量子力学中的微扰论

简要介绍了量子力学中三种定态微扰方法：一是通常的Ｒａｙｌｅｉｇｈ－Ｓｃｈｒｏｄｉｎｇｅｒ（Ｒ－Ｓ）微扰，但其缺点是高阶微扰项的表达式很繁，且无一般的通项公式，这里介介绍了一个能量和波函数的递推公式，便于计算机编程计算，二是由Ｂｒｉｌｌｏｕｉｎ－Ｗｉｇｎｅｒ（Ｂ－Ｗ）微扰论演变出发迭代微扰法，对于有限维不变子空间或在有限维空间截断近似下计算，收敛是很快的，收敛范围大于Ｒ－２法，最便于上机计算，三是近     

ERROR ESTIMATES FOR THE FULL-DISCRETE MIXED FEM FOR NONLINEAR HYPERBOLIC PROBLEMS

1 IntroductionConsider the second order nonlinear hyPErbolic equationwhere D be a bounded domain in R' with Lipschitz boundary are, and J~ (0,TJ.The following regularity assumptions will be made on the functions a,c,f,g and solutionu of (1. I ):(1) there exist constantS c.,c*,a., and a' such that for all xos and ie R,(2) The functions a ~a (x, u),c ~ c(x, u ), f~f(x, u,t), g~g (x, t) are continuously differentiable with respect to u and t. Moreover, there exists a bound K= such that, for …     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

A fourth–order orthogonal spline collocation method for two-dimensional Helmholtz problems with interfaces

Orthogonal spline collocation is implemented for the numerical solution of two-dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N² log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.     

拟线性粘弹性方程混合有限元分析

主要讨论拟线性粘弹性方程的Bernadi-Raugel混合有限元方法,给出了逼近解和精确解的收敛性分析.同时,基于积分恒等式技巧导出了超逼近性质,并进一步利用插值后处理技术得到了整体超收敛结果.     

A Modified Polynomial Preserving Recovery and Its Applications to A Posteriori Error Estimates

<正>A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.     

Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities

We consider the Poisson equation with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain with re-entrant angles. A multigrid method for the computation of singular solutions and stress intensity factors using piecewise linear functions is analyzed. When , the rate of convergence to the singular solution in the energy norm is shown to be , and the rate of convergence to the stress intensity factors is shown to be , where is the largest re-entrant angle of the domain and can be arbitrarily small. The cost of the algorithm is . When , the algorithm can be modified so that the convergence rate to the stress intensity factors is . In this case the maximum error of the multigrid solution over the vertices of the triangulation is shown to be .

     

A review of two different approaches for superconvergence analysis

In 1995, Wahbin presented a method for superconvergence analysis called Interior symmetric method, and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different.     

Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation

It is well-known that on uniform meshes the piecewise linearconforming finite element solution of the Poisson equation approximatesthe interpolant to a higher order than the solution itself.In this paper, this type of superclose property is studied forthe canonical interpolant defined by the nodal functionals ofseveral non-conforming finite elements of lowest order. By givingexplicit examples we show that some non-conforming finite elementsdo not admit the superclose property. In particular, we discusstwo non-conforming finite elements which satisfy the supercloseproperty. Moreover, applying a postprocessing technique, wecan also state a superconvergence property for the discretizationerror of the postprocessed discrete solution to the solutionitself. Finally, we show that an extrapolation technique leadsto a further improvement of the accuracy of the finite elementsolution.