若干非线性微分方程的数值方法及超奇异积分计算
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摘要
非线性微分方程作为微分方程的一个重要分支,在众多领域都有广泛的应用,如流体力学、气体动力学、材料力学、电磁场等.伴随着计算机运行能力的快速发展,数值分析和模拟日益成为工程问题中必不可少的工具之-相继产生了一系列的数值方法,如：有限差分法(FDM)[53]、有限元法(FEM)[12,22,33]、有限体积法(FVM)[52]、混合有限元法(MFEM)[9,13].谱方法[76]、配置法等,其中有限差分法具有较高的精度,有限元方法具有较强的灵活性,已经成为求解实际问题的强有力工具,然而对非线性微分方程的数值分析和模拟仍然是一项极具挑战性的工作.
有限差分方法,简称差分法,是数值解微分方程的一种重要方法[53].它的基本思想是：把连续的定解区域用由有限个离散点构成的网格来代替,这些离散点称作网格的节点,把在连续定解区域上定义的连续变量函数用在网格上定义的离散函数来近似,用差商来近似原方程和定解条件中的微商,积分用离散积分和来近似,于是原方程和定解条件就近似地代之以代数方程组,解此代数方程组就得到原问题的近似解.有限差分方法简洁、实用、易于在计算机上实现,在工程计算中得到了广泛应用.
单元中心差分方法(CCFDM)是一种精度相对较高的差分方法,若剖分网格为矩形(或长方体),也被称为块中心差分方法,此方法可视为最低次RT混合元在特定数值积分下产生的格式Weiser等[89]研究了线性椭圆方程的块中心差分方法(BCFDM),Rui等[82]研究了Darcy-Forchheimer模型的块中心差分方法,他们都得到了二阶精度的误差估计Arbogast等[5,6]研究了具有张量系数的椭圆问题在四边形网格下的单元中心差分方法Shen[85]研究了具有间断系数的线性椭圆方程的块中心差分方法.
有限元方法,是在古典Ritz-Galerkin变分方法的基础上,以分片多项式为工具的一种求解微分方程及实际工程问题的数值方法.冯康[33]于20世纪60年代初独立于西方创立了有限元方法.从此,有限元方法被广泛的应用于船舶、机械、建筑、水利等的设计,后来又被广泛应用于流体力学、电磁场等问题的分析.20世纪70年代Babuska[9]和Brezzi[13]创立了混合有限元法的一般理论.混合有限元方法是一种基于限制或者约束条件的变分形式的有限元方法,其主要结果就是所谓的B-B条件.20世纪80年代初Falk和Osborn[31]又提出了一种改进的方法.混合有限元方法的主要优点是通过引入中间变量(一般它们也具有实际的物理意义),可以将高阶微分方程降阶,从而也就能够降低有限元空间的光滑性要求,例如Possion方程、Navier-Stokes方程、对流-扩散方程、Sobolev方程、Burgers、KdV、RLW、KdV-Burgers和双调和方程等问题,通过降阶能使有限元空间简化,同时可以求到一些有意义的中间变量,此方法方便且容易实现.在流体模拟等问题中,混合有限元方法由于能同时计算压力、速度(或流量)等物理量.被广泛采用.对于线性和半线性二阶椭圆方程的混合元方法的研究可见文献[36,80]Milner等[65]研究了拟线性二阶椭圆方程的混合元方法.Park等[49,66,74]研究了非线性椭圆方程混合元方法.
基于混合元方法,Chen[18,19]提出了扩展混合有限元方法(EMFEM),此方法可以同时逼近三个(或更多)物理量,他研究了线性和拟线性二阶椭圆方程的扩展混合有限元方法,Arbogast等[5,6]也提出了类似的技术.后来扩展方法还得到了进一步的延伸,相继提出了扩展混合有限体积法[81]、特征扩展混合元、多重网格扩展混合元等方法.
