The Finite Element Approximations For Elliptic Variational Problems And Their Numerical Computation

We consider the following problem: find u ∈ K such thatWith different K and F(u), the problems mentioned above correspond to linear or nonlinear equations, one-obstacled or two-obstacled problems. We will discuss both L∞-error estimates of finite element approximations for infinite dimensional problems and numerical computations for the solutions of the finite dimensional problems. The content mainly includes: 1. Studying the L∞-error estimate for the finite element solution of the two-obstacled linear elliptic problem. 2. Studying the L∞—error estimate for the finite element solution of the nonlinear elliptic problem with a source term. 3. Using the direct methods to solve the finite dimensional two-obstacled problem with an M-matrix. 4. Applying Schwarz methods to solve the nonlinear complementarity problem with an M-function.In chapter 1, we will expound the application backgrounds of the considered problems and of the considered methods.In chapter 2, we study L∞-error estimate for the finite element solution of the two-obstacled linear elliptic problem and obtain h2|logh| error bound. The result has been gotten by others, but the technique we used is different. The main thought of our technique is to construct two one-obstacled problems that have the same solution to the source problem and to apply the error bounds of the one-obstacled problems to get our result.In chapter 3, we study L∞-error estimate for the finite element solution of the nonlinear elliptic problem with a source term. The most general L∞ optimal convergence result was achieved by the method of weighted norms of Nitsche. For the linear elliptic problem, the L∞-error bound of its finite element solution can reach h2|logh|. In this chapter, we will introduce weighted norms into the finite element space and establish the relation between the weighted norms and the L∞-norm. With the error bounds of the elliptic problems and the dual space theory, we obtain h2|logh| error bound for the source problem. The difference between our result and those already existed is that we use the lumped mass elements to discrete the nonlinear term of the source problem.In chapter 4, we will construct some direct methods to solve the finite dimen-...