【作者】 李蔚；

【导师】 黄云清；

【作者基本信息】 湘潭大学 ， 计算数学， 2007， 博士

【摘要】 本文的工作分两部分。第一部分主要研究了一类基于局部多项式逼近空间的单位分解法的最优误差估计。近年来,无网格方法被大量应用到科学与工程计算中。与经典有限元方法相比较,这类方法的共同特征是不再需要网格结构,它们在处理具有复杂域的问题或区域在求解过程中变化的问题时非常有效。单位分解法是非常重要的无网格方法之一,其两大主要特点是:一方面,它允许使用支集不依赖于网格或依赖于不与问题域一致的简单网格的单位分解函数(例如Shepard函数[4]),在此意义下,PUM是一种无网格方法,这个特征免去了网格生成;另一方面,局部逼近空间可以包含非多项式函数,从而很好地局部逼近未知解。这两大特征使单位分解法得到了迅速发展。虽然有关单位分解法的文献较多,但大部分都侧重于工程应用,只有极少数的数学理论分析,主要以I.Babu(?)ka和他的合作者为代表做了很多奠基性工作。但I.Babu(?)ka等现有的关于单位分解法的插值误差估计还没有获得最优阶,本文第一部分的主要工作就是:通过构造一种特殊的局部多项式近似空间,以获得最优阶插值误差估计。为此,作者从有限元方法的误差收敛阶入手,针对一类特殊的单位分解方法(取通常的有限元基函数作单位分解)进行分析,构造了一个特殊的局部多项式逼近空间,给出了一维下高次单位分解插值格式和二维下低次单位分解插值格式,推导了相应的最优阶插值误差,并研究了一维下Galerkin解的误差估计。第二部分主要研究了求解一类椭圆型变分不等式的修正自适应代数多重网格法及其并行化。根据离散的椭圆型变分不等方程所具有的线性互补性质,提出了一个基于积极集策略之上的修正代数多重网格解法,求解具有对称二阶椭圆算子的变分不等式的有限元离散问题。数值实验表明了该算法在一致网格和h-自适应网格上的计算有效性和健壮性。为了减少计算时间,本文还根据该修正算法内在的并行度,提出了一个并行计算格式,数值结果给出了该并行的加速比和效率。更多还原

【Abstract】 This paper consists of two parts.In the first part,we provide optimal error estimates of a class of partition of unity method(PUM) with local polynomial approximation spaces.Recently,meshfree methods have attracted much interest in the scientific computation.In contrast to classical FEM,this new family of numerical methods shares a common feature that no mesh is needed.Those methods are designed to handle more effectively problems in complex domains,or in domains evolving with the problem solution.The partition of unity method is one of important meshfree methods.One of the important aspects of PUM is that it permits the use of partition of unity functions,whose supports may not depend on any mesh(e.g.Shepard functions,see[4]),or may depend on a simple mesh that does not conform to the geometry of the domain.In this sense,the PUM is also a meshless method and this feature allows us to avoid the use of a sophisticated mesh generator.Another important aspect of PUM is that local approximation spaces can have functions other than polynomials,which locally approximate the unknown solution well.Hence PUM made a great progress.There are many recent papers on PUM,but most of them are of engineering character,without any mathematical analysis.I.Babu(?)ka and his co-workers did much fundation work.But until now I.Babu(?)ka can not get optimal order error estimates for PUM interpolants in his papers.The goal of this part is to get optimal order error estimates for PUM interpolants by chosen of a kind of special polynomial local approximation space.For this purpose,we construct a special polynomial local approximation space according to the consistence and local approximation properties of PUM at first.Then the PUM interpolation scheme of higher degree in 1D and the PUM interpolation scheme of lower degree in 2D are given,and optimal error estimates for PUM interpolants are derived.The interpolation error estimates are used to obtain optimal order error estimates for Galerkin solution in 1D.In the second part,a modified adaptive algebraic multigrid algorithm for a class of elliptic variational inequalities and parallization are investigated.According to the linear complementarity of discrete elliptic variational inequalities,a modified algebraic multigrid(AMG) algorithm based on an active-set strategy is presented to solve the discrete problems of variational inequalities with symmetric second-order elliptic operators.The numerical experiments show the efficiency and robustness of the proposed algorithm both on the uniform mesh and on h-adaptive mesh.To shorten computation time,this part presents a parallel scheme for the modified adaptive AMG.Numerical experiments illustrate the speedup and efficiency of the parallel scheme.更多还原