许多科学和工程问题,如声学、电磁散射和断裂力学等,可以归结为边界积分方程[98],而奇异积分方程又是积分方程的一个重要分支,其积分核函数往往使得通常的Riemann积分或者Lebesgue积分定义失效Linz[60]最早研究了超奇异积分的Newton-Cotes公式,其收敛阶要比Riemann积分的相应数值积分公式低,该文献在奇异点与节点不重合的条件下给出了二阶超奇异积分的梯形公式和Simpson公式及误差分析,当奇点位于某子区间中间时证明了其误差分别为O(h)和O(h2).对于奇点与节点重合的情况,Yu[98]给出了修正的Newton-Cotes公式,使得当奇异点与剖分节点重合时也可以计算.近年来,Wu等[91,92,93]研究了超奇异积分的超收敛现象,并且证明了超收敛现象出现在某个特殊函数的零点处.
全文分为五章,组织结构如下：
第一章介绍一些预备知识,首先介绍Sobolev空间及其范数,其次给出了几个常用的引理.
第二章研究散度形式下的非线性二阶椭圆方程的扩展混合元方法.利用此方法可以同时有效逼近u(压力),▽u(压力梯度)和——a(u,▽u)(流量).在传统的混合元方法中,不得不把变量Vu从a(u,Vu)中分离出来,对某些复杂的隐函数而言这往往是不可能的,而扩展混合元方法可以解决这个问题.此方法还有一些其他优势,比如,能处理三个变量的不同边界条件,同时也适用于微分方程系数很小(接近于零)的情况,并且不需要求倒数.因此,这种方法适用于扩散较小或低渗透流体问题.某些传统混合元方法只能得到拟最优的误差估计,而本章得到了最优阶L2模误差估计,同时得到了负模(H-s)和Lq模误差估计,证明了非线性离散形式解的存在唯一性,这比线性问题要复杂.为了得到误差估计,利用Taylor展开对误差方程进行了处理,最后进行数值实验.
第三章研究非线性单调椭圆方程的扩展混合元方法.利用此方法可以同时逼近u,Vu和—K(x,|▽u|)▽u.证明了连续和离散的B-B条件和离散解的存在唯一性,得到了最优阶L2模误差估计,最后针对Darcy-Forchheimer模型进行了应用和数值实验.
第四章研究p-Laplacian和p-Laplacian方程的单元中心差分方法.此类方程出现在许多物理过程的数学模型中,如：冰川学、幂律材料问题、非线性扩散对流与过滤问题和拟牛顿流问题等.关于p-Laplacian方程的有限元逼近已有诸多理论结果,近年来,W.B. Liu和J.W. Barrett等[61,62,63]在该方程的有限元误差估计方面做了大量的工作,取得了很大的进展,他们提出了一个新的误差估计方法：拟范数方法,该方法巧妙地利用了该方程特殊的非线性结构,在一定的条件下得到了最优阶误差估计Huang等[41]研究了p-Laplacian方程预条件下降算法.目前所知,关于p-Laplacian方程和p(x)-Laplacian方程差分方法的研究还相对较少,本章针对这两个方程提出了单元中心差分方法,给出了理论分析并进行数值实验,此方法简洁,但具有二阶精度的误差,丰富的数值算例显示此方法适用于较小或较大的参数p(或p(x)),最后对于奇异p-Laplacian方程给出了数值格式和算例.
第五章研究超奇异积分复合Hermite公式的超收敛现象,进行了误差分析,得到了误差展开式,当展开式中的特殊函数等于零时,会出现超收敛现象,此时误差阶与Riemann积分的误差估计相同,得到了超收敛点的局部坐标为±0.5383.相应的数值实验验证了理论分析的正确性.
As an important research topic, nonlinear differential equations are wide-ly used in many fields, such as fluid mechanics, aerodynamics, mechanics of materials, electromagnetic field, and so on. With the rapid development of computer, numerical analysis and simulation in engineering have become an indispensable tool. Some numerical methods are proposed, such as finite d-ifference method (FDM)[53], finite element method (FEM)[12,22,33], finite volume method (FVM)[52], mixed finite element method (MFEM)[9,13], spectral method[76], collocation method and so on. Among these methods, FDM and FEM are powerful and versatile due to its high accuracy and flex-ibility. However, numerical analysis and simulation of nonlinear differential equations are still challenging works.
