MATRIX ANALYSIS TO ADDITIVE SCHWARZ METHODS

Matrix analysis on additive Schwarz methods as preconditioners is given in this paper. Both cases of with and without coarse mesh are considered. It is pointed out that an advantage of matrix analysis is to obtain more exact upper hound. Our numerical tests access the estimations.

Fourier analysis of Schwarz domain decomposition methods for the biharmonic equation

Schwarz methods are an important type of domain decomposition methods. Using the Fourier transform, we derive error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation in this paper. We prove the convergence of the Schwarz methods from a new point of view, and provide detailed information about the convergence speeds and their dependence on the overlapping size of subdomains. The obtained results are independent of any unknown constant and discretization method, showing that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.

MATRIX ANALYSIS ON LEADING TERM OF CONDITION NUMBER FOR ADDITIVE SCHWARZ METHODS

It is well known the order of preconditioned matrix by using additive Schwarz methods. In order to estimate the resulted PCG iteration counts, the related leading term brfore the order is given in this paper.

An Algebraic Approach to the Schwarz Alternating Methods

In this paper, the choice of the optimal parameters for a relaxation additive Schwarz alternating method in two subregions case is obtained by an algebraic method, which shows that the arithmetic average is the best. A counterexample illustrates that the same result is not true for many subregions case. In the last, this technique is applied to demonstrate some well known results ,, simply and intuitively.

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

We concentrate on the parallel,fully coupled and fully implicit solution of the sequence of 3-by-3 block-structured linear systems arising from the symmetrypreserving finite volume element discretization of the unsteady three-temperature radiation diffusion equations in high dimensions.In this article,motivated by[M.J.Gander,S.Loisel,D.B.Szyld,SIAM J.Matrix Anal.Appl.33(2012)653–680]and[S.Nardean,M.Ferronato,A.S.Abushaikha,J.Comput.Phys.442(2021)110513],we aim to develop the additive and multiplicative Schwarz preconditioners subdividing the physical quantities rather than the underlying domain,and consider their sequential and parallel implementations using a simplified explicit decoupling factor approximation and algebraic multigrid subsolves to address such linear systems.Robustness,computational efficiencies and parallel scalabilities of the proposed approaches are numerically tested in a number of representative real-world capsule implosion benchmarks.

Multiscale Domain Decomposition Methods for Elliptic Problems with High Aspect Ratios

In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.

SCHWARZ METHOD FOR FINANCIAL ENGINEERING

Schwarz method is put forward to solve second order backward stochastic di erential equations(2BSDEs)in this work.We will analyze uniqueness,convergence,stability and optimality of the proposed method.Moreover,several simulation results are presented to demonstrate the e ectiveness;several applications of the 2BSDEs are investigated.It is concluded from these results that the proposed the method is powerful to calculate the 2BSDEs listing from the nancial engineering.

MULTIPLICATIVE SCHWARZ ALGORITHM WITH TIME STEPPING ALONG CHARACTERISTIC FOR CONVECTION DIFFUSION EQUATIONS

Examines the convection diffusion problems using domain decomposition method. Presentation of continuous and discrete convection diffusion equations; Kinds of multiplicative Schwarz algorithms; Optimal order error estimate results.

RESTRICTED ADDITIVE SCHWARZ METHOD FOR A KIND OF NONLINEAR COMPLEMENTARITY PROBLEM

In this paper, a new Schwarz method called restricted additive Schwarz method （RAS） is presented and analyzed for a kind of nonlinear complementarity problem （NCP）. The method is proved to be convergent by using weighted maximum norm. Besides, the effect of overlap on RAS is also considered. Some preliminary numerical results are reported to compare the performance of RAS and other known methods for NCP.

Quasi-Optimized Overlapping Schwarz Waveform Relaxation Algorithm for PDEs with Time-Delay

Schwarzwaveformrelaxation(SWR)algorithmhas been investigated deeply and widely for regular time dependent problems.But for time delay problems,complete analysis of the algorithm is rare.In this paper,by using the reaction diffusion equations with a constant discrete delay as the underlying model problem,we investigate the convergence behavior of the overlapping SWR algorithm with Robin transmission condition.The key point of using this transmission condition is to determine a free parameter as better as possible and it is shown that the best choice of the parameter is determined by the solution of a min-max problem,which is more complex than the one arising for regular problems without delay.We propose new notion to solve the min-max problem and obtain a quasi-optimized choice of the parameter,which is shown efficient to accelerate the convergence of the SWR algorithm.Numerical results are provided to validate the theoretical conclusions.

RATE OF CONVERGENCE OF SCHWARZ ALTERNATING METHOD FOR TIME-DEPENDENT CONVECTION-DIFFUSION PROBLEM

Presents a study that examined the application of an overlapping domain decomposition method to the solution of time-dependent convection-diffusion problems. Background on the Schwartz alternating procedure; Application of two kinds of Schwartz alternating procedure to solve the numerical approximation problem; Numerical results.

Analytics study on the problem of two holes having arbitrary shapes and arrangements in plane elastostatics

By using Schwarz alternating method, this paper presents asimplified alternating algorithm for the problems of two holes having arbitrary shapes and arrangements in an isotropic homogeneous linear elastic infinite region, and obtains stress and displacement fields for random times of iteration. After precision analysis it is found that the results for twenty times of iteration are of very high precision, and those with higher precision can be acquired if the iteration solving is further conducted. The comparison of the results from FEM further proves the reliability of the simplified alternating algorithm presented by this paper.