Some Researches On Numerical Methods For H(div)-Elliptic Problem

In this dissertation, we study some numerical methods for H(div)-elliptic prob-lem. As we know, H(div)-elliptic problem is ubiquitous in solid and fluid mechanics, and have many important real applications. It may arises from, the first-order system least-squares formulation of H1-elliptic problem, the implementation of the sequential regularization method for the nonstationary incompressible Navier-Stokes equations, the mixed methods with augmented lagrangians, or the stabilized formulations of the Stokes equations. Therefore, it is so important and necessary to design numerical methods to solve this equation.In Chapter1, we provide some preliminaries which may be applied to the re-maining chapters throughout the dissertation.In Chapter2, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. The numerical results presented validate our theoretical analysis.In Chapter3, we present a parallel Robin-Robin domain decomposition method (DDM) for H(div)-elliptic problem. The convergence of this method is proved for the continuous problem and the finite element discrete approximation problem. Some numerical testes are presented to demonstrate the effectiveness of the method.In Chapter4, we propose some optimization-based domain decomposition meth-ods for H(div)-elliptic problem. Convergent properties are examined by choosing proper parameters. The effectiveness of the method is validated by some numerical tests.In Chapter5, we provide an unified framework for an a posteriori error analysis of non-standard finite element approximations of H(div)-elliptic problem. We apply it to the interior penalty discontinuous Galerkin method in Chapter2. Furthermore, we can apply it to the mortar finite element method.In Chapter6, we propose an unfitted penalty finite element method for H(div)-elliptic interface problem. An optimal a priori error estimate in energy norm is ob- tained.