The finite difference method, known as the difference method, is an im-portant numerical method for computation of differential equations. The basic idea is replacing the continuous region by a finite number of discrete points on the grid, the discrete points are called nodes, so the continuous solution for continuous variable region is approximated by the discrete grid nodes. Derivative and integral can be replaced by difference quotient and numerical integral, respectively. Further, the original equations and bound-ary conditions are approximately by algebraic equations. Through solving the algebraic equations, one can get the approximate solution of the origi-nal problem. FDM is simple, universal and easy to be implemented on the computer, so it is widely used in engineering computation.
Cell-centered difference method (CCFDM) is a relatively accurate dif-ference method. If the mesh is rectangle (or cuboid), it can also be called block-centered finite difference method. This method can be regarded as the lowest RT mixed finite element method in the special numerical integral rule. Weiser[89] introduced the block-centered difference method (BCFDM) for the linear elliptic problems. Rui[82] considered the block-centered finite differ-ence method for the Darcy-Forchheimer model, they all gave the second-order accuracy error estimates. Arbogast[5,6] studied the cell-centered difference method for elliptic problem with tensor coefficients under quadrilateral mesh. Shen[85] discussed the block-centered finite difference method for the linear elliptic problem with discontinuous coefficients.
Based on the classical Ritz-Galerkin variational method, FEM is an ef-fective method for solving differential equations with piecewise polynomial. In1960's, K. Feng[33] proposed the finite element method. Then, this method has been widely used in shipbuilding, machinery, architecture, water facilities and so on. Afterwards, it has been widely used in fluid dynamics, electromag-netic field problems and so on. The general theory of the mixed finite element method was introduced by Babuska[9] and Brezzi[13] in1970's. Mixed finite element method is a variational form finite element method with limit or constraint. The main result is the so-called B-B condition. In1980's, Falk and Osborn[31] proposed an improved method. By introducing the interme-diate variable (they also have physical significance), the order of differential equation is reduced, which can decrease the smooth requirements of finite element space, this is the main advantage of mixed method. Applying this method to some equations, such as Possion equation, Navier-Stokes equa-tion, convection-diffusion equation, Sobolev equation, Burgers, KdV, RLW, KdV-Burgers, biharmonic equation and so on, the finite element space can be reduced. MFEM is convenient and easy to be implemented. MFEM can approximate pressure, velocity (or flux) and other physical quantities, so it is widely used in fluid simulation problems. Research of the linear and semi-linear second-order elliptic equations can be seen in[36,80]. Mixed finite element method for quasi-linear second-order elliptic equations is researched by Milner[65]. In [49,66,74], the mixed finite element method for nonlinear elliptic equations is considered.
Based on the mixed finite element method, Chen[18,19] proposed the expanded mixed finite element method(EMFEM). This method can approach three (or more) physical quantity at the same time. He researched the linear and quasilinear second-order elliptic equations. Arbogast[5,6] proposed the similar technology. Afterwards, expanded method has been further extend-ed, some other methods were proposed, such as expanded mixed finite vol-ume method[81], expanded characteristic mixed method, multigrid expanded mixed finite element method and so on.
Many science and engineering problems, such as acoustic, electromag-netic scattering and fracture mechanics, can be attributed to the boundary integral equations[98]. Singular integral equation is an important branch of integral equations which always make the usual Riemann or Lebesgue inte-gral definition infeasible. Linz[60] first introduced the Newton-Cotes rules for hypersingular integrals, the convergence order is one order lower than the classic Riemann integrals. He gave the error estimates under the condition that singularity point do not coincident with nodes, if the singular point is located in a point of a subinterval, the error of trapezoid and Simpson rule are O(h) and O(h2), respectively. Yu[98] researched the Newton-Cotes rules under the situation that singular point coincident with nodes. In recent years, Wu[91,92,93] studied the superconvergence of the hypersingular in-tegrals, and prove the superconvergence phenomenon appears in some zeros of a special function.
The dissertation is divided into five chapters. The outline is as follows.
In chapter1, some preliminaries is introduced. The Sobolev space, cor-responding norms, and several useful lemmas are recommended.
