局部间断Galerkin方法的误差估计

发布时间：2017-12-27 04:18

本文关键词：局部间断Galerkin方法的误差估计　出处：《南京大学》2016年博士论文　论文类型：学位论文

【摘要】：发展型对流扩散方程具有广泛的应用背景,相应的数值求解方法研究一直备受关注。局部间断有限元(Local discontinuous Galerkin,简称LDG)方法是目前非常流行的数值方法之一,具有良好的数值稳定性和高阶精度。在本论文中,我们将考虑典型的一维和二维对流扩散方程,建立相应LDG方法的丰满阶误差估计。主要结论包括两个内容。其一是数值流通量的具体设置更具一般性。换言之,我们将考虑广义的交替型数值流通量。其二是“双丰满”的局部误差估计。论文共分七章。第一章是对流扩散方程及其LDG方法的简要回顾,最后一章是总结和展望。余下五章是本文的主体,具体内容如下：在第二章,我们将考虑一维的线性对流扩散问题,并假设相应的真解在全局区域上是充分光滑的。基于广义交替数值流通量,我们将证明相应的LDG方法依旧具有丰满阶的整体L2模误差估计。为此,我们将采用最新发展起来的一个整体投影,称之为广义Gauss-Radau (GGR)投影,给出完整的理论证明。在这个过程中,我们完善了GGR投影的最优逼近性质对投影函数所需的光滑性要求。在第三章,我们将前面的工作推广到二维对流扩散问题的LDG方法研究。为简单起见,设有限元空间是基于矩形网格的双k次分片多项式空间。若LDG方法采用广义的交替型数值流通量,我们将理论证明其依旧具有丰满阶的整体L2模误差估计。证明的主要工具依旧是二维的GGR投影,但是维数的增加,使得我们在误差估计中,不能将单元边界误差消去,也不能利用内部单元的投影正交性。为此,我们需要建立二维GGR投影对于整体DG空间离散算子的超收敛性质。同原始的局部Gauss-Radau投影相比,该结论的证明路线具有明显的区别。从第四章开始,我们将讨论LDG方法的局部误差估计。第四章考虑具有边界层的一维奇异摄动问题。由于真解在狭窄的边界层内呈现出大梯度的急剧变化,前面的整体误差估计失去理论指导价值。为了突出LDG方法的数值求解优势,我们需要开展相应的局部分析。本文的目标是建立LDG方法的“双最优”误差估计结果。换言之,受到边界层影响的污染区域具有拟最优的宽度,并且污染区域外的L2模误差依旧是丰满的。为完成相关证明,我们需要引进一个特殊的权函数,开展相应的带权能量分析。关键技术主要有三。其一是,借用局部L2投影技术,建立相应的加权L2模稳定性；其二是,利用Dirichlet边界条件下的GGR投影技术；其三是,利用真解的正则性假设,具体设置权函数中的参数。在第五章,我们考虑一维奇异摄动问题的全离散LDG方法,其中时间采用二阶和三阶全变差不增的显式Runge-Kutta方法。分析的关键是对时间离散信息的有效控制。由于稳定性机制略有不同,基于上述两种时间离散技术的LDG全离散方法,具有明显不同的局部误差估计过程。在第六章,我们考虑二维奇异摄动问题LDG方法的“双最优”局部误差估计,其中我们采用了具有完全交替数值流通量的半离散LDG方法。此时分析的关键是建立二维GR投影的加权超收敛性质。
[Abstract]:The development type convection diffusion equation has a wide application background, and the corresponding numerical solution research has been paid much attention. Local discontinuous Galerkin (LDG) is one of the most popular numerical methods at present. It has good numerical stability and high-order accuracy. In this paper, we will consider the typical one and two dimensional convection diffusion equations and establish the full order error estimates for the corresponding LDG method. The main conclusions include two contents. One is that the specific setting of numerical circulation is more general. In other words, we will consider the generalized alternating numerical flow. The second is the local error estimation of "double plump". The paper is divided into seven chapters. The first chapter is a brief review of the convection diffusion equation and its LDG method, and the last chapter is a summary and prospect. The remaining five chapters are the main body of this paper. The details are as follows: in the second chapter, we will consider the one-dimensional linear convection diffusion problem, and assume that the corresponding true solutions are sufficiently smooth in the global region. Based on the generalized alternating numerical flow, we will prove that the corresponding LDG method still has the overall L2 modulus error estimation of the full order. To this end, we will use the latest development of a global projection, called the generalized Gauss-Radau (GGR) projection, and give a complete theoretical proof. In this process, we have perfected the requirement of the smoothness required by the optimal approximation property of the GGR projection to the projection function. In the third chapter, we generalize the previous work to the LDG method for the two-dimensional convection diffusion problem. For simplicity, the finite element space is a double K subdivision polynomial space based on a rectangular grid. If the LDG method uses a generalized alternating numerical flow, we will prove that the theory still has the overall L2 modulus error estimation of the full order. The main tool of the proof is the two-dimensional GGR projection. However, the increase of dimension makes it impossible to eliminate the unit boundary error and to use the projection orthogonality of the internal element in error estimation. For this reason, we need to establish the superconvergence property of the two-dimensional GGR projection for the discrete operator of the whole DG space. Compared with the original local Gauss-Radau projection, the proof of the conclusion is distinctly different. Starting from the fourth chapter, we will discuss the local error estimation of the LDG method. In the fourth chapter, one dimensional singular perturbation problem with boundary layer is considered. As the true solution changes sharply in the narrow boundary layer, the overall error ahead is estimated to lose the theoretical guidance value. In order to highlight the numerical solution advantages of the LDG method, we need to carry out the corresponding local analysis. The aim of this paper is to establish the "double optimal" error estimation results of the LDG method. In other words, the contaminated area affected by the boundary layer has the quasi optimal width, and the L2 mode error outside the contaminated area is still plump. In order to complete the relevant proof, we need to introduce a special weight function to carry out the corresponding weighted energy analysis. There are three key technologies. One is to use the local L2 projection technique to establish the corresponding weighted L2 module stability. The second is to use the GGR projection technology under the Dirichlet boundary condition, and the third is to set the parameters in the weight function by assuming the regularity of the real solution. In the fifth chapter, we consider the fully discrete LDG method for one dimensional singularly perturbed problem, in which the time adopts the explicit Runge-Kutta method that does not increase the two order and the three order total variation. The key of the analysis is the effective control of the time discrete information. Because the stability mechanism is slightly different, the LDG full discrete method based on the two time discrete techniques mentioned above has obviously different local error estimation. In the sixth chapter, we consider the "double optimal" local error estimate of LDG method for two-dimensional singularly perturbed problem. We use semi discrete LDG method with completely alternating numerical flux. The key of this analysis is to establish the weighted superconvergence property of the two-dimensional GR projection.
【学位授予单位】：南京大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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