几类奇异摄动方程的有限元分析
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摘要
本文主要研究了几类稳态奇异摄动方程的有限元数值逼近。针对不同的奇异摄动方程，采用多种单元使用不同方法进行误差分析和数值验证，已经得到一些不依赖摄动参数的一致收敛性结果。
奇异摄动问题是一类广泛存在于物理工程，流体力学，化学力学等研究领域的重要问题。其特点就是由于受到奇异摄动参数的影响，真解往往会在求解区域的某个子区域内变化剧烈，通常称之为具有层效应。由于此类问题的重要性和其真解的不确定性，采用数值方法求解引起了越来越多学者的关注。
奇异摄动问题的数值逼近，如果采用通常的有限元方法，在真解剧烈变化的区域会出现数值振荡，得不到满意的收敛效果。同时由于正则性条件的限制，必须要求剖分的足够加密，必然极大地增加计算工作量。为了克服此困难，采用各向异性网格是一种很好的选择。目前常用的一类方法就是改进传统的网格剖分，使用特殊构造的层自适应网格，这里面包括著名的Bakhvalov型和Shishkin型网格，以及新近出现的分层网格。这些网格上的有限元逼近都是需要满足各向异性性质的有限单元。另外常用的方法之一是采用稳定化方法，人工添加黏性项，对检验函数做某种修正来改善使用标准Galerkin方法出现的数值振荡；它属于Petrov-Galerkin方法的范畴，其中最著名的就是流线扩散有限元方法(简记SDFEM)和余项自由泡方法(简记RFB)等等。理论分析的目标就是得到能量模意义下不依赖于摄动参数的一致收敛性分析结果。
本文的研究对象是稳态奇异摄动方程，包括反应占优的反应扩散型奇异摄动方程和半奇异摄动方程，对流占优的对流扩散型奇异摄动方程，四阶椭圆奇异摄动方程等等。作者分别从先验网格，各向异性单元(包括协调元和非协调元)，稳定化方法，高次单元等多个方面进行研究。首先作者给出本文主题的一个一般介绍和一些基础知识。第二章在一类特定的分层网格上，考虑双线性单元同时逼近反应占优的反应扩散型奇异摄动方程和半奇异摄动方程。利用分层网格的性质，通过逐个单元分析，得到了相应加权模意义下的全局一致收敛性逼近结果。然后构造一个适当的插值后处理算子进一步得到超收敛的结果，大量的数值实验也验证了理论结果。第三章考虑一个五节点低阶非协调单元，同时逼近反应占优的反应扩散型奇异摄动方程和半奇异摄动方程，利用分层网格的性质，通过逐个单元分析，得到了上述类似的全部结果。第四章考虑了双二次单元对对流扩散问题的逼近，克服了已有的困难得到了高阶的一致收敛性结果。第五章利用流线扩散有限元方法，将一个新近出现的双参数单元应用于对流扩散问题，得到了理想的收敛性结果。第六章考虑数值求解带有简支边界条件的四阶椭圆奇异摄动问题，由双参数方法构造了一个新的非协调单元，在各向异性网格下证明了关于摄动参数的一致收敛性，并且得到了最优的收敛阶，给出的数值试验也验证了理论分析的结果。最后一章作者给出本文工作的一个总结和未来工作的展望。
The finite element approximations of some steady-state singular perturbation equations are investigated in this paper. For different singular perturbed equations, applying different methods with distinct elements for the sake of error estimates and numerical test, some uniform convergence results are obtained which are independent of the singular perturbation parameter.
Singular perturbation problems (SPP) arise in ninny application areas, such as in physics engineering, fluid dynamics and chemical kinetics, etc. Its special character is that the solution of such problems undergo rapid changes within very thin subdomains, which is called layer, owing to the singular perturbation parameter. For the importance and indeterminacy of the true solutions, more and more scholars are interested in their numerical solutions.