In chapter2, we aim to present the expanded mixed element method for the nonlinear second-order elliptic equations in divergence form. Using this method, one can approximate u (pressure), Vu (gradient of pressure) and-a(u,(?)u)(flux) effectively. If standard mixed method is used, one have to separate (?)u from a(u,(?)u), which is always impossible to some im-plicit function, while the expanded method can avoid this problem. This method also has some other advantages except introducing three variables. It can treat individual boundary conditions. Also, it is suitable for differ-ential equation with small coefficient (close to zero) which does not need to be inverted. Consequently, this method works for the problems with small diffusion or low permeability terms in fluid flow. We get optimal order error estimates, while standard mixed method sometimes gives only suboptimal error estimates. The proof of existence and uniqueness for the nonlinear dis-crete problem is given, which is much more complicate than linear case. In deriving error estimates, Taylor's expansion is used to deal with the error equations.
In chapter3, we consider the expanded mixed finite element method for the nonlinear monotone elliptic equations. Using this method we can approximate u,(?)u and-K(x,|(?)u|)(?)u. The existence, uniqueness, discrete and continuous B-B conditions are proved. The optimal order error estimates are got. At last, the application to Darcy-Forchheimer model and numerical experiments are carried out.
In chapter4, cell-centered finite difference method for the p-Laplacian and p(x)-Laplacian equations is studied. These two equations appear in many mathematical models, such as power-law material, glaciology, non-linear diffusion convection and filter problems, quasi-Newton flow problems amd so on. The finite element approximation of p-Laplacian equation has been considered. In recent years, W.B. Liu and J.W. Barrett[61,62,63] did much work in the finite element approximation. They put forward a new technique for the error estimates:quasi-norm method. This method got the optimal order error estimates which used the special nonlinear structure of the equations under certain assumptions. Huang[41] researched the pre-conditioned descent algorithms for p-Laplacian. At present, for p-Laplacian and p(x)-Laplacian equations, the finite difference method is less researched. Applying the cell-centered difference method, we prove that the convergence order is second-order. The theoretical analysis and numerical experiments for the p-Laplacian and p(x)-Laplacian equations are given. A lot of numerical examples show that this method is suitable for both small or large param-eter p (or p(x)). At last, the difference scheme and numerical examples for singular p-Laplacian equation are given.
In chapter5, the superconvergence of composite Hermite rule for hyper-singular integrals on interval is investigated. Error analysis and error expan-sion are got. The local coordinate of superconvergence points is±0.5383. At last, numerical experiments are carried out to validate the theoretical anal-ysis.
有限差分方法,简称差分法,是数值解微分方程的一种重要方法[53].它的基本思想是：把连续的定解区域用由有限个离散点构成的网格来代替,这些离散点称作网格的节点,把在连续定解区域上定义的连续变量函数用在网格上定义的离散函数来近似,用差商来近似原方程和定解条件中的微商,积分用离散积分和来近似,于是原方程和定解条件就近似地代之以代数方程组,解此代数方程组就得到原问题的近似解.有限差分方法简洁、实用、易于在计算机上实现,在工程计算中得到了广泛应用.
单元中心差分方法(CCFDM)是一种精度相对较高的差分方法,若剖分网格为矩形(或长方体),也被称为块中心差分方法,此方法可视为最低次RT混合元在特定数值积分下产生的格式Weiser等[89]研究了线性椭圆方程的块中心差分方法(BCFDM),Rui等[82]研究了Darcy-Forchheimer模型的块中心差分方法,他们都得到了二阶精度的误差估计Arbogast等[5,6]研究了具有张量系数的椭圆问题在四边形网格下的单元中心差分方法Shen[85]研究了具有间断系数的线性椭圆方程的块中心差分方法.
