Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method

A new approach is proposed for constructing nonoverlapping domain decomposition procedures for solving a linear system related to a nodal finite element method. It applies to problems involving either positive semi-definite or complex indefinite local matrices. The main feature of the method is to preserve the continuity requirements on the unknowns and the finite element equations at the nodes shared by more than two subdomains and to suitably augment the local matrices. We prove that the corresponding algorithm can be seen as a converging iterative method for solving the finite element system and that it cannot break down. Each iteration is obtained by solving uncoupled local finite element systems posed in each subdomain and, in contrast to a strict domain decomposition method, is completed by solving a linear system whose unknowns are the degrees of freedom attached to the above special nodes.     

Nonoverlapping Domain Decomposition Method with Mixed Element for Elliptic Problems

1.IntroductionDomaindecompositionasanewmethodofcomputationalmathematics,waJsdevel-opedsincethedevelopmentofparallelcomputersandmultiprocessorsupercomputers-Usingdomaindecompositionwecandecreasethescaleoftheproblemandimplementthesub-problemsonparallelcomputer.Fromatechnicalpointofviewmostofdo-maindecompositionmethodsconsideredsofarhavebeendealingwithfiniteelementmethods.In[1,2]ZhangandHuanghavegivenakindofnonoverlappingdomaindecompositionprocedurewithpiecewiselinearfiniteelementapproximation.…     

Optimal Coarse Grid Size in Domain Decomposition

1.IntroductionDomaindecomposition(DD)isaclassoftechniquesforsolvingellipticboundaryvalueproblemsinwhichthesolutionisobtainedbyiterstivelysolvingsmallersub-domainproblems.Thesemethodshavereceivedalotofstudyinrecentyears(see[6,1,2,7,1o]).TheyareattractivebecauseoftheirinherentpaxallelismandtheirOPtimalconvergencerates(i.e.independentofthemeshsize).TheoptimalityoftheconvergenceraterequiresthesolutionofacoarsegridproblemateaChiteration.ThestudyofhowtoincorporatesuchacoarsegridsolveinaDDmethodh…     

区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

Optimal Control in Heterogeneous Domain Decomposition Methods for Advection-Diffusion Equations

New domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first and second order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem, and the convergence of iterative processes is analyzed. The work was supported by the Russian Foundation for Basic Research (04-01-00615) and it was partly carried out while the first author was visiting the IACS at EPFL.     

Stokes问题的区域分解法

本文主要讨论了Ｓｔｏｋｅｓ问题的非重迭型两仓区域性情形的区域分解算法，首先讨论了连续情形，然后将区域分解算法应用到Ｓｔｏｋｅｓ问题的非协调离散情形。     

Hamilton-Jacobi-Bellman方程的区域分解算法

对离散Hamilton-Jacobi-Bellman方程提出了一类区域分解算法,并在合理的假设下证明了该算法的单调收敛性,数值结果表明该算法的有效性与准确性．     

A Domain Decomposition Method for Frictional Contact Problem

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method. The global problem is transformed to a smaller problem on the contact surface. The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure.     

The Coupling of Finite Element Method and Boundary Element Method for Two-Dimensional Helmholtz Equation in an Exterior Domain

This paper presents a new coupling of the finite element method and the boundary element method to solve the two-dimensional exterior Helmholtz problems by using the asymptotic radiation conditions in [1], in which the coupling relations are the same as C. Johnson and J. C. Nedelec's. The error estimates are derived and results of numerical calculation in comparison with analytic solution verify the theoretical estimates.     

一种有限元-边界元耦合分域算法

提出了一种有限元-边界元耦合分域算法．该算法将所分析问题的区域分解成有限元和边界元子域,在满足两子域界面上位移和面力协调连续的条件下,通过迭代求解得到问题的解．在迭代求解过程中,引入动态松弛系数,使收敛得以加速．该方法在两子域界面上有限单元结点和边界单元结点的位置相互独立,无需协调一致,对诸如裂纹扩展过程的模拟具有独特的优势．用所提出的耦合算法分析算例,得到的结果与有限元法、边界元法和另一种耦合算法的数值计算结果一致,验证了这种算法的正确性和可行性．     

Non-Conforming Domain Decomposition with Hybrid Method

We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case.     