The numerical approximations for solving singular perturbation problems, applying the classical finite element methods, may arise severe oscillation across the layer, so one can not get the satisfying convergence. And due to the regular condition, the finite element meshes need to be refined enough, which will enlarge the computation works enormously. For conquering this, applying anisotropic mesh may be a finer choice. Now, one common kind of methods is improving the classical meshes, using the appropriately layer-adapted meshes, including the well known Bakhvalov-type and Shishkin-type meshes, and the graded meshes which is appeared recently. The finite element approximations on these meshes need anisotropic elements. Another kind of methods is the stabilized methods, adding the artificial viscosity terms and modifying the test functions, in order to improving the numerical oscillation arose by standard Galerkin methods, and these methods belong to the so called Petrov-Galerkin methods, including the well known streamline diffusion finite element methods (SDFEM) and residual free bubbles (RFB), etc. The main aim of theoretical analysis is to get the uniform convergence error estimates independent of the singular perturbation parameter in energy norm.
The topic of this thesis is some steady-state singular perturbation equations, including reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations, convection dominated convection-diffusion type singular per- turbed equations, and fourth order elliptic singular perturbation equations, etc. The author endeavored to improve the convergence results in many aspects, such as refining a priori meshes, using anisotropic element (conforming and nonconforming element), applying stabilized methods or higher order elements, and so on. Firstly, we give a general introduction on singular perturbation problems and some basic informations. In Chapter 2, on the appropriately graded meshes, the bilinear element approximations of the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations are considered at the same time, by using the special character of graded meshes and analyzing element by element, the global uniformly convergence results in the corresponding weighted norm are obtained. And by constructing a appropriately interpolation postprocessing operator, the superconvergence results are derived, then some numerical experiments are also given to confirm the theoretical analysis. In Chapter 3, a lower order five node nonconforming element is used to approximate the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations simultaneously, and by using the special character of graded meshes and analyzing element by element, all the above similar results are achieved. In Chapter, biquadratic element approximation of convection-diffusion-reaction equation is considered under graded meshes, overcoming some difficulties, and some higher order uniformly convergence results are obtained. By using the streamline diffusion finite element methods with a double set parameter nonconforming element constructed recently, are used for the convection dominated convection-diffusion-reaction equation in the following chapter, and convergence results are derived. In Chapter 6, a new nonconforming element constructed by double set parameter method, is applied to the fourth order elliptic singular perturbation problem with simple support boundary condition. The convergence uniformly in the perturbation parameter, is proved under the anisotropic meshes, and optimal convergence rate is obtained. Numerical results support the theoretical analysis. In the last chapter, the author give some conclusions about this paper and suggestions on future works.
奇异摄动问题是一类广泛存在于物理工程，流体力学，化学力学等研究领域的重要问题。其特点就是由于受到奇异摄动参数的影响，真解往往会在求解区域的某个子区域内变化剧烈，通常称之为具有层效应。由于此类问题的重要性和其真解的不确定性，采用数值方法求解引起了越来越多学者的关注。
奇异摄动问题的数值逼近，如果采用通常的有限元方法，在真解剧烈变化的区域会出现数值振荡，得不到满意的收敛效果。同时由于正则性条件的限制，必须要求剖分的足够加密，必然极大地增加计算工作量。为了克服此困难，采用各向异性网格是一种很好的选择。目前常用的一类方法就是改进传统的网格剖分，使用特殊构造的层自适应网格，这里面包括著名的Bakhvalov型和Shishkin型网格，以及新近出现的分层网格。这些网格上的有限元逼近都是需要满足各向异性性质的有限单元。另外常用的方法之一是采用稳定化方法，人工添加黏性项，对检验函数做某种修正来改善使用标准Galerkin方法出现的数值振荡；它属于Petrov-Galerkin方法的范畴，其中最著名的就是流线扩散有限元方法(简记SDFEM)和余项自由泡方法(简记RFB)等等。理论分析的目标就是得到能量模意义下不依赖于摄动参数的一致收敛性分析结果。
本文的研究对象是稳态奇异摄动方程，包括反应占优的反应扩散型奇异摄动方程和半奇异摄动方程，对流占优的对流扩散型奇异摄动方程，四阶椭圆奇异摄动方程等等。作者分别从先验网格，各向异性单元(包括协调元和非协调元)，稳定化方法，高次单元等多个方面进行研究。首先作者给出本文主题的一个一般介绍和一些基础知识。第二章在一类特定的分层网格上，考虑双线性单元同时逼近反应占优的反应扩散型奇异摄动方程和半奇异摄动方程。利用分层网格的性质，通过逐个单元分析，得到了相应加权模意义下的全局一致收敛性逼近结果。然后构造一个适当的插值后处理算子进一步得到超收敛的结果，大量的数值实验也验证了理论结果。第三章考虑一个五节点低阶非协调单元，同时逼近反应占优的反应扩散型奇异摄动方程和半奇异摄动方程，利用分层网格的性质，通过逐个单元分析，得到了上述类似的全部结果。第四章考虑了双二次单元对对流扩散问题的逼近，克服了已有的困难得到了高阶的一致收敛性结果。第五章利用流线扩散有限元方法，将一个新近出现的双参数单元应用于对流扩散问题，得到了理想的收敛性结果。第六章考虑数值求解带有简支边界条件的四阶椭圆奇异摄动问题，由双参数方法构造了一个新的非协调单元，在各向异性网格下证明了关于摄动参数的一致收敛性，并且得到了最优的收敛阶，给出的数值试验也验证了理论分析的结果。最后一章作者给出本文工作的一个总结和未来工作的展望。
The finite element approximations of some steady-state singular perturbation equations are investigated in this paper. For different singular perturbed equations, applying different methods with distinct elements for the sake of error estimates and numerical test, some uniform convergence results are obtained which are independent of the singular perturbation parameter.