有限元方法,是在古典Ritz-Galerkin变分方法的基础上,以分片多项式为工具的一种求解微分方程及实际工程问题的数值方法.冯康[33]于20世纪60年代初独立于西方创立了有限元方法.从此,有限元方法被广泛的应用于船舶、机械、建筑、水利等的设计,后来又被广泛应用于流体力学、电磁场等问题的分析.20世纪70年代Babuska[9]和Brezzi[13]创立了混合有限元法的一般理论.混合有限元方法是一种基于限制或者约束条件的变分形式的有限元方法,其主要结果就是所谓的B-B条件.20世纪80年代初Falk和Osborn[31]又提出了一种改进的方法.混合有限元方法的主要优点是通过引入中间变量(一般它们也具有实际的物理意义),可以将高阶微分方程降阶,从而也就能够降低有限元空间的光滑性要求,例如Possion方程、Navier-Stokes方程、对流-扩散方程、Sobolev方程、Burgers、KdV、RLW、KdV-Burgers和双调和方程等问题,通过降阶能使有限元空间简化,同时可以求到一些有意义的中间变量,此方法方便且容易实现.在流体模拟等问题中,混合有限元方法由于能同时计算压力、速度(或流量)等物理量.被广泛采用.对于线性和半线性二阶椭圆方程的混合元方法的研究可见文献[36,80]Milner等[65]研究了拟线性二阶椭圆方程的混合元方法.Park等[49,66,74]研究了非线性椭圆方程混合元方法.
基于混合元方法,Chen[18,19]提出了扩展混合有限元方法(EMFEM),此方法可以同时逼近三个(或更多)物理量,他研究了线性和拟线性二阶椭圆方程的扩展混合有限元方法,Arbogast等[5,6]也提出了类似的技术.后来扩展方法还得到了进一步的延伸,相继提出了扩展混合有限体积法[81]、特征扩展混合元、多重网格扩展混合元等方法.
许多科学和工程问题,如声学、电磁散射和断裂力学等,可以归结为边界积分方程[98],而奇异积分方程又是积分方程的一个重要分支,其积分核函数往往使得通常的Riemann积分或者Lebesgue积分定义失效Linz[60]最早研究了超奇异积分的Newton-Cotes公式,其收敛阶要比Riemann积分的相应数值积分公式低,该文献在奇异点与节点不重合的条件下给出了二阶超奇异积分的梯形公式和Simpson公式及误差分析,当奇点位于某子区间中间时证明了其误差分别为O(h)和O(h2).对于奇点与节点重合的情况,Yu[98]给出了修正的Newton-Cotes公式,使得当奇异点与剖分节点重合时也可以计算.近年来,Wu等[91,92,93]研究了超奇异积分的超收敛现象,并且证明了超收敛现象出现在某个特殊函数的零点处.
全文分为五章,组织结构如下：
第一章介绍一些预备知识,首先介绍Sobolev空间及其范数,其次给出了几个常用的引理.
第二章研究散度形式下的非线性二阶椭圆方程的扩展混合元方法.利用此方法可以同时有效逼近u(压力),▽u(压力梯度)和——a(u,▽u)(流量).在传统的混合元方法中,不得不把变量Vu从a(u,Vu)中分离出来,对某些复杂的隐函数而言这往往是不可能的,而扩展混合元方法可以解决这个问题.此方法还有一些其他优势,比如,能处理三个变量的不同边界条件,同时也适用于微分方程系数很小(接近于零)的情况,并且不需要求倒数.因此,这种方法适用于扩散较小或低渗透流体问题.某些传统混合元方法只能得到拟最优的误差估计,而本章得到了最优阶L2模误差估计,同时得到了负模(H-s)和Lq模误差估计,证明了非线性离散形式解的存在唯一性,这比线性问题要复杂.为了得到误差估计,利用Taylor展开对误差方程进行了处理,最后进行数值实验.
第三章研究非线性单调椭圆方程的扩展混合元方法.利用此方法可以同时逼近u,Vu和—K(x,|▽u|)▽u.证明了连续和离散的B-B条件和离散解的存在唯一性,得到了最优阶L2模误差估计,最后针对Darcy-Forchheimer模型进行了应用和数值实验.
