On the parallel version of the successive approximation method for quasilinear boundary value problem

The convergence properties of the successive approximation method to solve a quasilinear two points boundary value problem is studied. The successive approximation method is used to solve the parallel/multiple version of the problem. Conditions which assure the convergence of the method and error bound are given.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

A BOUNDARY ELEMENT METHOD FOR A NONLINEAR BOUNDARY VALUE PROBLEM IN STEADY-STATE HEAT TRANSFER IN DIMENSION THREE

In this paper, a new method of boundary reduction is proposed, which reduces thesteady-state heat transfer equation with radiation. Moreover, a boundary element method is pre-sented for its solution and the error estimates of the numerical approximations are given.     

A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems

The inverse-free preconditioned Krylov subspace method of Golub and Ye [G.H. Golub, Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM J. Sci. Comp. 24 (2002) 312-334] is an efficient algorithm for computing a few extreme eigenvalues of the symmetric generalized eigenvalue problem. In this paper, we first present an analysis of the preconditioning strategy based on incomplete factorizations. We then extend the method by developing a block generalization for computing multiple or severely clustered eigenvalues and develop a robust black-box implementation. Numerical examples are given to illustrate the analysis and the efficiency of the block algorithm.     

A numerical experimental study of inverse preconditioning for the parallel iterative solution to 3D finite element flow equations

Integration of the subsurface flow equation by finite elements (FE) in space and finite differences (FD) in time requires the repeated solution to sparse symmetric positive definite systems of linear equations. Iterative techniques based on preconditioned conjugate gradients (PCG) are one of the most attractive tool to solve the problem on sequential computers. A present challenge is to make PCG attractive in a parallel computing environment as well. To this aim a key factor is the development of an efficient parallel preconditioner. FSAI (factorized sparse approximate inverse) and enlarged FSAI relying on the approximate inverse of the coefficient matrix appears to be a most promising parallel preconditioner. In the present paper PCG using FSAI, diagonal and pARMS (parallel algebraic recursive multilevel solvers) preconditioners is implemented on the IBM SP4/512 and CLX/768 supercomputers with up to 32 processors to solve underground flow problems of a large size. The results show that FSAI may allow for a parallel relative efficiency larger than 50% on the largest problems with p=32 processors. Moreover, FSAI turns out to be significantly less expensive and more robust than pARMS. Finally, it is shown that for p in the upper range may be much improved if PCG–FSAI is implemented on CLX.     

Existence of solutions for three-point boundary value problems for second order equations

Shooting methods are employed to obtain solutions of the three-point boundary value problem for the second order equation, where is continuous, and and conditions are imposed implying that solutions of such problems are unique, when they exist.

     

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.     

Two-level hierarchically preconditioned conjugate gradient methods for solving linear elasticity finite element equations

In this paper we study and compare some preconditioned conjugate gradient methods for solving large-scale higher-order finite element schemes approximating two- and three-dimensional linear elasticity boundary value problems. The preconditioners discussed in this paper are derived from hierarchical splitting of the finite element space first proposed by O. Axelsson and I. Gustafsson. We especially focus our attention to the implicit construction of preconditioning operators by means of some fixpoint iteration process including multigrid techniques. Many numerical experiments confirm the efficiency of these preconditioners in comparison with classical direct methods most frequently used in practice up to now.     

A system of boundary integral equations for the transmission problem in acoustics

In this paper, we reduce the classical two-dimensional transmission problem in acoustic scattering to a system of coupled boundary integral equations (BIEs), and consider the weak formulation of the resulting equations. Uniqueness and existence results for the weak solution of corresponding variational equations are established. In contrast to the coupled system in Costabel and Stephan (1985) [4], we need to take into account exceptional frequencies to obtain the unique solvability. Boundary element methods (BEM) based on both the standard and a two-level fast multipole Galerkin schemes are employed to compute the solution of the variational equation. Numerical results are presented to verify the efficiency and accuracy of the numerical methods.     

