The Finite Element Methods And Convergence Analysis For Some Initial/Boundary Value Problems On Overlapping Non-matching Grids

In this paper, we mainly concerntrate on the finite element methods for some initial/boundary value problems of linear or nonlinear partial differential equations on overlapping nonmatching grids. The construction, convergence analysis and stability analysis for finite element method on nonmatching overlapping grids based on the partition of unity are demonstrated. These facts are shown by numerical examples, and examples of partition of unity are also given. We prove that new finite element method admits an optimal convergence rate. The nonoverlapping nonmatching case can be carried out as a consequence of the overlapping nonmatching case. Our results are valid for multiple subdomains and spatial dimensions.The main pupose of this paper is to study the overlapping nonmatching grids finite element method for linear or nonlinear elliptic boundary value problems, linear or nonlinear parabolic initial-boundary value problems, hyperbolic initial-boundary value problems, the biharmonic problem and Stokes problem. The following is an outline of the main results of the paper:In Chapter 1 and Chapter 2, the research society and advances of overlapping nonmatching grids finite element method are summarized. The theoretical framework is set up for the development of conforming finite element methods on overlapping nonmatching grids. In addition, some examples of partition of unity functions are given.In Chapter 3, the conforming finite element methods on overlapping non-matching grids for linear or nonlinear elliptic boundary value problems are discussed, respectively. Numerical examples to illustrate the result are also given.In Chapter 4, linear or a class of nonlinear parabolic initial-boundary value problems on a polygonal domain are investigated. The conforming finite element methods on overlapping nonmatching grids for the parabolic equations are discussed. The convergence analysis of the semi-discrete and the fully discrete finite element solution on overlapping nonmatching grids are studied respectively. Furthermore, some numericals example to illustrate the result are also given.In Chapter 5 and Chapter 6, the overlapping nonmatching grids finite element method for the hyperbolic equation, the biharmonic problem and Stokes problem are discussed, respectively. The stability analysis of Stokes problem is studied, and the convergence analysis of the other problems are studied.