四阶椭圆型方程若干有限元新方法和高效求解算法
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摘要
四阶椭圆型微分方程广泛应用于固体力学、材料科学和图像处理诸领域,因此对它的数值解研究不但具有重要的理论意义也具有直接应用价值.本文的主要工作是构造了重调和方程的一种基于Poisson求解子的快速求解器;对四阶椭圆偏微分方程的MWX元方法设计了两网格局部和并行求解算法并进行了误差分析;将C0间断有限元方法应用于四阶椭圆偏微分方程并给出求解由该方法离散得到的线性代数方程组的区域分解法;也提出了自适应有限元方法的一个抽象框架.
首先,构造了重调和方程的一种基于Poisson求解子的快速求解器.通过位能极小原理,建立了重调和方程和Stokes方程之间的等价性,即重调和方程等价于一个Stokes方程和两个Possion方程.类似地,在有限元离散情形,建立了重调和方程Morley元方法和Stokes方程非协调P1-P0元方法之间的等价性,即重调和方程Morley元方法等价于一个Stokes方程非协调P1-P0元方法和两个Possion方程Morley元方法.然后基于这个等价性,结合代数多重网格法,构造了重调和方程的一种基于Poisson求解子的快速求解器.最后,用数值试验验证了这个求解器的有效性.
其次,提出了三种用于求解任意空间维数的四阶椭圆偏微分方程的两重网格局部和并行的Morley-Wang-Xu (MWX)元方法.由于MWX元空间不是嵌套的,需要引进一些网格转移算子,以从全局的粗网格有限元解和局部的细网格修正来获得改进的全局有限元解.首先通过构造基于修正Argyris元的网格转移算子,提出了求解MWX元离散的四阶问题的三种局部和并行算法.然后,证明了相应的数值解的离散能量误差的阶为O(h + H2),其中H和h分别为有限元单形剖分的粗细网格的网格大小.同时,构造了基于算术平均网格转移算子的局部和并行算法,证明了相应的数值解的离散能量误差的阶为O(h + H2(H/h)(d?1)/2),其中d为求解区域的空间维数.进一步,我们提供了数值实验来说明这些算法的计算效果.
再次,提出了求解薄板弯曲问题的一类新的C0间断有限元方法,并给出了数值解在离散能量误差和H1范数下的误差估计.首先将四阶偏微分方程写成二阶系统的形式,获得了构造求解原四阶问题的C0间断有限元方法的框架.然后,建立一个离散的稳定性等式,并由此选择可行的数值迹,得到求解薄板弯曲问题的一类稳定的C0间断有限元方法.根据二阶椭圆问题间断有限元方法误差分析中的思想和推导技巧,我们得到了薄板弯曲问题LCDG方法数值解在离散能量范数和H1范数下的最优阶误差估计以及CDG方法数值解在离散能量范数和H1范数下的误差估计.最后,用数值实验验证了理论的收敛阶.
接着,构造了薄板弯曲问题的局部C0间断有限元方法的两水平加型Schwarz预条件子并估计了条件数.对于我们提出的求解薄板弯曲问题的LCDG方法,通过定义特殊的网格转移算子,给出了该方法的区域分解算法.进一步估计了经过预处理的线性代数方程组的系数矩阵的条件数,该上界估计在δ≈H时是最优的.在小部分重叠情形,即δ<< H时,利用更多的子区域的信息,改进了条件数的上界估计.
然后,提出了基于标准循环的自适应有限元方法的抽象框架并得到了拟最优收敛性和最优复杂度.首先提出一些假设,基于这些假设得到自适应有限元方法的拟最优收敛性.在作了进一步的假设后,通过引进总误差,证明了自适应有限元方法的最优复杂度.最后通过前面提出的假设,将这个抽象框架应用于各种问题,包括一般二阶椭圆偏微分方程的高阶有限元方法, 2m阶椭圆偏微分方程的Morley型有限元方法,不定时谐Maxwell方程和H(div)方程.
最后,对于薄板弯曲问题k = 0,1时的混合有限元方法,验证了自适应有限元方法抽象框架中提出的假设.利用Helmholtz分解,对所有的k≥0,构造了薄板弯曲问题的混合有限元方法(H-H-J方法)基于残差的后验误差估计子并证明了估计子的可靠性.利用泡函数技巧分析了误差估计子的有效性.而且根据力在内边上的跳跃可以被体积部分所控制的性质,改进了误差估计子.进一步,获得了离散的Helmoltz分解和离散的inf-sup条件,利用这两个工具和后验误差分析中得到的性质,建立了拟正交性和离散可靠性.对于其它的假设,在k = 0,1时容易得到.
