对流扩散问题非协调有限元方法后验误差估计

发布时间：2018-05-14 22:38

本文选题：后验误差估计 + 健壮性　；参考：《郑州大学》2017年博士论文

【摘要】：本文主要研究对流扩散问题非协调有限元逼近的残量型后验误差估计.针对一系列的稳定化有限元方法,我们在统一的框架下推导了半健壮的和健壮的后验误差估计,并将此理论结果推广到四边形单元情形.对于半健壮的后验误差估计,我们采用通常的能量范数来度量误差.在一个抽象的理论框架下,我们得到了对流扩散问题有限元逼近误差的一般分解式.在这个误差分解式中,误差被分解为三部分:残量误差项,相容误差项以及非协调误差项.事实上,对于各种协调和非协调离散格式,这三种类型的误差项是固定的,其中只有相容误差项的估计依赖于具体的离散格式,而其它项可以用统一的方式来估计.特别地,对于协调逼近,非协调误差项自动消失.我们在通常的能量范数意义下证明了残量型估计子的可靠性和有效性,但在误差下界中出现的常数因子与扩散系数和单元尺寸相关.只有当单元尺寸与扩散系数相比足够小时,该因子是有界的,因此所推导的误差估计子在通常的能量范数意义下是半健壮的.对于健壮的后验误差估计,我们需要引入一个合适的范数来度量误差.为此,我们在原始能量范数的基础上引入了对流项对应的离散对偶半范数以及非协调有限元解在网格单元边(或面)上的加权跳跃,其中权重与单元尺寸相关.与半健壮后验估计类似,我们在改进的能量范数意义下给出了误差的一般分解式,从而将误差分解为残量误差项,相容误差项以及非协调误差项三部分.我们在改进的能量范数意义下证明了残量型估计子的可靠性和有效性,并且误差上下界中出现的常数因子与扩散系数和单元尺寸都无关,因此所推导的误差估计子在改进的能量范数意义下是健壮的.以上所有的工作首先是针对单纯形网格展开的.所得到的后验误差估计理论既适用于多种协调的稳定化方法,包括流线-扩散方法,连续内部惩罚方法,子网格粘度方法等,也适用于多种非协调的稳定化方法,包括非协调流线-扩散方法,非协调面惩罚和内部惩罚方法,非协调子网格粘度方法等.然后我们将上述结果推广到四边形单元情形,并建立了统一的理论框架.在特定的条件下,这一理论框架可以得到残量型误差估计子在通常能量范数意义下的半健壮性,以及改进能量范数意义下的健壮性,能够同时适用于非协调三角形单元和四边形单元,例如Crouzeix-Raviart元,非协调旋转Q1元以及带约束的旋转Q1元等.基于不同范数意义下的误差分解,后验误差估计的关键是存在一个具有一些基本性质的有界线性算子以及在不同离散格式下相容误差项的估计.最后,数值实验表明了残量型误差估计子的可靠性,有效性以及健壮性.
[Abstract]:In this paper, a remanent posteriori error estimation of nonconforming finite element approximation for convection-diffusion problems is studied. For a series of stable finite element methods, we derive semi-robust and robust posteriori error estimates under the unified framework, and extend the theoretical results to quadrilateral element cases. For semi-robust posteriori error estimation, we use the usual energy norm to measure the error. In an abstract theoretical framework, we obtain a general decomposition formula of the finite element approximation error for convection-diffusion problems. In the error decomposition formula, the error is decomposed into three parts: residual error term, consistent error term and non-conforming error term. In fact, for all kinds of concordant and non-conforming discrete schemes, these three types of error terms are fixed, in which only the estimation of compatible error terms depends on the specific discrete scheme, while the other terms can be estimated in a uniform manner. In particular, for concordant approximation, the non-conforming error term automatically disappears. We prove the reliability and validity of the residual estimator in the sense of ordinary energy norm, but the constant factor in the error lower bound is related to the diffusion coefficient and the size of the element. This factor is bounded only if the element size is small enough compared with the diffusion coefficient, so the error estimator derived is semi-robust in the sense of the usual energy norm. For robust posteriori error estimation, we need to introduce an appropriate norm to measure the error. Based on the original energy norm, we introduce the discrete dual semi-norm corresponding to the convection term and the weighted jump of the non-conforming finite element solution on the edge (or surface) of the grid element, in which the weight is dependent on the size of the element. Similar to the semi-robust posteriori estimation, we give a general decomposition of the error in the sense of the improved energy norm, and then decompose the error into three parts: the residual error, the compatible error and the non-conforming error. In the sense of the improved energy norm, we prove the reliability and validity of the residual estimator, and the constant factor in the upper and lower bounds of the error is independent of both the diffusion coefficient and the unit size. Therefore, the error estimator derived is robust in the sense of improved energy norm. All the above work is based on simplex mesh. The theory of posteriori error estimation is applicable to many kinds of coordinated stabilization methods, including streamline diffusion method, continuous internal penalty method, subgrid viscosity method, etc. It includes non-conforming streamline diffusion method, non-conforming surface penalty method and internal penalty method, non-conforming sub-grid viscosity method and so on. Then we extend the above results to the quadrilateral element case and establish a unified theoretical framework. Under certain conditions, we can obtain the semi-robustness of the residual error estimator in the sense of ordinary energy norm, and improve the robustness in the sense of energy norm. It can be applied to both non-conforming triangular element and quadrilateral element, such as Crouzeix-Raviart element, non-conforming rotation Q1 element and constrained rotated Q1 element, etc. Based on the error decomposition in the sense of different norms, the key of posteriori error estimation is the existence of a bounded linear operator with some basic properties and the estimation of compatible error terms in different discrete schemes. Finally, numerical experiments show the reliability, validity and robustness of the residual error estimator.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82

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