provide numerical results to support our conclusions.

Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities

In this paper we discuss different transmission operators for the non-overlapping Schwarz method which are suited for solving the time-harmonic Helmholtz equation in cavities (i.e. closed domains which do not feature an outgoing wave condition). Such problems are heavily impacted by back-propagating waves which are often neglected when devising optimized transmission operators for the Schwarz method. This work explores new operators taking into account those back-propagating waves and compares them with well-established operators neglecting these contributions. Notably, this paper focuses on the case of rectangular cavities, as the optimal (non-local) transmission operator can be easily determined. Nonetheless, deviations from this ideal geometry are considered as well. In particular, computations of the acoustic noise in a three-dimensional model of the helium vessel of a beamline cryostat with optimized Schwarz schemes are discussed. Those computations show a reduction of 46% in the iteration count, when comparing an operator optimized for cavities with those optimized for unbounded problems.

Time domain decomposition of parabolic control problems based on discontinuous Galerkin semi-discretization

A parallel-in-time method for the optimal control problem governed by linear parabolic equations is analyzed. The corresponding optimal system consists of a state equation (forward in time) and an adjoint equation (backward in time) which are coupled by the necessary condition. The whole time interval is divided into non-overlapping subdomains which are associated by the Robin-type transmission conditions. We apply discontinuous Galerkin method for time discretization in each subdomain and keep continuous in space. In order to prove the convergence of the algorithm, the semi-discretization errors at initial and final points are obtained based on nodal stability estimates. We complete the convergence analysis for iterative solution of the nonoverlapping Schwarz method through energy estimates for two subdomains and present the convergence rate affected by the transmission parameters. Finally, numerical experiments verify the theoretical results.

Analysis of Schwarz waveform relaxation for the coupled Ekman boundary layer problem with continuously variable coefficients

In this paper, we present a global-in-time non-overlapping Schwarz method applied to the Ekman boundary layer problem. Such a coupled problem is representative of large-scale atmospheric and oceanic flows in the vicinity of the air-sea interface. Schwarz waveform relaxation (SWR) algorithms provide attractive methods for ensuring a “tight coupling” between the ocean and the atmosphere. However, the convergence study of such algorithms in this context raises a number of challenges. Numerous convergence studies of Schwarz methods have been carried out in idealized settings, but the underlying assumptions to make these studies tractable may prohibit them to be directly extended to the complexity of climate models. We illustrate this aspect on the coupled Ekman problem, which includes several essential features inherent to climate modeling while being simple enough for analytical results to be derived. We investigate its well-posedness and derive an appropriate SWR algorithm. Sufficient conditions for ensuring its convergence for different viscosity profiles are then established. Finally, we illustrate the relevance of our theoretical analysis with numerical results and suggest ways to improve the computational cost of the coupling. Our study emphasizes the fact that the convergence properties can be highly sensitive to some model characteristics such as the geometry of the problem and the use of continuously variable viscosity coefficients.

A non-overlapping optimized Schwarz method for the heat equation with non linear boundary conditions and with applications to de-icing

When simulating complex physical phenomena such as aircraft icing or de-icing, several dedicated solvers often need to be strongly coupled. In this work, a non-overlapping Schwarz method is constructed with the unsteady simulation of de-icing as the targeted application. To do so, optimized coupling coefficients are first derived for the one dimensional unsteady heat equation with linear boundary conditions and for the steady heat equation with non-linear boundary conditions. The choice of these coefficients is shown to guarantee the convergence of the method. Using a linearization of the boundary conditions, the method is then extended to the case of a general unsteady heat conduction problem. The method is tested on simple cases and the convergence properties are assessed theoretically and numerically. Finally the method is applied to the simulation of an aircraft electrothermal de-icing problem in two dimensions.

Optimized double sweep Schwarz method by complete radiation boundary conditions

We present an optimized double sweep nonoverlapping Schwarz method for solving the Helmholtz equation in semi-infinite waveguides. The domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high-order absorbing boundary conditions are solved forward and backward sequentially. For communication between subdomains, Neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and Dirichlet data are exploited in the backward direction. The complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find Neumann data without introducing errors that would be produced if finite difference formulas were used for computing Neumann data. The convergence of the double sweep Schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory.

Optimized Schwarz method without overlap for the gravitational potential equation on cluster of graphics processing unit

Many engineering and scientific problems need to solve boundary value problems for partial differential equations or systems of them. For most cases, to obtain the solution with desired precision and in acceptable time, the only practical way is to harness the power of parallel processing. In this paper, we present some effective applications of parallel processing based on hybrid CPU/GPU domain decomposition method. Within the family of domain decomposition methods, the so-called optimized Schwarz methods have proven to have good convergence behaviour compared to classical Schwarz methods. The price for this feature is the need to transfer more physical information between subdomain interfaces. For solving large systems of linear algebraic equations resulting from the finite element discretization of the subproblem for each subdomain, Krylov method is often a good choice. Since the overall efficiency of such methods depends on effective calculation of sparse matrix–vector product, approaches that use graphics processing unit (GPU) instead of central processing unit (CPU) for such task look very promising. In this paper, we discuss effective implementation of algebraic operations for iterative Krylov methods on GPU. In order to ensure good performance for the non-overlapping Schwarz method, we propose to use optimized conditions obtained by a stochastic technique based on the covariance matrix adaptation evolution strategy. The performance, robustness, and accuracy of the proposed approach are demonstrated for the solution of the gravitational potential equation for the data acquired from the geological survey of Chicxulub crater.