第四章研究p-Laplacian和p-Laplacian方程的单元中心差分方法.此类方程出现在许多物理过程的数学模型中,如：冰川学、幂律材料问题、非线性扩散对流与过滤问题和拟牛顿流问题等.关于p-Laplacian方程的有限元逼近已有诸多理论结果,近年来,W.B. Liu和J.W. Barrett等[61,62,63]在该方程的有限元误差估计方面做了大量的工作,取得了很大的进展,他们提出了一个新的误差估计方法：拟范数方法,该方法巧妙地利用了该方程特殊的非线性结构,在一定的条件下得到了最优阶误差估计Huang等[41]研究了p-Laplacian方程预条件下降算法.目前所知,关于p-Laplacian方程和p(x)-Laplacian方程差分方法的研究还相对较少,本章针对这两个方程提出了单元中心差分方法,给出了理论分析并进行数值实验,此方法简洁,但具有二阶精度的误差,丰富的数值算例显示此方法适用于较小或较大的参数p(或p(x)),最后对于奇异p-Laplacian方程给出了数值格式和算例.
第五章研究超奇异积分复合Hermite公式的超收敛现象,进行了误差分析,得到了误差展开式,当展开式中的特殊函数等于零时,会出现超收敛现象,此时误差阶与Riemann积分的误差估计相同,得到了超收敛点的局部坐标为±0.5383.相应的数值实验验证了理论分析的正确性.
As an important research topic, nonlinear differential equations are wide-ly used in many fields, such as fluid mechanics, aerodynamics, mechanics of materials, electromagnetic field, and so on. With the rapid development of computer, numerical analysis and simulation in engineering have become an indispensable tool. Some numerical methods are proposed, such as finite d-ifference method (FDM)[53], finite element method (FEM)[12,22,33], finite volume method (FVM)[52], mixed finite element method (MFEM)[9,13], spectral method[76], collocation method and so on. Among these methods, FDM and FEM are powerful and versatile due to its high accuracy and flex-ibility. However, numerical analysis and simulation of nonlinear differential equations are still challenging works.
The finite difference method, known as the difference method, is an im-portant numerical method for computation of differential equations. The basic idea is replacing the continuous region by a finite number of discrete points on the grid, the discrete points are called nodes, so the continuous solution for continuous variable region is approximated by the discrete grid nodes. Derivative and integral can be replaced by difference quotient and numerical integral, respectively. Further, the original equations and bound-ary conditions are approximately by algebraic equations. Through solving the algebraic equations, one can get the approximate solution of the origi-nal problem. FDM is simple, universal and easy to be implemented on the computer, so it is widely used in engineering computation.
Cell-centered difference method (CCFDM) is a relatively accurate dif-ference method. If the mesh is rectangle (or cuboid), it can also be called block-centered finite difference method. This method can be regarded as the lowest RT mixed finite element method in the special numerical integral rule. Weiser[89] introduced the block-centered difference method (BCFDM) for the linear elliptic problems. Rui[82] considered the block-centered finite differ-ence method for the Darcy-Forchheimer model, they all gave the second-order accuracy error estimates. Arbogast[5,6] studied the cell-centered difference method for elliptic problem with tensor coefficients under quadrilateral mesh. Shen[85] discussed the block-centered finite difference method for the linear elliptic problem with discontinuous coefficients.
Based on the classical Ritz-Galerkin variational method, FEM is an ef-fective method for solving differential equations with piecewise polynomial. In1960's, K. Feng[33] proposed the finite element method. Then, this method has been widely used in shipbuilding, machinery, architecture, water facilities and so on. Afterwards, it has been widely used in fluid dynamics, electromag-netic field problems and so on. The general theory of the mixed finite element method was introduced by Babuska[9] and Brezzi[13] in1970's. Mixed finite element method is a variational form finite element method with limit or constraint. The main result is the so-called B-B condition. In1980's, Falk and Osborn[31] proposed an improved method. By introducing the interme-diate variable (they also have physical significance), the order of differential equation is reduced, which can decrease the smooth requirements of finite element space, this is the main advantage of mixed method. Applying this method to some equations, such as Possion equation, Navier-Stokes equa-tion, convection-diffusion equation, Sobolev equation, Burgers, KdV, RLW, KdV-Burgers, biharmonic equation and so on, the finite element space can be reduced. MFEM is convenient and easy to be implemented. MFEM can approximate pressure, velocity (or flux) and other physical quantities, so it is widely used in fluid simulation problems. Research of the linear and semi-linear second-order elliptic equations can be seen in[36,80]. Mixed finite element method for quasi-linear second-order elliptic equations is researched by Milner[65]. In [49,66,74], the mixed finite element method for nonlinear elliptic equations is considered.
