A Parallel Preconditioned Iterative Realization of the Panel Method in 3D

The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCube-2 parallel computer showing that preconditioned iterative methods are very well suited also in three-dimensional cases for implementation on an MIMD computer and that they are much more efficient than usual direct solution techniques.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

On parallel solution of linear elasticity problems. Part III: higher order finite elements

This is the third part of a trilogy on parallel solution of the linear elasticity problem. We consider the separate displacement ordering for a plain isotropic problem with full Dirichlet boundary conditions. The parallel solution methods presented in the first two parts of the trilogy are here generalised to higher order by using hierarchical finite elements. We discuss node numberings on regular grids for high degree of parallelism and even processor load as well as the problem of stability of the modified incomplete Cholesky factorisations used. Several preconditioning techniques for the conjugate gradient method are studied and compared for quadratic finite elements. Bounds for the condition numbers of the corresponding preconditioning methods are derived, and computer experiments are performed in order to confirm the theory and give recommendations on the choice of method. The parallel implementation is performed by message passing interface. Copyright © 2012 John Wiley & Sons, Ltd.     

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chan-type and P-type preconditioners are introduced. The suggested method can reach higher order accuracy both in space and in time than the existing methods. When the used boundary value method is $A_{k1,k2}$-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.     

Upper and lower solution method for a fourth-order four-point boundary value problem on time scales

In this work, we consider a fourth-order four-point boundary value problem on time scales. We establish criteria for the existence of a solution by developing the upper and lower solution method and the monotone iterative technique.     

一种迭代格式的有限元并行算法*

本文提出了一种求解有限元方程的迭代格式的并行算法.该方法在线性代数方程迭代解法的基础上,引进并行运算步骤;并且运用加权残数方法,通过选择适当的权函数,推导了该并行算法的有限元基本格式.该方法在西安交通大学BLXSI-6400并行计算机上程序实现.计算结果表明它能有效地提高运算速度,减少计算时间,是一种有效的求解大型结构有限元方程的并行算法.     

Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.     

Positive solutions for a class of fractional differential equations with integral boundary conditions

The existence of positive solutions for a class of fractional equations involving the Riemann–Liouville fractional derivative with integral boundary conditions is investigated. By means of the monotone iteration method and some inequalities associated with the Green function, we obtain the existence of a positive solution and establish the iterative sequence for approximating the solution.     

Numerical solutions of a two-phase membrane problem

In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.     

Grid solution of problem with unilateral constraints

The present study deals with the solution of a problem, defined in a three-dimensional domain, arising in fluid mechanics. Such problem is modelled with unilateral constraints on the boundary. Then, the problem to solve consists in minimizing a functional in a closed convex set. The characterization of the solution leads to solve a time-dependent variational inequality. An implicit scheme is used for the discretization of the time-dependent part of the operator and so we have to solve a sequence of stationary elliptic problems. For the solution of each stationary problem, an equivalent form of a minimization problem is formulated as the solution of a multivalued equation, obtained by the perturbation of the previous stationary elliptic operator by a diagonal monotone maximal multivalued operator. The spatial discretization of such problem by appropriate scheme leads to the solution of large scale algebraic systems. According to the size of these systems, parallel iterative asynchronous and synchronous methods are carried out on distributed architectures; in the present study, methods without and with overlapping like Schwarz alternating methods are considered. The convergence of the parallel iterative algorithms is analysed by contraction approaches. Finally, the parallel experiments are presented.     

Parallel Solution of the Helmholtz Equation in a Multilayer Domain

We study time-harmonic wave propagation in layered, heterogeneous media. Solving this relatively complex application problem numerically is a challenging task. The full potential of algorithms, parallel programming models and computer architectures must be exploited. Our aim is to give a broad perspective on the various considerations that come into play. The basic parts of our algorithms consist of finite difference discretizations, domain decomposition and preconditioned iterative methods. We present two serial algorithms with different properties. Then, we discuss parallelization strategies using a local memory model, a shared memory model, or a combination of the two. The numerical experiments highlight the differences between the approaches and show results for three different combinations of algorithm and computer architecture that lead to viable solution methods.     

Solving differential equations using modified Picard iteration

Many classes of differential equation are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and engineering and include non-linear as well as linear differential equations. Examples involving partial as well as ordinary differential equations are presented. The method is easy to implement on a computer and the solutions so obtained are essentially power series. With its conceptual clarity (differential equations are integrated directly), its uniform methodology (the overall approach is the same in all cases) and its straightforward computer implementation (the integration and iteration procedures require only standard commercial software), the modified Picard methods offer obvious benefits for the teaching of differential equations as well as presenting a basic but flexible tool-kit for the solution process itself.     

Cascadic multilevel methods for ill-posed problems

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle.     

A parallel implementation of overlapping Schwarz method for the dual reciprocity method

We present a parallel algorithm for the overlapping domain decomposition boundary integral equation method for two dimensional partial differential equations. In addition to the improvement of the ill-conditioning and the computational efficiency achieved by domain partitioning, using a parallel computer with p processors can offer up to p times efficiency. Assuming direct solution is used throughout, partitioning the domain into p subregions and employing a processor for each subproblem, overall, result in p² times efficiency over using a single domain and a single processor, taking into account that a sequential algorithm of the underlying method can improve the computational efficiency at least p times over using a single domain. Some numerical results showing the efficiency of the parallel technique will be presented.     

Convergence of Three-Step Taylor Galerkin Finite Element Scheme Based Monotone Schwarz Iterative Method for Singularly Perturbed Differential-Difference Equation

Monotone Schwarz iterative methods for parabolic partial differential equations are well known for their advantage of eliminating the search for an initial solution. In this article, we propose a monotone Schwarz iterative method for singularly perturbed parabolic retarded differential-difference equations based on a three-step Taylor Galerkin finite element scheme. The stability and ε-uniform convergence of the three-step Taylor Galerkin finite element method have been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed monotone Schwarz iterative method has been established.     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.     

Almost block diagonal linear systems: sequential and parallel solution techniques,and applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two‐point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright © 2000 John Wiley & Sons, Ltd.