非标准混合元方法分析及数值模拟
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摘要
混合有限元方法在微分方程数值解法中扮演着重要的角色.本文主要围绕分裂正定混合有限元方法和H1-Galerkin混合有限元方法这两个方面开展研究工作.
羊丹平于2001年针对多孔介质中可压缩驱动问题的非线性抛物型压力方程提出新的分裂正定混合有限元方法.该方法具有如下优点：形成混合有限元系统的系数矩阵是对称正定的；流函数方程不依赖于压力方程,进而易于求得流函数的近似解.在这里,我们利用分裂正定混合元方法研究了一些发展方程,取得的主要结果如下：
.研究一类二阶伪双曲方程的分裂正定混合有限元方法,依照不同的物理量,提出两种分裂混合格式.在所提出的程序中,辅助变量σ=a(x)▽u或σ=α(χ)(▽ut+▽u)的逼近解能够不依赖于未知纯量函数u的逼近解而独立求解,不需要求解方程组的耦合系统.证明了半离散混合有限元解的存在唯一性,并得到了空间半离散和全离散格式的误差估计.
·通过引入两个变换q=ut和σ=a(x)▽u+b(χ)▽ut,并求解关于▽u的常微分方程,进而提出了粘弹性波动方程的一个新的分裂正定混合有限元方法.与传统的混合有限元相比有如下优势：所提出的格式能够将变量σ独立于u和q而求解；含有σ的方程的系数矩阵是对称正定的,易于程序实现.证明了半离散和全离散格式误差估计,并证明了半离散混合元解的存在唯一性.最后,数值结果表明所提出的格式是可行的.我们不难发现该格式具有普遍性,可以用来求解如Sobolev方程和伪双曲方程等重要的发展方程.
1998年Pani针对抛物型偏微分方程提出了H1-Galerkin混合有限元方法,该方法较传统的混合有限元方法有如下优势：避免了LBB相容性条件的限制；混合有限元空间Vh和Wh的选取比较自由,空间中的多项式次数可以不同；再者,对流量的L2模估计可以得到较好的阶数.本文中,我们将应用H1-Galerkin混合元方法求解一些重要的发展方程,同时基于H1-Galerkin混合元方法提出一些新的数值格式,获得以下一些结果：
.利用H1-Galerkin混合有限元方法研究三类非线性发展方程(RLW-Burgers方程,Burgers-Huxley方程,SRLW方程).针对RLW-Burgers方程给出了完善的半离散和全离散格式的误差分析,证明了混合有限元解的存在唯一性.最后,些数值结果表明H1-Galerkin混合有限元方法对于这三类非线性发展方程都是行之有效的.
引入时空辅助变量q=▽ut,提出半线性强阻尼波动方程的新型H1-Galerkin混合有限元数值格式.该方法能够将原方程在时空两个方向同时降阶,得到一阶积分-微分混合系统.证明了一维情况下半离散格式和全离散格式的最优收敛阶误差估计,并将该数值格式推广应用到了多维情形.
.到目前为止,H1-Galerkin混合有限元方法研究的问题仅局限于二阶发展方程问题.然而对于高阶发展方程,特别是重要的四阶发展方程问题的研究却没有出现.本文首次提出四阶发展方程的H1-Galerkin混合有限元方法,为了给出理论分析的需要,我们考虑四阶抛物型发展方程.通过引进三个适当的中间辅助变量,形成四个一阶方程组成的方程组系统,提出四阶抛物型方程的H1-Galerkin混合有限元方法.得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计,并采用迭代方法证明了全离散格式的稳定性.最后,通过数值例子验证了提出算法的可行性.在一维情况下我们能够同时得到未知纯量函数、一阶导数、二阶导数和三阶导数的最优逼近解,这一点是以往混合元方法所不能得到的,并且该算法可以应用于四阶双曲方程等高阶发展方程问题.