Based on the mixed finite element method, Chen[18,19] proposed the expanded mixed finite element method(EMFEM). This method can approach three (or more) physical quantity at the same time. He researched the linear and quasilinear second-order elliptic equations. Arbogast[5,6] proposed the similar technology. Afterwards, expanded method has been further extend-ed, some other methods were proposed, such as expanded mixed finite vol-ume method[81], expanded characteristic mixed method, multigrid expanded mixed finite element method and so on.
Many science and engineering problems, such as acoustic, electromag-netic scattering and fracture mechanics, can be attributed to the boundary integral equations[98]. Singular integral equation is an important branch of integral equations which always make the usual Riemann or Lebesgue inte-gral definition infeasible. Linz[60] first introduced the Newton-Cotes rules for hypersingular integrals, the convergence order is one order lower than the classic Riemann integrals. He gave the error estimates under the condition that singularity point do not coincident with nodes, if the singular point is located in a point of a subinterval, the error of trapezoid and Simpson rule are O(h) and O(h2), respectively. Yu[98] researched the Newton-Cotes rules under the situation that singular point coincident with nodes. In recent years, Wu[91,92,93] studied the superconvergence of the hypersingular in-tegrals, and prove the superconvergence phenomenon appears in some zeros of a special function.
The dissertation is divided into five chapters. The outline is as follows.
In chapter1, some preliminaries is introduced. The Sobolev space, cor-responding norms, and several useful lemmas are recommended.
In chapter2, we aim to present the expanded mixed element method for the nonlinear second-order elliptic equations in divergence form. Using this method, one can approximate u (pressure), Vu (gradient of pressure) and-a(u,(?)u)(flux) effectively. If standard mixed method is used, one have to separate (?)u from a(u,(?)u), which is always impossible to some im-plicit function, while the expanded method can avoid this problem. This method also has some other advantages except introducing three variables. It can treat individual boundary conditions. Also, it is suitable for differ-ential equation with small coefficient (close to zero) which does not need to be inverted. Consequently, this method works for the problems with small diffusion or low permeability terms in fluid flow. We get optimal order error estimates, while standard mixed method sometimes gives only suboptimal error estimates. The proof of existence and uniqueness for the nonlinear dis-crete problem is given, which is much more complicate than linear case. In deriving error estimates, Taylor's expansion is used to deal with the error equations.
In chapter3, we consider the expanded mixed finite element method for the nonlinear monotone elliptic equations. Using this method we can approximate u,(?)u and-K(x,|(?)u|)(?)u. The existence, uniqueness, discrete and continuous B-B conditions are proved. The optimal order error estimates are got. At last, the application to Darcy-Forchheimer model and numerical experiments are carried out.
In chapter4, cell-centered finite difference method for the p-Laplacian and p(x)-Laplacian equations is studied. These two equations appear in many mathematical models, such as power-law material, glaciology, non-linear diffusion convection and filter problems, quasi-Newton flow problems amd so on. The finite element approximation of p-Laplacian equation has been considered. In recent years, W.B. Liu and J.W. Barrett[61,62,63] did much work in the finite element approximation. They put forward a new technique for the error estimates:quasi-norm method. This method got the optimal order error estimates which used the special nonlinear structure of the equations under certain assumptions. Huang[41] researched the pre-conditioned descent algorithms for p-Laplacian. At present, for p-Laplacian and p(x)-Laplacian equations, the finite difference method is less researched. Applying the cell-centered difference method, we prove that the convergence order is second-order. The theoretical analysis and numerical experiments for the p-Laplacian and p(x)-Laplacian equations are given. A lot of numerical examples show that this method is suitable for both small or large param-eter p (or p(x)). At last, the difference scheme and numerical examples for singular p-Laplacian equation are given.
In chapter5, the superconvergence of composite Hermite rule for hyper-singular integrals on interval is investigated. Error analysis and error expan-sion are got. The local coordinate of superconvergence points is±0.5383. At last, numerical experiments are carried out to validate the theoretical anal-ysis.
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