Easily implementable iterative methods for variational inequalities with nonlinear diffusion–convection operator and constraints to the gradient of solution

2015 ◽  
Vol 30 (1) ◽  
Author(s):  
Erkki Laitinen ◽  
Alexander Lapin ◽  
Sergey Lapin
Keyword(s):  
Variational Inequalities ◽  
Iterative Methods ◽  
Nonlinear Diffusion ◽  
Linear Equations ◽  
Finite Element Approximation ◽  
Iterative Solution ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Element Approximation ◽  
Diffusion Convection

AbstractNew iterative solution methods are proposed for the finite element approximation of a class of variational inequalities with nonlinear diffusion-convection operator and constraints to the gradient of solution. Implementation of every iteration of these methods reduces to the solution of a system of linear equations and a set of two-dimensional minimization problems. Convergence is proved by the application of a general result on the convergence of the iterative methods for a nonlinear constrained saddle point problem.

scholarly journals A relaxed non-linear inexact Uzawa algorithm for stokes problem

2019 ◽  
Vol 23 (4) ◽  
pp. 2323-2331
Author(s):  
Shao-Qing Zheng ◽  
Jun-Feng Lu
Keyword(s):  
Fluid Dynamics ◽  
Numerical Experiments ◽  
Stokes Problem ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Element Approximation ◽  
Uzawa Algorithm ◽  
Non Linear

In this paper, we consider a Stokes problem arising in fluid dynamics and thermal science, which can be transformed to a symmetric saddle point problem by using the mixed finite element approximation. A relaxed non-linear inexact Uzawa algorithm is proposed for solving the problem, and the convergence of this algorithm is also considered. Numerical experiments are presented to show the efficiency of relaxed non-linear inexact Uzawa algorithm.

scholarly journals Krylov Subspace Solvers and Preconditioners

2018 ◽  
Vol 63 ◽  
pp. 1-43
Author(s):  
C. Vuik
Keyword(s):  
Iterative Methods ◽  
Krylov Subspace ◽  
State Of The Art ◽  
Linear Equations ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Starting Point ◽  
Lecture Notes

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

scholarly journals Generalized Iterative Methods for Solving Double Saddle Point Problem

2020 ◽  
Vol 5 (2) ◽  
pp. 137-150
Author(s):  
Michele Benzi ◽  
Fatemeh Panjeh Ali Beik ◽  
Sayyed–Hasan Azizi Chaparpordi ◽  
Zohreh Rouygar ◽  
◽  
...  
Keyword(s):  
Saddle Point ◽  
Iterative Methods ◽  
Saddle Point Problem ◽  
Point Problem

scholarly journals A New Approach to Asymptotic Behavior for a Finite Element Approximation in Parabolic Variational Inequalities

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Salah Boulaaras ◽  
Mohamed Haiour
Keyword(s):  
Finite Element ◽  
Asymptotic Behavior ◽  
Variational Inequalities ◽  
Finite Element Approximation ◽  
Uniform Norm ◽  
Element Approximation ◽  
New Approach ◽  
Elliptic Variational Inequalities ◽  
Energy Behavior ◽  
Spatial Approximation

The paper deals with the theta time scheme combined with a finite element spatial approximation of parabolic variational inequalities. The parabolic variational inequalities are transformed into noncoercive elliptic variational inequalities. A simple result to time energy behavior is proved, and a new iterative discrete algorithm is proposed to show the existence and uniqueness. Moreover, its convergence is established. Furthermore, a simple proof to asymptotic behavior in uniform norm is given.

Sign in / Sign up

Export Citation Format

Share Document