Solving boundary value problems of ordinary differential equations with non-separated boundary conditions

This paper focuses on solving the two point boundary value problem, in which boundary conditions are systems of nonlinear equations. The shooting method was used together with a combination of Newton’s method and Broyden’s method, to update the initial values of the differential equations. The experiments showed that the proposed method performed well, in the sense that the overall amount of work was less than that of the Newton Shooting method.     

Positive solutions for boundary value problems of second order difference equations and their computation

We consider the following two classes of second order boundary value problems for difference equation:

     

Uniqueness implies existence for three-point boundary value problems for dynamic equations

Shooting methods are used to obtain solutions of the three-point boundary value problem for the second-order dynamic equation, y^ΔΔ = f (x, y, y^Δ), y(x₁) = y₁, y(x₃) − y(x₂) = y₂, where f : (a, b)_T × ² → is continuous, x₁ < x₂ < x₃ in (a, b)_T, y₁, y₂ ε , and T is a time scale. It is assumed such solutions are unique when they exist.     

On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem

     

The versions of the finite element method for problems with boundary layers

We study the uniform approximation of boundary layer functions for , , by the and versions of the finite element method. For the version (with fixed mesh), we prove super-exponential convergence in the range . We also establish, for this version, an overall convergence rate of in the energy norm error which is uniform in , and show that this rate is sharp (up to the term) when robust estimates uniform in are considered. For the version with variable mesh (i.e., the version), we show that exponential convergence, uniform in , is achieved by taking the first element at the boundary layer to be of size . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when is as small as, e.g., . They also illustrate the superiority of the approach over other methods, including a low-order version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

     

DRIC: A dynamic version of the RIC method

A new incomplete factorization method is proposed, differing from previous ones by the way in which the diagonal entries of the triangular factors are defined. A comparison is given with the dynamic modified incomplete factorization methods of Axelsson–Barker and Beauwens, and with the relaxed incomplete Cholesky method of Axelsson and Lindskog. Theoretical arguments show that the new method is at least as robust as both previous ones, while numerical experiments made in the discrete PDE context show an effective improvement in many practical circumstances, particularly for anisotropic problems.     

Cluster robustness of preconditioned gradient subspace iteration eigensolvers

The paper presents convergence estimates for a class of iterative methods for solving partial generalized symmetric eigenvalue problems whereby a sequence of subspaces containing approximations to eigenvectors is generated by combining the Rayleigh-Ritz and the preconditioned steepest descent/ascent methods. The paper uses a novel approach of studying the convergence of groups of eigenvalues, rather than individual ones, to obtain new convergence estimates for this class of methods that are cluster robust, i.e. do not involve distances between computed eigenvalues.     

A globally convergent version of the Polak-Ribière conjugate gradient method

In this paper we propose a new line search algorithm that ensures global convergence of the Polak-Ribière conjugate gradient method for the unconstrained minimization of nonconvex differentiable functions. In particular, we show that with this line search every limit point produced by the Polak-Ribière iteration is a stationary point of the objective function. Moreover, we define adaptive rules for the choice of the parameters in a way that the first stationary point along a search direction can be eventually accepted when the algorithm is converging to a minimum point with positive definite Hessian matrix. Under strong convexity assumptions, the known global convergence results can be reobtained as a special case. From a computational point of view, we may expect that an algorithm incorporating the step-size acceptance rules proposed here will retain the same good features of the Polak-Ribière method, while avoiding pathological situations. This research was supported by Agenzia Spaziale Italiana, Rome, Italy.     

Preconditioners for the p-version of the Galerkin method for a coupled finite element/boundary element system

We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log² p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log³ p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998     

Eigenvalue comparisons for boundary value problems for second order difference equations

We consider the eigenvalue problems for boundary value problems of second order difference equations

(1)

and

(2)

Comparison results for the eigenvalues of the problem (1) and the problem (2) are established.