Adaptive iterative linearization Galerkin methods for nonlinear problems

A wide variety of (fixed-point) iterative methods for the solution of nonlinear equations (in Hilbert spaces) exists. In many cases, such schemes can be interpreted as iterative local linearization methods, which, as will be shown, can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative schemes. In particular, for Lipschitz continuous and strongly monotone operators, we derive a general convergence analysis. Furthermore, in the context of numerical solution schemes for nonlinear partial differential equations, we propose a combination of the iterative linearization approach and the classical Galerkin discretization method, thereby giving rise to the so-called iterative linearization Galerkin (ILG) methodology. Moreover, still on an abstract level, based on two different elliptic reconstruction techniques, we derive a posteriori error estimates which separately take into account the discretization and linearization errors. Furthermore, we propose an adaptive algorithm, which provides an efficient interplay between these two effects. In addition, the ILG approach will be applied to the specific context of finite element discretizations of quasilinear elliptic equations, and some numerical experiments will be performed

Numerical Methods for Systems of Nonlinear Parabolic Equations with Time Delays

AbstractThe purpose of this paper is to investigate some numerical aspects of a class of coupled nonlinear parabolic systems with time delays. The system of parabolic equations is discretized by the finite difference method which yields a coupled system of nonlinear algebraic equations. The mathematical analysis of the nonlinear system is by the method of upper and lower solutions and its associated monotone iterations. Three monotone iterative schemes are presented and it is shown that the sequence of iterations from each one of these iterative schemes converges monotonically to a unique solution of the finite difference system. A theoretical comparison result for the various monotone sequences and error estimates for the three monotone iterative schemes are obtained. It is also shown that the finite difference solution converges to the classical solution of the parabolic system as the mesh size decreases to zero

On the Convergence of Adaptive Iterative Linearized Galerkin Methods

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work that covers some prominent procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main purpose of this paper is the derivation of an abstract convergence theory for the unified ILG approach (based on general adaptive Galerkin discretization methods) proposed in our previous work. The theoretical results will be tested and compared for the aforementioned three iterative linearization schemes in the context of adaptive finite element discretizations of strongly monotone stationary conservation laws

New high-order methods for solving non-linear equations

In this paper, we present two families of third and fourth order iterative methods for solving nonlinear equations. The efficiency index of the proposed schemes is 1.442 and 1.587. In order to compare the performance with some of the existing schemes, several numerical examples are furnished here

Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

We study a kind of nonlinear elliptic boundary value problems with generalized p-Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work

Some novel schemes by using multiplicative calculus for nonlinear equations

This research is supported by the Higher Education Commission of Pakistan.In this paper, we suggest and analyze a new family of two-step predictor corrector type iterative schemes for solving nonlinear equations in the framework of multiplicative calculus. We also discuss the convergence criteria of these newly developed iterative methods. Some numerical examples will be given to illustrate the efficiency and performance of derived methods.Publisher's Versio

Iterative schemes for finding all roots simultaneously of nonlinear equations

In this paper, we propose a procedure that can be added to any iterative scheme in order to turn it into an iterative method for approximating all roots simultaneously of any nonlinear equations. By applying this procedure to any iterative method of order p, we obtain a new scheme of order of convergence 2p. Some numerical tests allow us to confirm the theoretical results and to compare the proposed schemes with other known methods for simultaneous roots of polynomial and non-polynomial functions

A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007