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# Study on Numerical Methods for Some Nonlinear Equations and Equilibrium Problem

Author: CheHaiTao
Tutor: WangYiJu
School: Qufu Normal University
Course: Applied Mathematics
Keywords: H~1-Galerkin expanded mixed fnite element method Error esti-mates Nonlinear pseudo-parabolic integro-diferential equations Nonlinear vis-coelasticity equations Generalized BBM-Burgers equation Solvability Stability Fixed point Equilibrium problem Hybrid steepest descent method Nonexpan-sive semigroup Pseudocontractive mapping Variational inclusion Variationalinequality
CLC: O241.82
Type: PhD thesis
Year: 2013
Downloads: 14
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### Abstract

 Mixed fnite element method and iterative method play an important rolein the numerical methods for diferntial equations and equilibrium problem, re-spectively. In this thesis, we mainly study an H1-Galerkin expanded mixed fniteelement method of nonlinear equations and iterative method of equilibrium prob-lem.An H1-Galerkin expanded mixed fnite element procedure is proposed bycombining the H1-Galerkin formulation with the expanded mixed fnite elementmethods. The formulation inherits the advantages of H1-Galerkin and expandedmixed fnite element methods, it can solve the scalar unknown, its gradient and itsflux directly. Furthermore, the formulation is free of LBB condition as requiredby the mixed fnite element methods.Equilibrium problem has been extensively applied to physics, mechanics,morden control, nonlinear programming, economical equilibrium and some otherareas. The other aim of this thesis is to construct iterative algorithm for thecommon element of equilibrium problems, variational inequality and fxed piontproblem and to analyse the convergence of the iterative algorithm.We show our main results as follows.1. In Chapter I, we introduce the background of research and show thesummary of the work in this article.2. In Chapter II, an H1-Galerkin expanded mixed fnite element method isdiscussed for nonlinear pseudo-parabolic integro-diferential equations. We con-duct theoretical analysis to study the existence and uniqueness of numerical so-lutions to the discrete scheme. A priori error estimate is derived for the unknownfunction, gradient function, and flux. Numerical example is presented to illustratethe efectiveness of the proposed scheme.3. In Chapter III, we investigate an H1-Galerkin expanded mixed fnite ele-ment method for nonlinear viscoelasticity equations. The existence and unique- ness of solutions to the numerical scheme are proved. An optimal error estimateis derived for the unknown function, the gradient function, and the flux.4. In Chapter IV, a linear-implicit fnite diference scheme is given for theinitial-boundary problem of generalized Benjamin-Bona-Mahony-Burgers equa-tion, which is convergent and unconditionally stable. The unique solvability ofnumerical solutions is shown. A priori estimate and second-order convergenceof the fnite diference approximate solution are discussed using energy method.Numerical results demonstrate that the scheme is efcient and accurate.5. In Chapter V, we introduce a new iterative scheme based on viscosityapproximation method for fnding a common element of the set of common fxedpoints of an infnite family of ki-strictly pseudo-contractive mappings, the set ofsolutions of equilibrium problem in Hilbert spaces. We prove that the sequenceconverges strongly to a common element of the above two sets under some mildconditions. Furthermore, simple numerical examples are presented to illustratethe efectiveness of the proposed scheme.6. In Chapter VI, we introduce a new iterative scheme by using viscosityhybrid steepest descent method for fnding a common element of the set of so-lutions of a system of equilibrium problems, the set of fxed points of an infnitefamily of strictly pseudocontractive mappings, the set of solutions of fxed pointsfor nonexpansive semigroup, and two sets of solutions of variational inequalityproblems with relaxed cocoercive mapping in a real Hilbert space. We prove thatthe sequence converges strongly to a common element of the above fve sets undersome mild conditions. The results shown in this chapter improve and extend therecent ones announced by many others.7. In Chapter VII, we introduce a new iterative scheme base on a hybridsteepest descent method for fnding a common element of the set of solutionsof a system of equilibrium problems, the set of fxed points of an infnite familyof strictly pseudocontractive mappings, the set of solutions of fxed points fornonexpansive semigroup, the set of solutions of system of variational inclusions,the set of fxed points of a pseudo-contractive mapping and the set of solutionsof the variational inequality with Lipschitzian relaxed cocoercive mappings in areal Hilbert space. We prove that the sequence converges strongly to a commonelement of the above six sets under some mind conditions. The results shown in this chapter improve and extend the recent ones announced by many others.

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