Singular perturbation problems (SPP) arise in ninny application areas, such as in physics engineering, fluid dynamics and chemical kinetics, etc. Its special character is that the solution of such problems undergo rapid changes within very thin subdomains, which is called layer, owing to the singular perturbation parameter. For the importance and indeterminacy of the true solutions, more and more scholars are interested in their numerical solutions.
The numerical approximations for solving singular perturbation problems, applying the classical finite element methods, may arise severe oscillation across the layer, so one can not get the satisfying convergence. And due to the regular condition, the finite element meshes need to be refined enough, which will enlarge the computation works enormously. For conquering this, applying anisotropic mesh may be a finer choice. Now, one common kind of methods is improving the classical meshes, using the appropriately layer-adapted meshes, including the well known Bakhvalov-type and Shishkin-type meshes, and the graded meshes which is appeared recently. The finite element approximations on these meshes need anisotropic elements. Another kind of methods is the stabilized methods, adding the artificial viscosity terms and modifying the test functions, in order to improving the numerical oscillation arose by standard Galerkin methods, and these methods belong to the so called Petrov-Galerkin methods, including the well known streamline diffusion finite element methods (SDFEM) and residual free bubbles (RFB), etc. The main aim of theoretical analysis is to get the uniform convergence error estimates independent of the singular perturbation parameter in energy norm.
The topic of this thesis is some steady-state singular perturbation equations, including reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations, convection dominated convection-diffusion type singular per- turbed equations, and fourth order elliptic singular perturbation equations, etc. The author endeavored to improve the convergence results in many aspects, such as refining a priori meshes, using anisotropic element (conforming and nonconforming element), applying stabilized methods or higher order elements, and so on. Firstly, we give a general introduction on singular perturbation problems and some basic informations. In Chapter 2, on the appropriately graded meshes, the bilinear element approximations of the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations are considered at the same time, by using the special character of graded meshes and analyzing element by element, the global uniformly convergence results in the corresponding weighted norm are obtained. And by constructing a appropriately interpolation postprocessing operator, the superconvergence results are derived, then some numerical experiments are also given to confirm the theoretical analysis. In Chapter 3, a lower order five node nonconforming element is used to approximate the reaction dominated reaction-diffusion type singular perturbed equations and semisingular perturbed equations simultaneously, and by using the special character of graded meshes and analyzing element by element, all the above similar results are achieved. In Chapter, biquadratic element approximation of convection-diffusion-reaction equation is considered under graded meshes, overcoming some difficulties, and some higher order uniformly convergence results are obtained. By using the streamline diffusion finite element methods with a double set parameter nonconforming element constructed recently, are used for the convection dominated convection-diffusion-reaction equation in the following chapter, and convergence results are derived. In Chapter 6, a new nonconforming element constructed by double set parameter method, is applied to the fourth order elliptic singular perturbation problem with simple support boundary condition. The convergence uniformly in the perturbation parameter, is proved under the anisotropic meshes, and optimal convergence rate is obtained. Numerical results support the theoretical analysis. In the last chapter, the author give some conclusions about this paper and suggestions on future works.
引文
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