Domain Decomposition Method Modified by Characteristic Finite Element Procedure for System of Parabolic Equations with Discontinuous Coefficients

In this work, system of parabolic equations with discontinuous coefficients is studied. The domain decomposition method modified by a characteristic finite element procedure is applied. A function is defined to approximate the fluxes on inner boundaries by using the solution at the previous level. Thus the parallelism is achieved. Convergence analysis and error estimate are also presented.     

Accelerated Non-Overlapping Domain Decomposition Method for Total Variation Minimization

We concern with fast domain decomposition methods for solving the total variation minimization problems in image processing. By decomposing the image domain into non-overlapping subdomains and interfaces, we consider the primal-dual problem on the interfaces such that the subdomain problems become independent problems and can be solved in parallel. Suppose both the interfaces and subdomain problems are uniformly convex, we can apply the acceleration method to achieve an $\mathcal{O}(1 / n^2)$ convergent domain decomposition algorithm. The convergence analysis is provided as well. Numerical results on image denoising, inpainting, deblurring, and segmentation are provided and comparison results with existing methods are discussed, which not only demonstrate the advantages of our method but also support the theoretical convergence rate.     

Some Domain Decomposition and Iterative Refinement Algorithms for Elliptic Finite Element Problems

In this contribution, we report on some results recently obtained in joint work with Maksymilian Dryja. We first study an additive variant of Schwarz' alternating algorithm and establish that a fast method of this kind can be devised which is optimal in the sense that the number of conjugate gradient iterations required, to reach a certain tolerance, is independent of the mesh size as well as the number of subregions.     

Domain Decomposition of Optimal Control Problems for Dynamic Networks of Elastic Strings

We consider optimal control problems related to exact- and approximate controllability of dynamic networks of elastic strings. In this note we concentrate on problems with linear dynamics, no state and no control constraints. The emphasis is on approximating target states and velocities in part of the network using a dynamic domain decomposition method (d³m) for the optimality system on the network. The decomposition is established via a Uzawa-type saddle-point iteration associated with an augmented Lagrangian relaxation of the transmission conditions at multiple joints. We consider various cost functions and prove convergence of the infinite dimensional scheme for an exemplaric choice of the cost. We also give numerical evidence in the case of simple exemplaric networks.     

一元四次整系数多项式的因式分解法

仅对一元四次整系数多项式在实数域内分解问题进行了研究,根据分解后其系数应为二次代数整数的特点,以及导出的二次方程判别式的完全平方性质,得出了一元四次整系数多项式在实数域内能分解成两个二次因式乘积的条件及方法,从而解决了一元四次整系数多项式在实数域内的因式分解问题.     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

基于对称化原理的对称区域分裂法

本文利用对称化原理,讨论了一种只需在子区域上计算两个完全独立子问题就可得到原问题解的对称区域分裂法,并用此方法求解线性算子方程和线性透射问题.此方法可作为并行算法在MIMD计算机上使用.     

A Source Transfer Domain Decomposition Method for Helmholtz Equations in Unbounded Domain Part II: Extensions

In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz problems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.     

Improving the Reconstruction of Vector Fields Using Mixed Finite Element Methods and Optimal Preconditioning

In this article, we study numerically a diagnostic model, based on mass conservation, to recover solenoidal vector fields from experimental data. Based on a reformulation of the mathematical model as a saddle‐point problem, we introduce an iterative preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To obtain a stable algorithm, we use a second‐order mixed finite element approximation for discretization. We show, using synthetic vector fields, that this new approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1137–1154, 2016