H（div）-椭圆问题的若干数值方法研究
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摘要
在本文中,我们研究H(div)-椭圆问题的若干数值方法。据我们所知,H(div)-椭圆问题在固体和流体力学中是一个很普遍的问题,并且在实际问题中有很重要的应用。对二阶椭圆问题做一阶最小二乘时,在实现对于非静态不可压Navier-Stokes方程组序列正则化的过程中,在增广Lagrangian混合有限元方法中,在Stokes方程组的有限元稳定化过程中,都有可能遇到H(div)-椭圆问题的求解。因此,设计数值算法离散和求解该问题是有必要的,也是有意义的。
第一章,我们给出文中其他章节会采用的一些预备知识。
第二章,我们设计了内罚型间断Galerkin方法离散H(div)、椭圆问题。首先给出了在能量范数意义下的最优先验误差估计。并且,我们给出了一类残量型后验误差估计子,证明了该误差估计子的可靠性和有效性。最后,数值实验验证了我们的理论结果。
第三章,我们设计了一类并行的Robin-Robin区域分解算法求解H(div)-椭圆问题。对偏微分方程和有限元逼近问题,都给出了收敛性分析。最后,数值实验说明了该算法的有效性。
第四章,我们设计了最优型区域分解算法求解H(div)-椭圆问题。通过适当参数的选取,论证了算法的收敛性。数值实验验证了算法的有效性。
第五章,我们提供了关于非标准有限元离散H(div)-椭圆问题的后验误差框架。我们将该框架应用于第二章的内罚型间断Galerkin方法。并且说明该框架也可以应用于Mortar有限元方法。
第六章,我们设计了非拟合罚有限元方法离散H(div)-椭圆界面问题。给出了能量范数意义下的最优先验误差估计。
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.
第一章,我们给出文中其他章节会采用的一些预备知识。
第二章,我们设计了内罚型间断Galerkin方法离散H(div)、椭圆问题。首先给出了在能量范数意义下的最优先验误差估计。并且,我们给出了一类残量型后验误差估计子,证明了该误差估计子的可靠性和有效性。最后,数值实验验证了我们的理论结果。
第三章,我们设计了一类并行的Robin-Robin区域分解算法求解H(div)-椭圆问题。对偏微分方程和有限元逼近问题,都给出了收敛性分析。最后,数值实验说明了该算法的有效性。
第四章,我们设计了最优型区域分解算法求解H(div)-椭圆问题。通过适当参数的选取,论证了算法的收敛性。数值实验验证了算法的有效性。
第五章,我们提供了关于非标准有限元离散H(div)-椭圆问题的后验误差框架。我们将该框架应用于第二章的内罚型间断Galerkin方法。并且说明该框架也可以应用于Mortar有限元方法。
第六章,我们设计了非拟合罚有限元方法离散H(div)-椭圆界面问题。给出了能量范数意义下的最优先验误差估计。
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.
引文
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