Homeomorphisms of triangulations with applications to computing fixed points

In the past decade various complementary pivoting algorithms have been developed to search for fixed points of certain functions and point to set maps. All these methods generate a sequence of simplexes which are shrinking to a point. This paper proposes a new method for shrinking the simplexes. It is shown that under certain conditions, the function whose fixed point is sought may be used to control this shrinking process. A computational method for implementing these ideas is also suggested and several examples are solved using this approach.An abstract appears in the November, 1978 issue of Notices of the American Mathematical Society.     

Geometric structures associated to triangulations as fixed point sets of involutions

We establish that hyperbolic structures and spherical CR structures on a three-dimensional manifold are contained in fixed point sets of a larger class of structures associated to a triangulation of the manifold. We generalize the 5 term relation to this setting.     

On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian

Let λ be a positive number, and let be a fixed Riesz-basis sequence, namely, (x_j) is strictly increasing, and the set of functions is a Riesz basis (i.e., unconditional basis) for L₂[−π,π]. Given a function whose Fourier transform is zero almost everywhere outside the interval [−π,π], there is a unique sequence in , depending on λ and f, such that the function

is continuous and square integrable on (−∞,∞), and satisfies the interpolatory conditions I_λ(f)(x_j)=f(x_j), . It is shown that I_λ(f)converges to f in , and also uniformly on , as λ→0⁺. In addition, the fundamental functions for the univariate interpolation process are defined, and some of their basic properties, including their exponential decay for large argument, are established. It is further shown that the associated interpolation operators are bounded on for every p[1,∞].     

A new multivariate spline based on mixed partial derivatives and its finite element approximation

We present a new multivariate spline using mixed partial derivatives. We show the existence and uniqueness of the proposed multivariate spline problem, and propose a simple finite element approximation.     

Sharp estimates for finite element approximations to elliptic problems with Neumann boundary data of low regularity

Consider a second order homogeneous elliptic problem with smooth coefficients, , on a smooth domain, , but with Neumann boundary data of low regularity. Interior maximum norm error estimates are given for finite element approximations to this problem. When the Neumann data is not in , these local estimates are not of optimal order but are nevertheless shown to be sharp. A method for ameliorating this sub-optimality by preliminary smoothing of the boundary data is given. Numerical examples illustrate the findings.

     

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.     

On the design of an upwind scheme for compressible flow on general triangulations

We discuss the design features and mathematical background of an explicit upwind finite-volume method to simulate non-stationary flow of a compressible, inviscid fluid. One of the design goals was the rigorous mathematical justification of each ingredient of the method. The method itself contains elements from finite-difference methods as well as finite-element methods and is formulated in a finite volume framework. The use of well-known algorithmic ingredients in a new framework results in a robust time-accurate scheme. To be able to easily handle complex geometries as well as adaption algorithms a tringale-based formulation was chosen. Numerical tests for two-dimensional flow are presented.     

SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT SUPERCONVERGENCE ANALYSIS FOR CUBIC TRIANGULAR ELEMENT OF THE FINITE ELEMENT

1. IntroductionAlthough we bed proved the superconvergence of quadratic triangular elem6ntsbefore 1985, the superconvergence reseaxch of k(k 2 3)-degree triangulax elemeds Onlyhas a few advances, e.g.,.Lin, Yan and Zhou (see [15]) Prove that the three degreeHerlnite elements possess superconvergence and Walilbin (see [5-7]) obtains a roughresult by using a fine interinr estimation, that is, the placement fUnCtion or its gradestmay have weak superconvergence in the lOcal syUUnetric points,e…     

有限元多尺度小波

利用有限元插值和多尺度分析理论构造出了有限元多尺度小波.这些小波函数集许多优良性质于一身,如固定的短支集、高阶的消失矩、半正交性及正则性等.     

TIME-DISCRETIZATION PROCEDURE FOR FINITE ELEMENT APPROXIMATION OF COMPRESSIBLE MISCIBLE DISPLACEMENT IN POROUS MEDIA

We'll consider the model of two-phase compressible miscible displacement in porous media which includes molecular diffusion and dispersion in one dimensional space. Time-discretization procedure is established and analysed. The optimal error estimate in L2 norm is proved by introducing a new interpolation operator R.     

