Three Numerical Methods Analysis And Applications For Solving Elliptic Variational Inequality

The elliptic variational inequality problem plays an important role in nonlinear problems. At the same time, it is one of the important ways to study many free boundary problems in mechanics, physics and engineering. With in-depth development of numerical methods and the rapid increase of the computer speed in numerical calculation, it is possible that we can solve variational inequalities by numerical methods and solve many practical problems by simulation.On the basis of the research results of the majority of academic, the paper firstly solves a class of variational inequality problem (P) using the dual method and penalty method, as well as the pros and cons of the two algorithms were compared. Secondly, the duality theory will be applied to typical examples of the Bingham fluid problem in cylindrical pipe, which takes the advantage for the numerical solution. Finally, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years,the paper conducts a study.Full-text is divided into five chapters. The first chapter mainly outlines the elliptic variational inequalities and their status and shows the source and significance of the subject. The second chapter describes some of the important basic concepts and conclusions involved by the papers, which have laid a theoretical foundation for the studied contents in the paper.The third chapter firstly conducts analysis for a class of variational inequalities problem (P) using dual methods and penalty methods, giving the formula to solve and proving convergence of the algorithms. Secondly, according to numerical examples the paper does an analysis for various parameters in methods on the impact of the numerical results and comparison of the pros and cons of the dual method and penalty method on numerical solving variational inequality problem (P) .The fourth chapter puts duality theory into the Bingham fluid problems. Firstly the variational inequality problem derived by the problem is conducted a conversion through the dual theory. Secondly, the transformation dual problem is solved directly using relaxation method. Finally, numerical examples are given, and numerical results and relative error is analyzed to verify the feasibility of the method.The fifth chapter, for the elastic-plastic torsion problem of a cylinder bar widespread concerned by scholars in recent years, the paper conducts a study. The variational inequality problem derived by the problem actually is a special case of the problem (P). The numerical solution of the variational inequality problem will be solved using this important tool--the relaxation method with a projection, and numerical examples are given, reflecting the flexibility of the method.