Parallel iterative methods for parabolic equations

For the problems of the parabolic equations in one- and two-dimensional space, the parallel iterative methods are presented to solve the fully implicit difference schemes. The methods presented are based on the idea of domain decomposition in which we divide the linear system of equations into some non-overlapping sub-systems, which are easy to solve in different processors at the same time. The iterative value is proved to be convergent to the difference solution resulted from the implicit difference schemes. Numerical experiments for both one- and two-dimensional problems show that the methods are convergent and may reach the linear speed-up.     

Parallelizing implicit algorithms for time-dependent problems by parabolic domain decomposition

Applications of the multidomain Local Fourier Basis method [1], for the solution of PDEs on parallel computers are described. The present approach utilizes, in an explicit way, the rapid (exponential) decay of the fundamental solutions of elliptic operators resulting from semi-implicit discretizations of parabolic time-dependent problems. As a result, the global matching relations for the elemental solutions are decoupled into local interactions between pairs of solutions in neighboring domains. Such interactions require only local communications between processors with short communication links. Thus the present algorithm overcomes the global coupling, inherent both in the use of the spectral Fourier method and implicit time discretization scheme.This research is supported partly by a grant from the French-Israeli Binational Foundation for 1991–1992.     

A parallel algorithm for solving the implicit diffusion difference equations

This paper presents a parallel algorithm for solving the implicit diffusion difference equations. The basic idea is based on vectorization of the tridiagonal Toeplitz difference equations. This method is superior to the algorithm showed by H. Stone [8]. We computed some examples on an NEC SX-3/44R supercomputer by our method. The results showed a good parallelism with this algorithm.     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.     

Hierarchical matrix techniques for a domain decomposition algorithm

Summary In this paper, we investigate the effectiveness of hierarchical matrix techniques when used as the linear solver in a certain domain decomposition algorithm. In particular, we provide a direct performance comparison between an algebraic multigrid solver and a hierarchical matrix solver which is based on nested dissection clustering within the software package PLTMG.      

Adaptive domain decomposition algorithms and finite volume/finite element approximation for advection-diffusion equations

Poor performances can be obtained from classical domain decomposition algorithms to solve advection-diffusion equations in the case of convection dominated flows. Therefore, adaptive domain decomposition have been developed for such flows. We investigate the properties of some algorithms of this kind in the framework of a finite volume/finite element discretization.This research was carried out while the author was visiting the Group of Applied Mathematics and Simulation of CRS4, and was supported by an HCM fellowship.     

An optimal compact sixth-order finite difference scheme for the Helmholtz equation

In this paper, we present an optimal compact finite difference scheme for solving the 2D Helmholtz equation. A convergence analysis is given to show that the scheme is sixth-order in accuracy. Based on minimizing the numerical dispersion, a refined optimization rule for choosing the scheme’s weight parameters is proposed. Numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters.     

Explicit/implicit and Crank–Nicolson domain decomposition methods for parabolic partial differential equations

Two parallel non-overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed. The algorithms combine Crank–Nicolson scheme with implicit Galerkin finite element methods in sub-domains and explicit flux approximation along inner boundaries at each time step. Thus, parallelism can be easily achieved. $L^2$-norm error estimates for these explicit/implicit procedures are presented, in which time step constraints are proved to be less severe than that of fully explicit schemes. Numerical experiments are also performed to verify the theoretical analysis.     

Non-overlapping domain decomposition algorithm based on modified transmission conditions for the Helmholtz equation

A square-root based transmission conditions domain decomposition method was recently introduced for the Helmholtz equation. It produces an effective algorithm where the convergence is independent of the wavenumber and the mesh discretization. We modify here these conditions in order to guarantee well-posedness of local problems and further improve the efficiency of the whole method. Numerical results, in particular in the three dimensional case, show significant reduction of the computational time needed in the iterative procedure while preserving the iteration number when compared with the original algorithm.     

