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.

DOMAIN DECOMPOSITION METHODS FOR SOLVING PDE's ON MULTI-PROCESSORS

In this paper, we discuss the parallel domain decomposition method(DDM)for solving PDE's on parallel computers. Three types of DDM: DDM with overlapping, DDM without overlapping and DDM with fictitious component are discussed in a uniform framework. The eonvergence of the asynchronous parallel algorithms based on DDM are discussed.

Optimal Boundary Control Method for Domain Decomposition Algorithm

To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.

A Fully Nonlinear HOBEM with the Domain Decomposition Method for Simulation of Wave Propagation and Diffraction

A higher-order boundary element method(HOBEM) for simulating the fully nonlinear regular wave propagation and diffraction around a fixed vertical circular cylinder is investigated. The domain decomposition method with continuity conditions enforced on the interfaces between the adjacent sub-domains is implemented for reducing the computational cost. By adjusting the algorithm of iterative procedure on the interfaces, four types of coupling strategies are established, that is, Dirchlet/Dirchlet-Neumman/Neumman(D/D-N/N), Dirchlet-Neumman(D-N),Neumman-Dirchlet(N-D) and Mixed Dirchlet-Neumman/Neumman-Dirchlet(Mixed D-N/N-D). Numerical simulations indicate that the domain decomposition methods can provide accurate results compared with that of the single domain method. According to the comparisons of computational efficiency, the D/D-N/N coupling strategy is recommended for the wave propagation problem. As for the wave-body interaction problem, the Mixed D-N/N-D coupling strategy can obtain the highest computational efficiency.

ANALYSIS OF WAVEGUIDE PROBLEMS USING A RELAXED ITERATIVE DOMAIN DECOMPOSITION METHOD COMBINED WITH MULTIFRONTAL ALGORITHM

In this paper, an absorbing Fictitious Boundary Condition (FBC) is presented to generate an iterative Domain Decomposition Method (DDM) for analyzing waveguide problems.The relaxed algorithm is introduced to improve the iterative convergence. And the matrix equations are solved using the multifrontal algorithm. The resulting CPU time is greatly reduced.Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.

APPLICATION OF DOMAIN DECOMPOSITION IN ACOUSTIC AND STRUCTURAL ACOUSTIC ANALYSIS

Conventional element based methods for modeling acoustic problems are limited to low-frequency applications due to the huge computational efforts. For high-frequency applications, probabilistic techniques, such as statistical energy analysis （SEA）, are used. For mid-frequency range, currently no adequate and mature simulation methods exist. Recently, wave based method has been developed which is based on the indirect TREFFTZ approach and has shown to be able to tackle problems in the mid-frequency range. In contrast with the element based methods, no discretization is required. A sufficient, but not necessary, condition for convergence of this method is that the acoustic problem domain is convex. Non-convex domains have to be partitioned into a number of （convex） subdomains. At the interfaces between subdomains, specific coupling conditions have to be imposed. The considered two-dimensional coupled vibro-acoustic problem illustrates the beneficial convergence rate of the proposed wave based prediction technique with high accuracy. The results show the new technique can be applied up to much higher frequencies.

A NON-OVERLAPPING DOMAIN DECOMPOSITION ALGORITHM BASED ON THE NATURAL BOUNDARY REDUCTION FOR WAVE EQUATIONS IN AN UNBOUNDED DOMAIN

In this paper, a new domain decomposition method based on the natural boundary reduction, which solves wave problems over an unbounded domain, is suggestted. An circular artifcial boundary is introduced. The original unbounded domain is divided into two subdomains, an internal bounded region and external unbounded region outside the artificial boundary. A Dirichlet-Neumann(D-N) alternating iteration algorithm is constructed. We prove that the algorithm is equavilent to preconditional Richardson iteration method. Numerical studies are performed by finite element method. The numerical results show that the convergence rate of the discrete D-N iteration is independent of the fnite element mesh size.