.引入两个新的辅助变量,提出伪双曲方程的基于H1-Galerkin混合有限元方法的新的数值格式.所提出的格式能够形成三个微分子格式,不需要求解方程组的耦合系统.给出了一维情况下半离散和Crank-Nicolson-Galerkin全离散格式最优收敛阶误差估计.并且该格式对LBB相容性条件不作要求.最后,通过数值算例验证了所提出算法的有效性.
.利用扩展混合有限元方法和H1-Galerkin混合有限元方法相结合的技巧,研究一维RLW-Burgers方程的H1-Galerkin扩展混合有限元方法.该方法同时保持了扩展混合元方法和H1-Galerkin混合元方法的优点.证明了半离散混合有限元解的存在唯一性和格式的稳定性,并得到了未知纯量函数、梯度和流量的半离散和全离散格式最优收敛阶误差估计,最后,一些数值结果表明了算法的可行性.
本文结构安排如下：第一章简述了混合有限元方法发展状况和本文的主要结果；第二章我们研究了伪双曲方程的两类分裂正定混合元方法；第三章提出粘弹性波动方程的新型分裂正定混合有限元方法；第四章给出三类非线性发展方程(RLW-Burgers方程,Burgers-Huxley方程,SRLW方程)的一些H1-Galerkin混合元程序的数值结果；第五章通过引入一个新的辅助变量,研究四阶半线性强阻尼波方程的新型H1-Galerkin混合有限元方法；第六章我们首次提出四阶发展方程的H1-Galerkin混合有限元方法,并给出了一些数值结果.第七章提出伪双曲方程的新的分裂式H1-Galerkin混合有限元方法；最后,第八章我们利用H1-Galerkin扩展混合元方法研究了RLW-Burgers方程问题.
Mixed finite element method plays an important role in the numerical methods for differential equations. In this thesis, we mainly study the splitting positive mixed finite element method and H1-Galerkin mixed finite element method.
Danping Yang (in 2001) proposed a new mixed finite element method called the splitting positive definite mixed finite element procedure to treat the pressure equation of parabolic type in a nonlinear parabolic system describing a model for compressible flow displacement in a porous medium. The proposed procedure has the following advantages: the coefficient matrix of the mixed element system is symmetric positive definite; the flux equation is separated from the pressure equation so that we can obtain an approximate solution of the flux function easily. Here, we study the splitting definite positive mixed element method for some evolution equations, and get the following results:
●Splitting positive definite mixed finite element methods are discussed for a class of second-order pseudo-hyperbolic equations, depending on the different physical quantities of interest, two kinds of splitting mixed schemes are proposed. The approximate solution ofσ= a(x)Vυorσ= a(x)(▽υt+▽υ), which does not depend on the approximate solution ofυ, can be solved, and the proposed procedures do not need to solve a coupled system of equations. The existence and uniqueness of mixed element solutions for semidiscrete schemes are proved, and error estimates are derived for both semidiscrete and fully discrete schemes.
●Introducing two transformations q=υt andσ= a(x)▽υ+b(x)▽υt, and solving ordinary differential equation for▽υ, a new splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. Compared to standard mixed methods, the proposed method has several attractive features:σ, which dose not depend onυand q, can be solved; the coefficient matrix of the equation forσis symmetric positive definite, and the procedure is implemented easily. Error estimates are derived for both semidiscrete and fully discrete schemes, and the existence and uniqueness of mixed element solutions for semidiscrete schemes are proved. Finally, some numerical results are provided to illustrate the effectiveness of our method. It is easy to see that the proposed scheme can be applied to many important evolution equations such as Sobolev equation and pseudo-hyperbolic equation and so on.