The fourth order elliptic equations frequently occur in the fields of solid mechanics, ma-terial science, image processing and so on. It is both theoretically and practically importantto investigate numerical methods for such equations. This thesis is intended to construct anefficient Poisson based solver for the biharmonic equation discretized by Morley’s element,to design local and parallel algorithms for fourth order problems discretized by the MWXelement method, to present general C0 discontinuous Galerkin method for plate bendingproblems, and to give a two-level additive Schwarz preconditioner for local C0 discontinu-ous Galerkin method of plate bending problems. Also, a general framework is establishedfor adaptive finite element methods, which can cover almost all existing typical AFEMs.
Firstly, an efficient Poisson based solver is presented for the biharmonic equation dis-cretized by Morley’s element. By the minimum principle of total potential energy, an equiva-lence relation is established between the biharmonic equation and the Stokes equation. Afterthat, a similar equivalence analogue is obtained between the numerical method for the bihar-monic equation discretized by Morley’s element and that of the Stokes equation discretizedby the nonconforming P1 ? P0 element. Then, by use of this equivalence together withsome algebraic multigrid methods, an efficient Poisson based solver is given for solving thebiharmonic equation discretized by Morley’s element. The efficiency of the new solver isillustrated by some numerical experiments.
Secondly, three two-grid local and parallel algorithms are designed for solving fourthorder problems discretized by the MWX element method in any dimensions. Since the MWXelement spaces are nonnested, some intergrid transfer operators have to be introduced, in or-der to get an improved global solution from the global coarse grid solution and the local finegrid corrections. Concretely speaking, three two-grid local and parallel algorithms are pro-posed for solving fourth order problems discretized by the MWX element method, throughintroducing a modified Argyris element based intergrid transfer operator. It is shown that thediscrete energy error of the numerical solution is bounded by O(h + H2). Then, based oncertain arithmetic average intergrid transfer operators, three two-grid local and parallel algo-rithms are proposed, and the discrete energy error of the numerical solution is bounded by O(h + H~2(H/h)~((d-1)/2)). Furthermore, a number of numerical results are reported to showthe computational performance of the methods just mentioned.
Thirdly, we develop the general C0 discontinuous Galerkin method for plate bendingproblems and give the error estimate for the numerical solution in broken energy norm andH1 norm. We first reformulate the original fourth-order partial differential equation as asecond-order system and obtain a framework of constructing CDG methods for solving theoriginal problem. Then, we establish a discrete stability identity, from which we derive fea-sible choices of numerical traces and get a class of stable CDG methods for plate bendingproblems. Following some ideas on error analysis of DG methods for second order ellipticproblems and detailed technical derivation, we derive error estimates of the numerical solu-tion in certain broken energy norm and H1 norm for the LCDG method and CDG method.Some numerical results are included to confirm our theoretical convergence orders.
Moreover, a two-level additive Schwarz preconditioner is constructed for the above-mentioned local C0 discontinuous Galerkin method of plate bending problems. By con-structing a special intergrid transfer operator, a domain decomposition method is developedfor plate bending problems discretized by our LCDG method. And we estimate the conditionnumber of the coefficient matrix of the preconditioned linear system. The upper bound ofthe condition number will be optimal whenδ≈H. In the case of small overlap, i.e.δ<< H,the upper bound of the condition number can be improved in view of some more carefulmathematical reasoning.
Then, a general framework is developed for adaptive finite element methods. Undersome assumptions, we get the qausi-optimal convergence rate of adaptive finite elementmethods. With some more assumptions, by introducing the total error, the optimal com-plexity of adaptive finite element methods is achieved too. At last, the general framework isapplied to several partial differential equations, including general second order elliptic equa-tions with high order finite elements, 2m-th order elliptic problems with Morley-type finiteelements, classical time-harmonic Maxwell’s equations, and H(div) equations.
Finally, the assumptions in the general framework of adaptive finite element methodsare verified for adaptive mixed finite element method with k = 0, 1 for plate bending prob-lems. We make use of the Helmholtz decomposition to construct a residual-based a posteriorierror estimator for a mixed finite element method, namely the H-H-J method with all ordersk≥0, and prove its reliability. Then the efficiency of the error estimator is achieved bythe technique of the bubble functions. Furthermore, an improved version is provided for thesubsequent adaptive algorithm based on the property that the force jump across an interior edge is dominated by the volume part. Moreover, a discrete Helmholtz decomposition and adiscrete inf- sup condition are derived, and then by means of these two tools and the prop-erty obtained in the a posteriori analysis, we are able to establish the quasi-orthogonality andeven the discrete reliability. Then, we develop the quasi-optimal convergence theory as wellas the optimal complexity for the H-H-J method.