Modified domain decomposition method for Hamilton-Jacobi-Bellman equations

This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.

A DOMAIN DECOMPOSITION ALGORITHM WITH FINITE ELEMENT-BOUNDARY ELEMENT COUPLING

A domain decomposition algorithm coupling the finite element and the boundary element was presented. It essentially involves subdivision of the analyzed domain into sub-regions being independently modeled by two methods, i.e., the finite element method （FEM） and the boundary element method （BEM）. The original problem was restored with continuity and equilibrium conditions being satisfied on the interface of the two sub-regions using an iterative algorithm. To speed up the convergence rate of the iterative algorithm, a dynamically changing relaxation parameter during iteration was introduced. An advantage of the proposed algorithm is that the locations of the nodes on the interface of the two sub-domains can be inconsistent. The validity of the algorithm is demonstrated by the consistence of the results of a numerical example obtained by the proposed method and those by the FEM, the BEM and a present finite element-boundary element （FE-BE） coupling method.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and in- compressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangu- lar sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method （CSM） is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neu- mann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet bound- ary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison be- tween the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

Eddy Current Analyses by Domain Decomposition Method Using Double-Double Precision

A matrix equation solved in an eddy current analysis,??-??method based on a domain decomposition method becomes a complex symmetric system.In general,iterative method is used as the solver.Convergence of iterative method in an interface problem is improved by increasing an accuracy of a solution of an iterative method of a subdomain problem.However,it is difficult to improve the convergence by using a small convergence criterion in the subdomain problem.Therefore,authors propose a method to introduce double-double precision into the interface problem and the subdomain problem.This proposed method improves the convergence of the interface problem.In this paper,first,we describe proposed method.Second,we confirm validity of the method by using Team Workshop Problem 7,standard model for eddy current analysis.Finally,we show effectiveness of the method from two numerical results.

CHEBYSHEV PSEUDOSPECTRAL DOMAIN DECOMPOSITION METHOD OF ONE－DIMENSIONAL ELLIPTIC PROBLEMS

This paper is devoted to the Chebyshev pseudospectral domain decomposition method of one-dimensional elliptic problems,it is easily applied to complex geometry.The approximate accuracy can be increased by increasing the order of approximation in fixed number of subdomains,rather than by resorting to a further partitioning.The stability and the convergence of this method are proved.

DOMAIN DECOMPOSITION PRECONDITIONERS FOR SECOND-ORDER HYPERBOLIC EQUATIONS ON L-SHAPED REGIONS

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations on L-shaped region are considered. We analyse the use of domain deocmposilion preconditioner.s for the solution of linear systems via the preconditioned conjugate gradient method. For the constant-coefficient second-order hyperbolic equaions with initial and Dirichlet boundary conditions,we prove that the conditionnumber of the preconditioned interface system is bounded by 2+x2 2+0.46x2 where x is the quo-tient between the lime and space steps. Such condition number produces a convergence rale that is independent of gridsize and aspect ratios. The results could be extended to parabolic equations.

A COMBINED TECHNIQUE FOR SOLUTION OF PDE＇s VIA THE GENERALIZED DOMAIN DECOMPOSITION METHOD

The Domain Decomposition Method(DDM) is a powerful approach to solving maily types of PDE's. DDM is especially suitable for massively Parallel computers. In the past, most research on DDM has focused on the domain splitting technique. In this paper. we focus our attention on use of a combination of techniques to solve each subproblem. The central question with DDM is that of how to doal with the pseodoboundary conditions. Here, we introduce a set of operators which act on the pseudo-boundaries in the solution process, referring to this new. procedure as the 'Generalized Domain Decomposition A.Jlethod(GDDM).' We have already obtained convergence factors for GDDM with certain classes of PDE's. These ctonvergence factors show that we can derive exact solutions of the whole problem for certain types of PDE's, and can get superior speed of convergence for other types.