In 1998, Pani proposed an H1-Galerkin mixed finite element method for parabolic partial differential equation, the proposed one has the following advantages:they are not subject to the LBB consistency condition; the mixed finite element spaces Vh and Wh may be of differing polynomial degrees; moreover, a better order of convergence for the flux in L2-norm is obtained. In this thesis, we apply the H1-Galerkin mixed finite element method to solve some important evolution equations, and propose some new numerical schemes based on H1-Galerkin mixed finite element method, and obtain the following results:
●The H1-Galerkin mixed finite element method is studied for three kinds of nonlinear equations (RLW-Burgers equation; Burgers-Huxley equation; SRLW-equation). Error estimates are derived for both semidiscrete and fully discrete schemes for RLW-Burgers equation, and the existence and uniqueness of mixed finite element solutions are proved. Finally, some numerical results are given to illustrate the effectiveness of H1-Galerkin mixed method for three kinds of nonlinear equations.
●By introducing a space-time auxiliary variables q= Vut, a new H1-Galerkin mixed finite element method is constructed for the semilinear strongly damped wave equation. We can get the lower equation for space-time direction and the integro-differential mixed systems. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variable are discussed.
●So far, the H1-Galerkin mixed finite element method was applied to many second-order evolution equations. However, the H1-Galerkin mixed method for the higher-order evolution equations, especially, for fourth-order evolution equations has not been studied in the literature. In this thesis, we first proposed the H1-Galerkin mixed method for fourth-order evolution equation. For the need of the analysis of theories, we consider the fourth-order parabolic evolution equation. By introducing three auxiliary variables, the first-order system of four equations is formulated, and the H1-Galerkin mixed finite element method for fourth-order parabolic equation is proposed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates are derived for semidiscrete scheme for several space di-mensions, and the stability for fully discrete scheme is proved by the iteration method. Finally, some numerical results are provided to illustrate the effectiveness of our method. Optimal approximate solutions for the scalar unknown, first derivative, second derivative and third derivative are obtained, which can't be derived by the other mixed methods, and the method can be applied to higher-order evolution equations such as the fourth-order hyperbolic partial differential equations.
●By introducing two new auxiliary variables, a new numerical scheme based on the H1-Galerkin mixed finite element method for a class of second-order pseudo-hyperbolic equations is constructed. The proposed procedure can be split into three independent differential sub-schemes and does not need to solve a coupled system of equations. Opti-mal error estimates are derived for both semidiscrete and Crank-Nicolson-Galerkin fully discrete schemes for problems in one space dimension. And the proposed method does not require the LBB consistency condition. Finally, some numerical results are provided to illustrate the effectiveness of our method.
●An H1-Galerkin expanded mixed finite element method which combines expanded mixed method and H1-Galerkin mixed method is studied for 1-D regularized long wave-Burgers (RLW-Burgers) equation. The formulation not only keeps the advantages of expanded mixed formulation but also keeps the advantages of H1-Galerkin mixed formu-lation. The existence, uniqueness and stability of semidiscrete mixed element solutions are proved, and the optimal error estimates of the scalar unknown, its gradient and its flux for semidiscrete and fully discrete schemes are derived. Finally, some numerical results are given to illustrate the effectiveness of the proposed method.
The layout of this thesis is as follows:In chapter I, we introduce the development for mixed finite element methods and the main results of this thesis; Chapter II study two kinds of splitting positive definite mixed element methods for the pseudo-hyperbolic type equation; We propose a new splitting positive definite mixed element method for second-order viscoelasticity wave equation in chapter III; Chapter IV give some numerical results of H1-Galerkin mixed element procedure for three classes of nonlinear evolution equations (RLW-Burgers equation, Burgers-Huxley equation, SRLW equation); By introducing a new auxiliary variable, we present a new H1-Galerkin mixed element method for fourth-order strongly damped wave equation in chapter V; In chapter VI, we first propose the H1-Galerkin mixed element method for the fourth-order evolution equation, and present some numerical results; Chapter VII propose a new splitting H1-Galerkin mixed element method for the pseudo-hyperbolic equation; Finally, we study the H1-Galerkin expanded mixed finite element method for RLW-Burgers equation in chapter VIII.
