Research On H-, P-, Hp-Adaptive Element Methods

In this paper, we study two important types of adaptive boundary element method, including h- and p- adaptive boundary element method and we do the detailed research about them. After that, we get their functions stratified respectively, then the adaptive h- and p- hierarchical refinement methods are given. We also do the reaearch about their corresponding error analysis.At the same time we give the example to compare the adaptive h- and p- hierarchical boundary element method on their superiority, combine with the advantages of two algorithms for a new combination method, called adaptive hp- hierarchical boundary element method. Then, we applicate it to the classic elasticity problem. Due to its high efficiency, it develops adaptive boundary element technique more further.The paper includes five chapters. The chapter 1 is a introduction, which summarizes its development of the boundary element method, and adaptive boundary element method, then, presented the source, significance of the task and the major work of the paper.In chapter 2, we set up the frame of Sobelev space,in which we study the theory of posteriori error estimates through which the adaptive method is realized. At the same time, we present the expression for error estimates of ABEM and its especial adaptive process. Then, through including the process of establishing the boundary integral equation, it proved the theory basis of the last chapters.In chapter 3, we introduce the adaptive h- hierarchical boundary element method, including iterative process and its corresponding error analysis.In chapter 4, we give detailed research on the adaptive p- hierarchical boundary element method, according to its adaptive feature gives two simple calculation error analysis. In chapter 5, we applicate the adaptive h-and p- hierarchical boundary element method on the comparison superiority,through comparing analysis and the calculation on the structure in the similar points, developed a new self-adaptive hp- hierarchical iterative scheme of the boundary element method. Then given a concrete example to prove the method is effective and superiority.