Meshless Method with Domain Decomposition for Submerged Porous Breakwaters in Waves

Based on the improved version of the meshless singular boundary method(ISBM)in multi domain(MD),a numerical method is proposed in this paper to study the interaction of submerged permeable breakwaters and regular waves at normal incidence.To account for fluid flow inside the porous breakwaters,the conventional model of Sollitt and Cross for porous media is adopted.Both single and dual trapezoidal breakwaters are examined.The physical problem is formulated in the context of the linear potential wave theory.The domain decomposition method(DDM)is employed,in which the full computational domain is decomposed into separate domains,that is,the fluid domain and the domains of the breakwaters.Respectively,appropriate mixed type boundary and continuity conditions are applied for each subdomain and at the interfaces between domains.The solution is approximated in each subdomain by the ISBM.The discretized algebraic equations are combined,resulting in an overdetermined full system that is solved using a least-square solution procedure.The numerical results are presented in terms of the hydrodynamic quantities of reflection,transmission,and wave-energy dissipation.The relevance of the results of the present numerical procedure is first validated against data of previous studies,and then selected computations are discussed for various structural conditions.The proposed method is demonstrated to be highly accurate and computationally efficient.

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker-Planck Equations

Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application,a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.

PBSV-DDM在电大尺寸柱体电磁散射中的应用

提出了一种高效率的基于部分基础解向量的区域分解算法(PBSV DDM).它首先求出关于连接边界上节点的部分基础解向量,在迭代过程中,只需要对部分基础解向量作简单的线性组合就可以获得整个求解区域的最终解,极大地提高了计算效率,降低了存储量.PBSV DDM不但适合于快速高效地计算任意电大尺寸柱体的电磁散射,还特别适合于求解具有几何重复性特征的结构,如天线阵列、有限周期频率选择表面、PBG EBG等的电磁仿真问题.数值算例验证了该方法的准确性和有效性.

电大尺寸开口腔体电磁散射特性的DDM/FEM-BIE混合法分析

该文将超松弛重叠及非重叠区域分裂法(DDM)与矢量有限元方法(EB-FEM)、边界积分方程(BIE)法相结合对三维电大尺寸开口腔体的电磁(EM)散射特性进行分析。通过DDM将原腔体分解成若干子腔体,在各子腔体内应用EB-FEM进行分析。腔体间应用矢量传输条件进行耦合,最终腔体内电场分布通过迭代获得。在原腔体口面,运用积分方程进行描述.在分析过程中,将传输条件和BIE统一成第三类边界条件形式。最后给出的数值结果验证了DDM/FEM-BIE混合方法分析腔体的电磁散射特性的精确性及高效性。

矩形波导E面不连续性的DDM/FEM分析

应用区域分解法(DDM)结合有限元法(FEM)分析矩形波导E面不连续性问题,将原求解区域分解为若干个非重叠的子域,在子域的虚拟边界上采用吸收虚拟边界条件,以保证相邻子域间的波传播。分别计算了矩形波导中加载双金属膜片和双介质柱时的散射系数,结果与有关文献一致。采用这种技术,大大地减少了对计算机内存的需求。

基于DDM技术的FEM/PO-PTD方法在深腔导体目标RCS分析中的应用

采用一种基于区域分解法 (DDM)的混合方法———FEM/PO PTD法 ,分析带有深腔的三维电大尺寸导体目标的电磁散射特性。应用DDM将原有深腔分成若干个较小的子域。对于腔体开口面所在子域 ,应用等效原理 ,结合物理光学法 (PO)与物理绕射理论 (PTD) ,得到腔体开口面的边界积分方程 ;对于其它子域 ,通过传输条件实现各子域间的耦合。在各子域内推导出此边值问题的等价泛函 ,应用基于棱边的有限元法 (Edge basedFEM)进行分析。理论分析与计算结果表明 ,该混合方法与其它计算同类问题的方法相比 ,有较高的计算精度 ,同时能减少对计算机内存的需求。