羊丹平于2001年针对多孔介质中可压缩驱动问题的非线性抛物型压力方程提出新的分裂正定混合有限元方法.该方法具有如下优点：形成混合有限元系统的系数矩阵是对称正定的；流函数方程不依赖于压力方程,进而易于求得流函数的近似解.在这里,我们利用分裂正定混合元方法研究了一些发展方程,取得的主要结果如下：
.研究一类二阶伪双曲方程的分裂正定混合有限元方法,依照不同的物理量,提出两种分裂混合格式.在所提出的程序中,辅助变量σ=a(x)▽u或σ=α(χ)(▽ut+▽u)的逼近解能够不依赖于未知纯量函数u的逼近解而独立求解,不需要求解方程组的耦合系统.证明了半离散混合有限元解的存在唯一性,并得到了空间半离散和全离散格式的误差估计.
·通过引入两个变换q=ut和σ=a(x)▽u+b(χ)▽ut,并求解关于▽u的常微分方程,进而提出了粘弹性波动方程的一个新的分裂正定混合有限元方法.与传统的混合有限元相比有如下优势：所提出的格式能够将变量σ独立于u和q而求解；含有σ的方程的系数矩阵是对称正定的,易于程序实现.证明了半离散和全离散格式误差估计,并证明了半离散混合元解的存在唯一性.最后,数值结果表明所提出的格式是可行的.我们不难发现该格式具有普遍性,可以用来求解如Sobolev方程和伪双曲方程等重要的发展方程.
1998年Pani针对抛物型偏微分方程提出了H1-Galerkin混合有限元方法,该方法较传统的混合有限元方法有如下优势：避免了LBB相容性条件的限制；混合有限元空间Vh和Wh的选取比较自由,空间中的多项式次数可以不同；再者,对流量的L2模估计可以得到较好的阶数.本文中,我们将应用H1-Galerkin混合元方法求解一些重要的发展方程,同时基于H1-Galerkin混合元方法提出一些新的数值格式,获得以下一些结果：
.利用H1-Galerkin混合有限元方法研究三类非线性发展方程(RLW-Burgers方程,Burgers-Huxley方程,SRLW方程).针对RLW-Burgers方程给出了完善的半离散和全离散格式的误差分析,证明了混合有限元解的存在唯一性.最后,些数值结果表明H1-Galerkin混合有限元方法对于这三类非线性发展方程都是行之有效的.
引入时空辅助变量q=▽ut,提出半线性强阻尼波动方程的新型H1-Galerkin混合有限元数值格式.该方法能够将原方程在时空两个方向同时降阶,得到一阶积分-微分混合系统.证明了一维情况下半离散格式和全离散格式的最优收敛阶误差估计,并将该数值格式推广应用到了多维情形.
.到目前为止,H1-Galerkin混合有限元方法研究的问题仅局限于二阶发展方程问题.然而对于高阶发展方程,特别是重要的四阶发展方程问题的研究却没有出现.本文首次提出四阶发展方程的H1-Galerkin混合有限元方法,为了给出理论分析的需要,我们考虑四阶抛物型发展方程.通过引进三个适当的中间辅助变量,形成四个一阶方程组成的方程组系统,提出四阶抛物型方程的H1-Galerkin混合有限元方法.得到了一维情形下的半离散和全离散格式的最优收敛阶误差估计和多维情形的半离散格式误差估计,并采用迭代方法证明了全离散格式的稳定性.最后,通过数值例子验证了提出算法的可行性.在一维情况下我们能够同时得到未知纯量函数、一阶导数、二阶导数和三阶导数的最优逼近解,这一点是以往混合元方法所不能得到的,并且该算法可以应用于四阶双曲方程等高阶发展方程问题.