首先,构造了重调和方程的一种基于Poisson求解子的快速求解器.通过位能极小原理,建立了重调和方程和Stokes方程之间的等价性,即重调和方程等价于一个Stokes方程和两个Possion方程.类似地,在有限元离散情形,建立了重调和方程Morley元方法和Stokes方程非协调P1-P0元方法之间的等价性,即重调和方程Morley元方法等价于一个Stokes方程非协调P1-P0元方法和两个Possion方程Morley元方法.然后基于这个等价性,结合代数多重网格法,构造了重调和方程的一种基于Poisson求解子的快速求解器.最后,用数值试验验证了这个求解器的有效性.
其次,提出了三种用于求解任意空间维数的四阶椭圆偏微分方程的两重网格局部和并行的Morley-Wang-Xu (MWX)元方法.由于MWX元空间不是嵌套的,需要引进一些网格转移算子,以从全局的粗网格有限元解和局部的细网格修正来获得改进的全局有限元解.首先通过构造基于修正Argyris元的网格转移算子,提出了求解MWX元离散的四阶问题的三种局部和并行算法.然后,证明了相应的数值解的离散能量误差的阶为O(h + H2),其中H和h分别为有限元单形剖分的粗细网格的网格大小.同时,构造了基于算术平均网格转移算子的局部和并行算法,证明了相应的数值解的离散能量误差的阶为O(h + H2(H/h)(d?1)/2),其中d为求解区域的空间维数.进一步,我们提供了数值实验来说明这些算法的计算效果.
再次,提出了求解薄板弯曲问题的一类新的C0间断有限元方法,并给出了数值解在离散能量误差和H1范数下的误差估计.首先将四阶偏微分方程写成二阶系统的形式,获得了构造求解原四阶问题的C0间断有限元方法的框架.然后,建立一个离散的稳定性等式,并由此选择可行的数值迹,得到求解薄板弯曲问题的一类稳定的C0间断有限元方法.根据二阶椭圆问题间断有限元方法误差分析中的思想和推导技巧,我们得到了薄板弯曲问题LCDG方法数值解在离散能量范数和H1范数下的最优阶误差估计以及CDG方法数值解在离散能量范数和H1范数下的误差估计.最后,用数值实验验证了理论的收敛阶.
接着,构造了薄板弯曲问题的局部C0间断有限元方法的两水平加型Schwarz预条件子并估计了条件数.对于我们提出的求解薄板弯曲问题的LCDG方法,通过定义特殊的网格转移算子,给出了该方法的区域分解算法.进一步估计了经过预处理的线性代数方程组的系数矩阵的条件数,该上界估计在δ≈H时是最优的.在小部分重叠情形,即δ<< H时,利用更多的子区域的信息,改进了条件数的上界估计.
然后,提出了基于标准循环的自适应有限元方法的抽象框架并得到了拟最优收敛性和最优复杂度.首先提出一些假设,基于这些假设得到自适应有限元方法的拟最优收敛性.在作了进一步的假设后,通过引进总误差,证明了自适应有限元方法的最优复杂度.最后通过前面提出的假设,将这个抽象框架应用于各种问题,包括一般二阶椭圆偏微分方程的高阶有限元方法, 2m阶椭圆偏微分方程的Morley型有限元方法,不定时谐Maxwell方程和H(div)方程.
最后,对于薄板弯曲问题k = 0,1时的混合有限元方法,验证了自适应有限元方法抽象框架中提出的假设.利用Helmholtz分解,对所有的k≥0,构造了薄板弯曲问题的混合有限元方法(H-H-J方法)基于残差的后验误差估计子并证明了估计子的可靠性.利用泡函数技巧分析了误差估计子的有效性.而且根据力在内边上的跳跃可以被体积部分所控制的性质,改进了误差估计子.进一步,获得了离散的Helmoltz分解和离散的inf-sup条件,利用这两个工具和后验误差分析中得到的性质,建立了拟正交性和离散可靠性.对于其它的假设,在k = 0,1时容易得到.
