Optimized Schwarz Method Research Articles

NonLocal Optimized Schwarz Method for the Helmholtz Equation with Physical Boundaries

NonLocal Optimized Schwarz Method for the Helmholtz Equation with Physical Boundaries

An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method.

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective.

How to Best Choose the Outer Coarse Mesh in the Domain Decomposition Method of Bank and Jimack

In Ciaramella et al. (2020) we defined a new partition of unity for the Bank–Jimack domain decomposition method in 1D and proved that with the new partition of unity, the Bank–Jimack method is an optimal Schwarz method in 1D and thus converges in two iterations for two subdomains: it becomes a direct solver, and this independently of the outer coarse mesh one uses! In this paper, we show that the Bank–Jimack method in 2D is an optimized Schwarz method and its convergence behavior depends on the structure of the outer coarse mesh each subdomain is using. For an equally spaced coarse mesh its convergence behavior is not as good as the convergence behavior of optimized Schwarz. However, if a stretched coarse mesh is used, then the Bank–Jimack method becomes faster then optimized Schwarz with Robin or Ventcell transmission conditions. Our analysis leads to a conjecture stating that the convergence factor of the Bank–Jimack method with overlap L and m geometrically stretched outer coarse mesh cells is 1-O(L^{frac {1}{2 m}}).

Robust treatment of cross-points in optimized Schwarz methods

In the field of domain decomposition, the optimized Schwarz method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between subdomains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two subdomains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross-points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the proposed method.

Wave-heat coupling in one-dimensional unbounded domains: artificial boundary conditions and an optimized Schwarz method

This paper deals with the coupling between one-dimensional heat and wave equations in unbounded subdomains, as a simplified prototype of fluid-structure interaction problems. First we apply appropriate artificial boundary conditions that yield an equivalent problem, but with bounded subdomains, and we carry out the stability analysis for this coupled problem in truncated domains. Then we devise an optimized Schwarz-in-time (or Schwarz Waveform Relaxation) method for the numerical solving of the coupled equations. Particular emphasis is made on the design of optimized transmission conditions. Notably, for this setting, the optimal transmission conditions can be expressed analytically in a very simple manner. This result is illustrated by some numerical experiments.

On the Choice of Robin Parameters for the Optimized Schwarz Method for Domains with Non-Conforming Heterogeneities

On the Choice of Robin Parameters for the Optimized Schwarz Method for Domains with Non-Conforming Heterogeneities

On the Choice of Interface Parameters in Robin–Robin Loosely Coupled Schemes for Fluid–Structure Interaction

We consider two loosely coupled schemes for the solution of the fluid–structure interaction problem in the presence of large added mass effect. In particular, we introduce the Robin–Robin and Robin–Neumann explicit schemes where suitable interface conditions of Robin type are used. For the estimate of interface Robin parameters which guarantee stability of the numerical solution, we propose a new strategy based on the optimization of the reduction factor of the corresponding strongly coupled (implicit) scheme, by means of the optimized Schwarz method. To check the suitability of our proposals, we show numerical results both in an ideal cylindrical domain and in a real human carotid. Our results showed the effectiveness of our proposal for the calibration of interface parameters, which leads to stable results and shows how the explicit solution tends to the implicit one for decreasing values of the time discretization parameter.

Non-local variant of the optimised Schwarz method for arbitrary non-overlapping subdomain partitions

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form “identity + non-expansive”.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

Abstract We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

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 Schwarz Methods for Spherical Interfaces With Application to Fluid-Structure Interaction

In this work we consider the optimized Schwarz method designed for computational domains that feature spherical or almost spherical interfaces. In the first part, we consider the diffusion-reaction problem. We provide a convergence analysis of the generalized Schwarz method and, following [G. Gigante and C. Vergara, Numer. Math., 131 (2015), pp. 369--404], we discuss an optimization procedure for constant interface parameters leading to a Robin--Robin scheme. Finally, we present some numerical results both in spherical and in ellipsoidal domains. In the second part of the work, we address the fluid-structure interaction problem. Again, we provide a convergence analysis and discuss optimized choices of constant interface parameters. Finally, we present three-dimensional numerical results inspired by hemodynamic applications, to validate the proposed choices in the presence of large added mass effect. In particular, we consider numerical experiments both in an ideal spherical domain and in a realistic abdominal aortic aneurysm.