Optimized Schwarz methods for a diffusion problem with discontinuous coefficient

We study non-overlapping Schwarz methods for solving a steady-state diffusion problem in heterogeneous media. Various optimized transmission conditions are determined by solving the corresponding min-max problems; we consider different choices of Robin conditions and second order conditions. To compare the resulting methods, we analyze the convergence in two separate asymptotic regimes: when the mesh size is small, and when the jump in the coefficient is large. It is shown that optimized two-sided Robin transmission conditions are very effective in both regimes; in particular they give mesh independent convergence. Numerical experiments are presented to illustrate and confirm the theoretical results.

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 domain decomposition method for concurrent coupling of multiscale models

SUMMARYMotivated by atomistic‐to‐continuum coupling, we consider a fine‐scale problem defined on a small region embedded in a much larger coarse‐scale domain and propose an efficient solution technique on the basis of the domain decomposition framework. Specifically, we develop a nonoverlapping Schwarz method with two important features: (i) the use of an efficient approximation of the Dirichlet‐to‐Neumann map for the interface conditions; and (ii) the utilization of the inherent scale separation in the solution. The paper includes a detailed formulation of the proposed interface condition, along with the illustration of its effectiveness by using simple but representative numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

Optimized Schwarz Methods for Maxwell's Equations

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.

OPTIMAL CONVERGENCE OF NON-OVERLAPPING SCHWARZ METHODS FOR THE HELMHOLTZ EQUATION

The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.

Two-Level Non-Overlapping Schwarz Preconditioners for a Discontinuous Galerkin Approximation of the Biharmonic Equation

We present some two-level non-overlapping additive and multiplicative Schwarz methods for a discontinuous Galerkin method for solving the biharmonic equation. We show that the condition numbers of the preconditioned systems are of the order O( H3/h3) for the non-overlapping Schwarz methods, where h and H stand for the fine mesh size and the coarse mesh size, respectively. The analysis requires establishing an interpolation result for Sobolev norms and Poincare--Friedrichs type inequalities for totally discontinuous piecewise polynomial functions. It also requires showing some approximation properties of the multilevel hierarchy of discontinuous Galerkin finite element spaces.

Non-overlapping Schwarz methods with optimized transmission conditions for the Helmholtz equation

The development of efficient transmission conditions for the Schwarz method without overlap is an important area of current research. The techniques developed in recent years mainly consist of algebraic approximations of the discrete transparent operator or optimized approximations of the continuous transparent operator. These optimized approximations are currently based on an analytical approach involving a minimization problem over all the available frequencies and using the L ∞ -norm. In this paper, new numerical optimization techniques are investigated. These original approaches allow selection of the frequency range for the minimization problem which is based on the L p -norm. New numerical experiments performed on a model problem demonstrate the performance and the robustness of the proposed optimization techniques in the particular cases p=1 and p=+∞. Applications to industrial automotive problems confirm the efficiency of these techniques when using arbitrary mesh partitions.

Hierarchical Vector Finite Elements with p-Type non-Overlapping Schwarz Method for Modeling Waveguide Discontinuities

Hierarchical Vector Finite Elements with p-Type non-Overlapping Schwarz Method for Modeling Waveguide Discontinuities

On accelerated convergence of nonoverlapping Schwarz methods

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. Schwarz methods for nonoverlapping subdomains also exist but they have not been popular because of their slow convergence. These methods contain a free parameter in the Robin boundary condition of each subdomain problem. The slow convergence can be attributed to an improper choice of this parameter. In this paper, two models are proposed to give guidance to the choice of this parameter. For the Poisson equation on rectangular domains, these models suggest very simple expressions for the parameter in terms of the dimensions of the subdomain. Numerical experiments verify their effectiveness. When used as a preconditioner, it is demonstrated numerically in some examples that the algorithm is quite efficient.

Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems

We present some two-level nonoverlapping and overlapping additive Schwarz methods for a discontinuous Galerkin method for solving second order elliptic problems. It is shown that the condition numbers of the preconditioned systems are of the order $O(\frac{H}{h})$ for the nonoverlapping Schwarz methods and of the order $O(\frac{H}{\delta})$ for the overlapping Schwarz methods, where h and H stand for the fine-mesh size and the coarse-mesh size, respectively, and $\delta$ denotes the size of the overlaps between subdomains. Numerical experiments are provided to gauge the efficiency of the methods and to validate the theory.

An iterative substructuring method for the $p$ -version of the boundary element method for hypersingular integral operators in three dimensions

We study preconditioners for the \(p\)-version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by \(C(1+\log p)\). Here, \(p\) is the polynomial degree of the ansatz functions and \(C\) is a constant which is independent of \(p\) and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported.