A class of iterative methods for solving nonlinear projection equations

A class of globally convergent iterative methods for solving nonlinear projection equations is provided under a continuity condition of the mappingF. WhenF is pseudomonotone, a necessary and sufficient condition on the nonemptiness of the solution set is obtained.The author would like to thank two referees for their useful comments on this paper and one of them, in particular, for bringing Ref. 15 to his attention. The author also thanks Professor He for sending him Ref. 23.     

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     

Global Hölder regularity for discontinuous elliptic equations in the plane

-regularity up to the boundary is proved for solutions of boundary value problems for elliptic equations with discontinuous coefficients in the plane.

In particular, we deal with the Dirichlet boundary condition

where , 2$">, or with the following normal derivative boundary conditions:

where , 2$">, 0$"> and is the unit outward normal to the boundary .

     

Various Newton-type iterative methods for solving nonlinear equations

The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.     

Numerical solutions of coupled systems of nonlinear elliptic equations

This article deals with numerical solutions of a general class of coupled nonlinear elliptic equations. Using the method of upper and lower solutions, monotone sequences are constructed for difference schemes which approximate coupled systems of nonlinear elliptic equations. This monotone convergence leads to existence‐uniqueness theorems for solutions to problems with reaction functions of quasi‐monotone nondecreasing, quasi‐monotone nonincreasing and mixed quasi‐monotone types. A monotone domain decomposition algorithm which combines the monotone approach and an iterative domain decomposition method based on the Schwarz alternating, is proposed. An application to a reaction‐diffusion model in chemical engineering is given. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 621–640, 2012     

Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems

In this paper, a two-dimensional quasilinear elliptic problem of the form -divF(x,▽u)=g(x)$-\mathit{divF}(\mathbf{x}\text{,}▽u)=g(\mathbf{x})$ is considered. This problem is ill-conditioned and we therefore propose a modified iterative algorithm based on coupling of the Sobolev space gradient method and WEB-spline finite element method. Applying the preconditioned iterative method, which has been already provided by Farago and Karatson (2001) [1] reduces the our considered problem to a sequence of linear Poisson’s problems. Then the WEB-spline finite element method is applied to the approximate solution of these Poisson’s problems. In this sense, a convergence theorem is proved and the advantages of this technique than the gradient finite element method (GFEM) is also described. Finally, the presented method is tested on some examples and compared with GFEM. It is shown that the gradient WEB-spline finite element method gives better test results.     

A new efficient parametric family of iterative methods for solving nonlinear systems

ABSTRACT

A bi-parametric family of iterative schemes for solving nonlinear systems is presented. We prove for any value of parameters the sixth-order of convergence of any members of the class. The efficiency and computational efficiency indices are studied for this family and compared with that of the other known schemes with similar structure. In the numerical section, we solve, after discretizating, the nonlinear boundary problem described by the Fisher's equation. This numerical example confirms the theoretical results and show the performance of the proposed schemes.     

Adaptive wavelet frame domain decomposition methods for nonlinear elliptic equations

In this article, we are concerned with the numerical treatment of nonlinear elliptic boundary value problems. Our method of choice is a domain decomposition strategy. Partially following the lines from (Cohen, Dahmen and deVore, SIAM J Numer Anal 41 (2003), 1785–1823; Kappei, Appl Anal J Sci 90 (2011), 1323–1353; Lui, SIAM J Sci Comput 21 (2000), 1506–1523; Stevenson and Werner, Math Comp 78 (2009), 619–644), we develop an adaptive additive Schwarz method using wavelet frames. We show that the method converges with an asymptotically optimal rate and support our theoretical results with numerical tests in one and two space dimensions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n-1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

A new eighth-order iterative method for solving nonlinear equations

In this paper we present an improvement of the fourth-order Newton-type method for solving a nonlinear equation. The new Newton-type method is shown to converge of the order eight. Per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of , which is better than the well known Newton-type methods of lower order. We shall examine the effectiveness of the new eighth-order Newton-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons are made with several other existing methods to show the performance of the presented method.     

On modification of certain methods of the conjugate direction type for solving systems of linear algebraic equations

A modification of certain well-known methods of the conjugate direction type is proposed and examined. The modified methods are more stable with respect to the accumulation of round-off errors. Moreover, these methods are applicable for solving ill-conditioned systems of linear algebraic equations that, in particular, arise as approximations of ill-posed problems. Numerical results illustrating the advantages of the proposed modification are presented.     

Direct and approximate algorithmic methods for solving self-adjoint elliptic PDEs in three-space dimensions

The numerical implementation of the extended to the limit sparse LDL^T factorization solution methods for three-dimensional self-adjoint elliptic partial differential equations [3] is given. Two FORTRAN routines for the approximate (or exact) factorization of the coefficient matrix and solution of the resulting finite difference equations are supplied. The amount of fill-in terms can be controlled by the user through parameters R1, R2 the limiting case being when the matrix is factorized exactly.     

Minimax methods for singular elliptic equations with an application to a jumping problem

A jumping problem for a class of singular semilinear elliptic equations is considered. Minimax methods in the framework of nonsmooth critical point theory are applied.     

On modification of certain methods of the conjugate direction type for solving rectangular systems of linear algebraic equations

The use of modifications of certain well-known methods of the conjugate direction type for solving systems of linear algebraic equations with rectangular matrices is examined. The modified methods are shown to be superior to the original versions with respect to the round-off accumulation; the advantage is especially large for ill-conditioned matrices. Examples are given of the efficient use of the modified methods for solving certain fairly large ill-conditioned problems.     

Concentrated terms and varying domains in elliptic equations: Lipschitz case

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in H¹an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.