基于有效并行求解策略的显式有限元分析并行算法

针对大规模结构非线性动力问题的有限元分析非常耗时,基于消息传递接口（MPI）机群环境,提出多种基于并行求解策略的显式有限元并行算法。基于显式消息传递的区域分解技术,采取重叠、非重叠区域分解技术及动态任务分配方法,通过将计算与通信重叠,优化处理器间的通信,对非重叠通信区域分解并行算法、重叠通信区域分解并行算法、群动态任务分配算法、动态任务分配算法及动态负载平衡算法进行研究。为在机群环境下实现非线性动力有限元分析,开发了基于有效并行求解策略的显式有限元并行算法。编写了基于消息传递编程模式的并行有限元程序,在工作站机群上实现了数值算例,分析了算法的性能,并与传统的Newmark算法进行了比较。算例表明：群动态任务分配算法的性能优于动态任务分配算法,低于区域分解算法的性能,动态负载平衡算法最优。对相同规模的问题提出的算法比Newmark算法快,优于Newmark算法。对结构非线性动力问题的有限元分析,所提出的并行算法是可行有效的。     

热传导方程有限差分区域分解算法的若干注记

51.引言由于受到并行计算的推动,十多年来,抛物型方程有限差分并行算法设计与分析一直得到关注.应     

Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

Optimization with time-dependent partial differential equations (PDEs) as constraints appears in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward time-dependent PDE coupled with optimality conditions. An optimization process by using the one-shot method determines the optimal control, state and adjoint state at once, with the cost of solving a large scale, fully discrete optimality system. Hence, such a one-shot method could easily become computationally prohibitive when the time span is long or time step is small. To overcome this difficulty, we propose several time domain decomposition algorithms for improving the computational efficiency of the one-shot method. In these algorithms, the optimality system is split into many small subsystems over a much smaller time interval, which are coupled by appropriate continuity matching conditions. Both one-level and two-level multiplicative and additive Schwarz algorithms are developed for iteratively solving the decomposed subsystems in parallel. In particular, the convergence of the one-level, non-overlapping algorithms is proved. The effectiveness of our proposed algorithms is demonstrated by both 1D and 2D numerical experiments, where the developed two-level algorithms show convergence rates that are scalable with respect to the number of subdomains.     

Parameter extraction of interdigital slow‐wave coplanar waveguide circuits using finite difference frequency domain algorithm

The slow wave effect can be obtained by a capacitively loaded structure with a symmetrical interdigital line connected on both sides of the coplanar waveguide (CPW) central line. The ferroelectric thin film with high dielectric constant can reduce the size of circuit and make it possible to realize tunable devices such as filter by applying voltage on it. Actually, this kind of slow wave structure is a periodic guided‐wave structure and can be analyzed by using classic finite difference frequency domain (FDFD) method for periodic guided‐wave structures. However, the very compact slow‐wave structures will usually result in simulation errors when the classic FDFD method is adopted, which will lead to a nonsymmetrical generalized eigenvalue problem. In this article, the shift‐and‐invert (SI) Arnoldi method is used to directly resolve this nonsymmetrical generalized eigenvalue problem. As a result, the accuracy of FDFD algorithm is improved. Especially for the large scale eigenvalue problem, SI method can also have a very fast speed of calculation. By means of its complex propagation constant obtained from simulation, one can extract circuit parameters of the interdigital capacitor. Consequently, one can analyze and design relevant resonators and filters in a quick and accurate manner, which are constructed with such interdigital slow wave structures. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.     

A domain decomposition technique for finite element based parametric sweep and tolerance analyses of microwave passive devices

A domain decomposition approach is here applied to the finite element solution of a multiport waveguide passive device. The approach allows separating the problem in multiple, coupled subproblems which can be solved individually. By appropriately defining one of these subdomains as containing all the possible variations to be studied it is hence possible to restrict the tolerance analysis to this latter, smaller domain. Numerical results showing the gain in computing time are presented. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.     

Domain decomposition operator splitting for mimetic finite difference discretizations of non-stationary problems

In this paper, we propose reliable and efficient numerical methods for solving semilinear, time-dependent partial differential equations of reaction–diffusion type. The original problem is first integrated in time by using a linearly implicit fractional step Runge–Kutta method. This method takes advantage of a suitable partitioning of the diffusion operator based on domain decomposition techniques. The resulting semidiscrete problem is fully discretized by means of a mimetic finite difference method on quadrilateral meshes. Due to the previous splitting, the totally discrete scheme can be reduced to a set of uncoupled linear systems which can be solved in parallel. The overall algorithm is unconditionally stable and second-order convergent in both time and space. These properties are confirmed by numerical experiments.