The fourth order elliptic equations frequently occur in the fields of solid mechanics, ma-terial science, image processing and so on. It is both theoretically and practically importantto investigate numerical methods for such equations. This thesis is intended to construct anefficient Poisson based solver for the biharmonic equation discretized by Morley’s element,to design local and parallel algorithms for fourth order problems discretized by the MWXelement method, to present general C0 discontinuous Galerkin method for plate bendingproblems, and to give a two-level additive Schwarz preconditioner for local C0 discontinu-ous Galerkin method of plate bending problems. Also, a general framework is establishedfor adaptive finite element methods, which can cover almost all existing typical AFEMs.
Firstly, an efficient Poisson based solver is presented for the biharmonic equation dis-cretized by Morley’s element. By the minimum principle of total potential energy, an equiva-lence relation is established between the biharmonic equation and the Stokes equation. Afterthat, a similar equivalence analogue is obtained between the numerical method for the bihar-monic equation discretized by Morley’s element and that of the Stokes equation discretizedby the nonconforming P1 ? P0 element. Then, by use of this equivalence together withsome algebraic multigrid methods, an efficient Poisson based solver is given for solving thebiharmonic equation discretized by Morley’s element. The efficiency of the new solver isillustrated by some numerical experiments.
Secondly, three two-grid local and parallel algorithms are designed for solving fourthorder problems discretized by the MWX element method in any dimensions. Since the MWXelement spaces are nonnested, some intergrid transfer operators have to be introduced, in or-der to get an improved global solution from the global coarse grid solution and the local finegrid corrections. Concretely speaking, three two-grid local and parallel algorithms are pro-posed for solving fourth order problems discretized by the MWX element method, throughintroducing a modified Argyris element based intergrid transfer operator. It is shown that thediscrete energy error of the numerical solution is bounded by O(h + H2). Then, based oncertain arithmetic average intergrid transfer operators, three two-grid local and parallel algo-rithms are proposed, and the discrete energy error of the numerical solution is bounded by O(h + H~2(H/h)~((d-1)/2)). Furthermore, a number of numerical results are reported to showthe computational performance of the methods just mentioned.
Thirdly, we develop the general C0 discontinuous Galerkin method for plate bendingproblems and give the error estimate for the numerical solution in broken energy norm andH1 norm. We first reformulate the original fourth-order partial differential equation as asecond-order system and obtain a framework of constructing CDG methods for solving theoriginal problem. Then, we establish a discrete stability identity, from which we derive fea-sible choices of numerical traces and get a class of stable CDG methods for plate bendingproblems. Following some ideas on error analysis of DG methods for second order ellipticproblems and detailed technical derivation, we derive error estimates of the numerical solu-tion in certain broken energy norm and H1 norm for the LCDG method and CDG method.Some numerical results are included to confirm our theoretical convergence orders.
Moreover, a two-level additive Schwarz preconditioner is constructed for the above-mentioned local C0 discontinuous Galerkin method of plate bending problems. By con-structing a special intergrid transfer operator, a domain decomposition method is developedfor plate bending problems discretized by our LCDG method. And we estimate the conditionnumber of the coefficient matrix of the preconditioned linear system. The upper bound ofthe condition number will be optimal whenδ≈H. In the case of small overlap, i.e.δ<< H,the upper bound of the condition number can be improved in view of some more carefulmathematical reasoning.
Then, a general framework is developed for adaptive finite element methods. Undersome assumptions, we get the qausi-optimal convergence rate of adaptive finite elementmethods. With some more assumptions, by introducing the total error, the optimal com-plexity of adaptive finite element methods is achieved too. At last, the general framework isapplied to several partial differential equations, including general second order elliptic equa-tions with high order finite elements, 2m-th order elliptic problems with Morley-type finiteelements, classical time-harmonic Maxwell’s equations, and H(div) equations.
Finally, the assumptions in the general framework of adaptive finite element methodsare verified for adaptive mixed finite element method with k = 0, 1 for plate bending prob-lems. We make use of the Helmholtz decomposition to construct a residual-based a posteriorierror estimator for a mixed finite element method, namely the H-H-J method with all ordersk≥0, and prove its reliability. Then the efficiency of the error estimator is achieved bythe technique of the bubble functions. Furthermore, an improved version is provided for thesubsequent adaptive algorithm based on the property that the force jump across an interior edge is dominated by the volume part. Moreover, a discrete Helmholtz decomposition and adiscrete inf- sup condition are derived, and then by means of these two tools and the prop-erty obtained in the a posteriori analysis, we are able to establish the quasi-orthogonality andeven the discrete reliability. Then, we develop the quasi-optimal convergence theory as wellas the optimal complexity for the H-H-J method.
引文
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