An optimized Schwarz method with relaxation for the Helmholtz equation: the negative impact of overlap

In this paper we study how the overlapping size influences the convergence rate of an optimized Schwarz domain decomposition (DD) method with relaxation in the two subdomain case for the Helmholtz equation. Through choosing suitable parameters, we find that the convergence rate is independent of the wave number k and mesh size h, but sensitively depends on the overlapping size. Furthermore, by careful analysis, we obtain that the convergence behavior deteriorates with the increase of the overlapping size. Numerical results which confirm our theory are given.

On the Scalability of Classical One-Level Domain-Decomposition Methods

One-level domain-decomposition methods are in general not scalable, and coarse corrections are needed to obtain scalability. It has however recently been observed in applications in computational chemistry that the classical one-level parallel Schwarz method is surprizingly scalable for the solution of one- and two-dimensional chains of fixed-sized subdomains. We first review some of these recent scalability results of the classical one-level parallel Schwarz method, and then prove similar results for other classical one-level domain-decomposition methods, namely the optimized Schwarz method, the Dirichlet–Neumann method, and the Neumann–Neumann method. We show that the scalability of one-level domain decomposition methods depends critically on the geometry of the domain-decomposition and the boundary conditions imposed on the original problem. We illustrate all our results also with numerical experiments.

Optimized Schwarz methods for the coupling of cylindrical geometries along the axial direction

In this work, we focus on the Optimized Schwarz Method for circular flat interfaces and geometric heterogeneous coupling arising when cylindrical geometries are coupled along the axial direction. In the first case, we provide a convergence analysis for the diffusion-reaction problem and jumping coefficients and we apply the general optimization procedure developed in Gigante and Vergara (Numer. Math. 131 (2015) 369–404). In the numerical simulations, we discuss how to choose the range of frequencies in the optimization and the influence of the Finite Element and projection errors on the convergence. In the second case, we consider the coupling between a three-dimensional and a one-dimensional diffusion-reaction problem and we develop a new optimization procedure. The numerical results highlight the suitability of the theoretical findings.

Asynchronous optimized Schwarz methods with and without overlap

An asynchronous version of the optimized Schwarz method for the solution of differential equations on a parallel computational environment is studied. In a one-way subdivision of the computational domain, with and without overlap, the method is shown to converge when the optimal artificial interface conditions are used. Convergence is also proved for the Laplacian operator under very mild conditions on the size of the subdomains, when approximate (non-optimal) interface conditions are utilized. Numerical results are presented on a large three-dimensional problem on modern parallel clusters and supercomputers illustrating the efficiency of the asynchronous approach.

Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose–Einstein condensates

In this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrodinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions.

A two‐level domain decomposition method with accurate interface conditions for the Helmholtz problem

SummaryA new and efficient two‐level, non‐overlapping domain decomposition (DD) method is developed for the Helmholtz equation in the two Lagrange multiplier framework. The transmission conditions are designed by utilizing perfectly matched discrete layers (PMDLs), which are a more accurate representation of the exterior Dirichlet‐to‐Neumann map than the polynomial approximations used in the optimized Schwarz method. Another important ingredient affecting the convergence of a DD method, namely, the coarse space augmentation, is also revisited. Specifically, the widely successful approach based on plane waves is modified to that based on interface waves, defined directly on the subdomain boundaries, hence ensuring linear independence and facilitating the estimation of the optimal size for the coarse problem. The effectiveness of both PMDL‐based transmission conditions and interface‐wave‐based coarse space augmentation is illustrated with an array of numerical experiments that include comprehensive scalability studies with respect to frequency, mesh size and the number of subdomains. Copyright © 2015 John Wiley & Sons, Ltd.

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 Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

In this paper, we present a nonoverlapping optimized Schwarz algorithm with a high-order transmission condition for the Helmholtz equation posed in waveguides. We introduce the new high-order transmission conditions based on the complete radiation boundary conditions (CRBCs) that have been developed for high-order absorbing boundary conditions. We obtain the convergence rate of the algorithm in terms of reflection coefficients of CRBCs, which decrease exponentially with the order of CRBCs. It will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the Schwarz algorithm. Also, this algorithm can be employed efficiently for a preconditioner in GMRES implementations based on the substructured form as in [M. J. Gander, F. Magoulès, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60]. Finally, numerical examples confirming the theory will be presented.

Nearwell local space and time refinement in reservoir simulation

In reservoir simulations, nearwell regions usually require finer space and time scales compared with the remaining of the reservoir domain. We present a domain decomposition algorithm for a two phase Darcy flow model coupling nearwell regions locally refined in space and time with a coarser reservoir discretization. The algorithm is based on an optimized Schwarz method using a full overlap at the coarse level. The main advantage of this approach is to apply to fully implicit discretizations of general multiphase flow models and to allow a simple optimization of the interface conditions based on a single phase flow equation.