.引入两个新的辅助变量,提出伪双曲方程的基于H1-Galerkin混合有限元方法的新的数值格式.所提出的格式能够形成三个微分子格式,不需要求解方程组的耦合系统.给出了一维情况下半离散和Crank-Nicolson-Galerkin全离散格式最优收敛阶误差估计.并且该格式对LBB相容性条件不作要求.最后,通过数值算例验证了所提出算法的有效性.
.利用扩展混合有限元方法和H1-Galerkin混合有限元方法相结合的技巧,研究一维RLW-Burgers方程的H1-Galerkin扩展混合有限元方法.该方法同时保持了扩展混合元方法和H1-Galerkin混合元方法的优点.证明了半离散混合有限元解的存在唯一性和格式的稳定性,并得到了未知纯量函数、梯度和流量的半离散和全离散格式最优收敛阶误差估计,最后,一些数值结果表明了算法的可行性.
本文结构安排如下：第一章简述了混合有限元方法发展状况和本文的主要结果；第二章我们研究了伪双曲方程的两类分裂正定混合元方法；第三章提出粘弹性波动方程的新型分裂正定混合有限元方法；第四章给出三类非线性发展方程(RLW-Burgers方程,Burgers-Huxley方程,SRLW方程)的一些H1-Galerkin混合元程序的数值结果；第五章通过引入一个新的辅助变量,研究四阶半线性强阻尼波方程的新型H1-Galerkin混合有限元方法；第六章我们首次提出四阶发展方程的H1-Galerkin混合有限元方法,并给出了一些数值结果.第七章提出伪双曲方程的新的分裂式H1-Galerkin混合有限元方法；最后,第八章我们利用H1-Galerkin扩展混合元方法研究了RLW-Burgers方程问题.
Mixed finite element method plays an important role in the numerical methods for differential equations. In this thesis, we mainly study the splitting positive mixed finite element method and H1-Galerkin mixed finite element method.
Danping Yang (in 2001) proposed a new mixed finite element method called the splitting positive definite mixed finite element procedure to treat the pressure equation of parabolic type in a nonlinear parabolic system describing a model for compressible flow displacement in a porous medium. The proposed procedure has the following advantages: the coefficient matrix of the mixed element system is symmetric positive definite; the flux equation is separated from the pressure equation so that we can obtain an approximate solution of the flux function easily. Here, we study the splitting definite positive mixed element method for some evolution equations, and get the following results:
●Splitting positive definite mixed finite element methods are discussed for a class of second-order pseudo-hyperbolic equations, depending on the different physical quantities of interest, two kinds of splitting mixed schemes are proposed. The approximate solution ofσ= a(x)Vυorσ= a(x)(▽υt+▽υ), which does not depend on the approximate solution ofυ, can be solved, and the proposed procedures do not need to solve a coupled system of equations. The existence and uniqueness of mixed element solutions for semidiscrete schemes are proved, and error estimates are derived for both semidiscrete and fully discrete schemes.
●Introducing two transformations q=υt andσ= a(x)▽υ+b(x)▽υt, and solving ordinary differential equation for▽υ, a new splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. Compared to standard mixed methods, the proposed method has several attractive features:σ, which dose not depend onυand q, can be solved; the coefficient matrix of the equation forσis symmetric positive definite, and the procedure is implemented easily. Error estimates are derived for both semidiscrete and fully discrete schemes, and the existence and uniqueness of mixed element solutions for semidiscrete schemes are proved. Finally, some numerical results are provided to illustrate the effectiveness of our method. It is easy to see that the proposed scheme can be applied to many important evolution equations such as Sobolev equation and pseudo-hyperbolic equation and so on.
