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

Abstract

New iterative solution methods are proposed for the nite element approximation of a class of variational inequalities with nonlinear di usion-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.The development of the e cient numerical methods for the variational inequalities with constraints to gradient of a solution is a problem of a great practical importance. The size and complexity of the discrete variational inequalities constructed by using nite element and nite di erence methods pose signi cant problems for the iterative solution methods. The most popular iterative methods for this class of problems are so-called Algorithms 2 to 6 [5, 6] based on the Augmented Lagrangian technique. This technique leads to the constrained saddle point problems with ' lled operators' acting on the primal variables, and the proposed algorithms can be treated as a combination of block relaxation methods and gradient method in dual variables.Another approach to constructing iterative methods for constrained saddle point problems are proposed and investigated in the previous articles of the authors [8,9]. Namely, the saddle point problems are constructed after appropriate equivalent transformations of the discrete inclusion (equivalent to discrete variational inequality). These transformations lead to the constrained saddle point problems with block triangle operators acting on the primal variables, and these problems can be e ectively solved by the Uzawa-type methods (preconditioned gradient methods in dual variables).All aforementioned algorithms can be e ectively implemented when applying to the discrete variational inequalities with potential operators. This is not the case when solving a variational inequality with nonlinear di usion-convection operator because on every step of any of these iterative methods we have to solve a system of nonlinear equations, and this is the most time consuming part of the algorithms.In this article we prove the convergence of the easily implementable 'linearized' Uzawa-type methods. Their implementation reduces to the solution of a system of linear equations and a set of two-dimensional minimization problems. We investigate two methods -with exact solution of the system of linear equations and with its iterative solution by an inner iterative algorithm (the two-stage method). The convergence of the proposed methods follows from a general result for the iterative solution methods for a nite dimensional constrained saddle point problem.Brought to you by | Florida International University Libraries Authenticated Download Date | 6/8/15 12:50 AM