Research On Variational Inequality For A Boundary Value Problem About Elliptic Differential Equation With Variable Coefficients

Many problems in the engineering practice may be transformed as boundary value problems about elliptic differential equations. Variational inequality method, developed in the resent years, provides an united frame and powerful facility for the problem. It induces all the boundary conditions and the contact conditions to a variational inequality, which makes the theoretical analysis simple. The main numerical methods of variational inequality problem contain finite difference method, finite element method and boundary element method. Many researches about numerical methods solving the boundary value problems about elliptic differential equations are made in home and abroad, but most of the researches are aimed at differential equations with constant coefficients, and the discussions on problems with variable coefficients are few.The boundary value problem of the two-dimensional inhomogeneous Helmholtz equation is solved by BEM, using the fundamental solution of the Laplace equation first in the paper. Then a boundary value problem about elliptic differential equation with variable coefficients is discussed in the paper, by transforming it as a variational inequality problem, and then the unique existence of the solution are proved. The variational inequality is dispersed by the finite element method and the theoretical analyses about the stability and the error estimates of the solution are taken.This paper includes five chapters. In the first chapter, the development of variational inequality, finite element method and solving elliptic variational inequality by the finite element method are mainly introduced; and research dynamic of scholar of home and abroad is emphasized.In the second chapter, a complete set of theories are given in Sobolev space, such as variation principle of the boundary problem, generalized derivative, Lax-Milgram theory, embedding theorem and trace theorem, equivalent module theory, finite element method discrete and so on.In the third chapter, the Helmholtz equation that is applied widely is taken as the study object. The numerical solution is presented using boundary element method. The course of the solution shows the commonly process solving problems using boundary element method. Finally the numerical example shows feasibility of the method.In the fourth chapter, it is presented the variational inequality problem that is equal to the boundary value problem about an elliptic differential equation with variable coefficients, and the proof of the unique existence of the solution of the variation inequality. Finally, regularization for the indifferentiable item of the variational inequality is taken and the variational inequality can be formulated as a differentiable variational equation.In the last chapter, numerical approximation about the variational equation is taken by the finite element method. The stability and the error estimates of the solution are presented.