In 1998, Pani proposed an H1-Galerkin mixed finite element method for parabolic partial differential equation, the proposed one has the following advantages:they are not subject to the LBB consistency condition; the mixed finite element spaces Vh and Wh may be of differing polynomial degrees; moreover, a better order of convergence for the flux in L2-norm is obtained. In this thesis, we apply the H1-Galerkin mixed finite element method to solve some important evolution equations, and propose some new numerical schemes based on H1-Galerkin mixed finite element method, and obtain the following results:
●The H1-Galerkin mixed finite element method is studied for three kinds of nonlinear equations (RLW-Burgers equation; Burgers-Huxley equation; SRLW-equation). Error estimates are derived for both semidiscrete and fully discrete schemes for RLW-Burgers equation, and the existence and uniqueness of mixed finite element solutions are proved. Finally, some numerical results are given to illustrate the effectiveness of H1-Galerkin mixed method for three kinds of nonlinear equations.
●By introducing a space-time auxiliary variables q= Vut, a new H1-Galerkin mixed finite element method is constructed for the semilinear strongly damped wave equation. We can get the lower equation for space-time direction and the integro-differential mixed systems. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variable are discussed.
●So far, the H1-Galerkin mixed finite element method was applied to many second-order evolution equations. However, the H1-Galerkin mixed method for the higher-order evolution equations, especially, for fourth-order evolution equations has not been studied in the literature. In this thesis, we first proposed the H1-Galerkin mixed method for fourth-order evolution equation. For the need of the analysis of theories, we consider the fourth-order parabolic evolution equation. By introducing three auxiliary variables, the first-order system of four equations is formulated, and the H1-Galerkin mixed finite element method for fourth-order parabolic equation is proposed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension, and error estimates are derived for semidiscrete scheme for several space di-mensions, and the stability for fully discrete scheme is proved by the iteration method. Finally, some numerical results are provided to illustrate the effectiveness of our method. Optimal approximate solutions for the scalar unknown, first derivative, second derivative and third derivative are obtained, which can't be derived by the other mixed methods, and the method can be applied to higher-order evolution equations such as the fourth-order hyperbolic partial differential equations.
●By introducing two new auxiliary variables, a new numerical scheme based on the H1-Galerkin mixed finite element method for a class of second-order pseudo-hyperbolic equations is constructed. The proposed procedure can be split into three independent differential sub-schemes and does not need to solve a coupled system of equations. Opti-mal error estimates are derived for both semidiscrete and Crank-Nicolson-Galerkin fully discrete schemes for problems in one space dimension. And the proposed method does not require the LBB consistency condition. Finally, some numerical results are provided to illustrate the effectiveness of our method.
●An H1-Galerkin expanded mixed finite element method which combines expanded mixed method and H1-Galerkin mixed method is studied for 1-D regularized long wave-Burgers (RLW-Burgers) equation. The formulation not only keeps the advantages of expanded mixed formulation but also keeps the advantages of H1-Galerkin mixed formu-lation. The existence, uniqueness and stability of semidiscrete mixed element solutions are proved, and the optimal error estimates of the scalar unknown, its gradient and its flux for semidiscrete and fully discrete schemes are derived. Finally, some numerical results are given to illustrate the effectiveness of the proposed method.
The layout of this thesis is as follows:In chapter I, we introduce the development for mixed finite element methods and the main results of this thesis; Chapter II study two kinds of splitting positive definite mixed element methods for the pseudo-hyperbolic type equation; We propose a new splitting positive definite mixed element method for second-order viscoelasticity wave equation in chapter III; Chapter IV give some numerical results of H1-Galerkin mixed element procedure for three classes of nonlinear evolution equations (RLW-Burgers equation, Burgers-Huxley equation, SRLW equation); By introducing a new auxiliary variable, we present a new H1-Galerkin mixed element method for fourth-order strongly damped wave equation in chapter V; In chapter VI, we first propose the H1-Galerkin mixed element method for the fourth-order evolution equation, and present some numerical results; Chapter VII propose a new splitting H1-Galerkin mixed element method for the pseudo-hyperbolic equation; Finally, we study the H1-Galerkin expanded mixed finite element method for RLW-Burgers equation in chapter VIII.
引文
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