Application of spectral filtering to discontinuous Galerkin methods on triangulations

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a discontinuous Galerkin (DG) method solving hyperbolic conservation laws on triangular grids. The connection between SV and spectral filtering, which is undertaken for the first time in the context of DG methods on unstructured grids, allows to specify conditions on the filter strength regarding time step choice and mesh refinement. A crucial advantage of this novel damping strategy is its low computational cost. We furthermore obtain new error bounds for filtered Dubiner expansions of smooth functions. While high order accuracy with respect to the polynomial degree N is proven for the filtering procedure in this case, an adaptive application is proposed to retain the high spatial approximation order. Although spectral filtering stabilizes the scheme, it leaves weaker oscillations. Therefore, as a postprocessing step, we apply the image processing technique of digital total variation (DTV) filtering in the new context of DG solutions and prove conservativity in the limit for this filtering procedure. Numerical experiments for scalar conservation laws confirm the designed order of accuracy of the DG scheme with adaptive modal filtering for polynomial degrees up to 8 and the viability of spectral and DTV filtering in case of shocks. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Error estimates for scattered data interpolation on spheres

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the -sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

     

On the cibstryctuib of discretizations of elliptic partial differential equations

Algorithmic aspects of a class of finite element collocation methods for the approximate numerical solution of elliptic partial differential equations are described Locall for each finite element the approximate solution is a polynomial. polynomials corresponding toadjacent finite elements need not match continuously but their values and noumal derivatives match at a discrete set of points on the common boundary.High order accuracy can be attained by increasing the number of mathching points and the number of colloction points for each finite element.Forlinear equations the collocation methods can be equivalently definde as generlized finite difference methods. The linear (or linearzed )equations that arise from the discretization lend themselves well to solution by the methods of the methods nested dissection.An implememtation is described and some numerical results are givevn.     

Enhancing credit default swap valuation with meshfree methods

In this paper, we apply the meshfree radial basis function (RBF) interpolation to numerically approximate zero-coupon bond prices and survival probabilities in order to price credit default swap (CDS) contracts. We assume that the interest rate follows a Cox-Ingersoll-Ross process while the default intensity is described by the Exponential-Vasicek model. Several numerical experiments are conducted to evaluate the approximations by the RBF interpolation for one- and two-factor models. The results are compared with those estimated by the finite difference method (FDM). We find that the RBF interpolation achieves more accurate and computationally efficient results than the FDM. Our results also suggest that the correlation between factors does not have a significant impact on CDS spreads.     

FINITE ELEMENT METHODS FOR THE NAVIER-STOKES EQUATIONS BY H（div） ELEMENTS

We derived and analyzed a new numerical scheme for the Navier-Stokes equations by using H（div） conforming finite elements. A great deal of effort was given to an establishment of some Sobolev-type inequalities for piecewise smooth functions. In particular, the newly derived Sobolev inequalities were employed to provide a mathematical theory for the H（div） finite element scheme. For example, it was proved that the new finite element scheme has solutions which admit a certain boundedness in terms of the input data. A solution uniqueness was also possible when the input data satisfies a certain smallness condition. Optimal-order error estimates for the corresponding finite element solutions were established in various Sobolev norms. The finite element solutions from the new scheme feature a full satisfaction of the continuity equation which is highly demanded in scientific computing.     

A locally conservative LDG method for the incompressible Navier-Stokes equations

In this paper a new local discontinuous Galerkin method for the incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in is obtained by simple, element-by-element post-processing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

     

GLOBAL SUPERCONVERGENCE ESTIMATES OF FINITE ELEMENT METHOD FOR SCHRODINGER EQUATION

1.IntroductionConsiderthefollowinginitialboundaryvalueproblemofSchr6dingerequationwheren~[0,1]',at~On/Ot,T>0isaconstant.Theequivalentvariationalformof(1.1)is:foralltE[0,T],findu(t)6Hi(n)satisfiesthefollowingvariationalequation:where(w,v)~IwvdxdenotestheinnerproductofL'(fl)anda(u,v)~(Vu,Vv),ibetheimaginaryunit.Weassumethatthefunctionsarecomplex--valuedandHibertspacesarecomplexspaces.LetThbeaquasiuniformrectangulationoffiwithmeshsizeh>0andS'(O)CHi(fi)bethecorrespondingpiecewisebilinearpol…     

散乱数据曲面拟合的局部加权最小二乘插值方法及权函数的选择讨论

本文首先用局部加权最小二乘法将三维空间内任意散乱数据点集均匀,再估计出立方体网格点上的偏导数值及混合偏导数值,最后仅用网格点数据进行快速光滑插值加密计算,从而可得到任意点处的函数值。通过对已知函数的随机数据点集进行计算,取得了令人满意的效果。同时,在最小二乘逼近过程中,本文提供了一种权函数,并与其它二种权函数进行分析比较,给出了各种情况下的误差。