Algebraic Properties of Subdivision Operators with Matrix Mask and Their Applications

Subdivision operators play an important role in wavelet analysis. This paper studies the algebraic properties of subdivision operators with matrix mask, especially their action on polynomial sequences and on some of their invariant subspaces. As an application, we characterize, under a mild condition, the approximation order provided by refinable vectors in terms of the eigenvalues and eigenvectors of polynomial sequences of the associated subdivision operator. Moreover, some necessary conditions, in terms of nondegeneracy and simplicity of eigenvalues of a matrix related to the subdivision operator for the refinable vector to be smooth are given. The main results are new even in the scalar case     

A Nyström interpolant for some weakly singular nonlinear Volterra integral equations

We consider a second kind weakly singular nonlinear Volterra–Hammerstein integral equation defined by a compact operator and derive a Nyström type interpolant of the solution based on Gauss–Radau nodes. We prove the convergence of the interpolant and derive convergence estimates. For equations with nonlinearity of algebraic kind, we improve the rate of convergence by using a smoothing transformation. Some numerical examples are given.     

A Nyström interpolant for some weakly singular linear Volterra integral equations

We consider a second kind weakly singular Volterra integral equation defined by a non-compact operator and derive a Nyström type interpolant of the solution based on Gauss-Radau nodes. Assuming the stability of the interpolant, which is confirmed by the numerical tests, we derive convergence estimates.     

Optimal convergence analysis of mixed finite element methods for fourth‐order elliptic and parabolic problems

By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A general radial quasi-interpolation operator on the sphere

Geodetic and meteorological data, collected via satellites for example, are genuinely scattered and not confined to any special set of points. In order to learn geodetic and meteorological rules, one needs to use these scattered data only to construct an approximant or interpolant. In this paper, we introduce a general distance generated from the scattered data, and, using this, construct a general radial quasi-interpolation operator on the sphere, and we study the convergence rate of this operator. We also show some potential applications of the results obtained here in satellite geodesy.     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

Bivariate interpolation based on univariate subdivision schemes

The paper presents a bivariate subdivision scheme interpolating data consisting of univariate functions along equidistant parallel lines by repeated refinements. This method can be applied to the construction of a surface passing through a given set of parametric curves. Following the methodology of polysplines and tension surfaces, we define a local interpolator of four consecutive univariate functions, from which we sample a univariate function at the mid-point. This refinement step is the basis to an extension of the 4-point subdivision scheme to our setting. The bivariate subdivision scheme can be reduced to a countable number of univariate, interpolatory, non-stationary subdivision schemes. Properties of the generated interpolant are derived, such as continuity, smoothness and approximation order.     

Supercloseness analysis and polynomial preserving Recovery for a class of weak Galerkin Methods

In this article, we analyze convergence and supercloseness properties of a class of weak Galerkin (WG) finite element methods for solving second‐order elliptic problems. It is shown that the WG solution is superclose to the Lagrange interpolant using Lobatto points. This supercloseness behavior is obtained through some newly designed stabilization terms. A postprocessing technique using polynomial preserving recovery (PPR) is introduced for the WG approximation. Superconvergence analysis is performed for the PPR recovered gradient. Numerical examples are provided to illustrate our theoretical results.     

Interpolation of polynomial operators in a Hilbert space

An interpolant with orthonormal nodes is constructed for a polynomial operator of a given degree defined on an abstract Hilbert space. An estimate for the interpolation accuracy is derived in the metric of the space of operator values, while in the case where nodes are elements of a basis, the pointwise convergence of the interpolational operator process in this metric is proved as the number of nodes increases. Bibliography: 7 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 3–9     

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high‐order quasi‐compact schemes for space fractional diffusion equations. Because of the quasi‐compactness of the derived schemes, no points beyond the domain are used for all the high‐order schemes including second‐order, third‐order, fourth‐order, and even higher‐order schemes; moreover, the algebraic equations for all the high‐order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the fractional derivatives with nonhomogeneous boundaries are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1345–1381, 2015     

Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge–Kutta and hp-Finite Element Methods

The convergence of full discretisations by implicit Runge–Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied. The scope of applications includes differential inclusions governed by a nonlinear operator that is monotone and fulfills a certain growth condition. A basic assumption on the considered class of stiffly accurate Runge–Kutta time discretisations is a stability criterion which is in particular satisfied by the Radau IIA and Lobatto IIIC methods. In order to allow nonconforming hp-finite element approximations of unilateral constraints, set convergence of convex subsets in the sense of Glowinski–Mosco–Stummel is utilised. An appropriate formulation of the fully discrete variational inequality is deduced on the basis of a characteristic example of use, a Signorini-type initial-boundary value problem. Under hypotheses close to the existence theory of nonlinear first-order evolutionary equations and inequalities involving a monotone main part, a convergence result for the piecewise constant in time interpolant is established.     

First applications of a formula for the error of finite sinc interpolation

In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the barycentric formula, rational sinc interpolants (with and without replacement of the (usually unknown) derivatives with finite differences), convergence acceleration through extrapolation and improvement of one-sided interpolants. Work partly supported by the Swiss National Science Foundation under grant Nr 200021-116122.     

Optimal local spline interpolants

A constructive proof is given of the existence of a local spline interpolant which also approximates optimally in the sense that its associated operator reproduces polynomials of maximal order. First, it is shown that such an interpolant does not exist for orders higher than the linear case if the partition points of the appropriate spline space coincide with the given interpolation points. Next, in the main result, the desired existence of an optimal local spline interpolant for all orders is proved by increasing, in a specified manner, the set of partition points. Although our interpolant reproduces a more restricted function space than its quasi-interpolant counterpart constructed by De Boor and Fix [1], it has the advantage of interpolating every real function at a given set of points. Finally, we do some explicit calculations in the quadratic case.     

A Construction of Multiresolution Analysis on Interval

We present a concrete method of constructing multiresolution analysis on interval. The method generalizes the corresponding results of Cohen, Daubechies and Vial [Appl. Comput. Harmonic Anal., 1（1993）, 54-81]. By the use of the subdivision operator, the expressions of the constructed functions are more compact. Furthermore, the method reveals more clearly some properties of multiresolution analysis with certain approximation order.     

一阶HÖlder条件下带参数的修正型Euler-Halley迭代族的局部收敛性

1引言设X和Y为实或复Banach空间,Ω■X是开凸子集,F:Ω■X→Y是一阶连续可微的非线性算子.非线性算子方程F(x)=0 (1.1) 的求解及收敛域问题是现代科学计算理论的基本问题.解方程(1.1)的最著名的迭代方法是Newton法,在适当的条件下,它是二阶收敛的,此即著名的Kantorovich定理.关于Newton法收敛球半径的估计由Traub和王兴华分别给出,见[2]和[3],而收敛性研究的进一步发展可参看[4,5,6]及综述文章[7].     

Convergence of a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection–diffusion equation in 2D

A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.     

第三型伯恩斯坦插值过程的双结点修正

In this work,the well-known problem put forward by S N Bernstein in 1930 is studied in a deep step.An operator is constructed by revising double interpolation nodes.It is proved that the operator converges to arbitrary continuous functions uniformly and the convergence order is the best.     

Set-valued quasi variational inclusions

In this paper, we introduce and study a new class of variational inclusions, called the set-valued quasi variational inclusions. The resolvent operator technique is used to establish the equivalence between the set-valued quasi variational inclusions and the fixed point problem. This equivalence is used to study the existence of a solution and to suggest a number of iterative algorithms for solving the set-valued variational inclusions. We also study the convergence criteria of these algorithms.     

求解非线性算子方程的加速迭代格式

研究非线性算子方程的近似求解方法.首先对通常的求解非线性方程加速迭代格式进行推广,得到高阶收敛速度的加速迭代格式,最后把这种加速迭代格式推广到非线性算子方程的求解中去,利用非线性算子的渐进展开,证明了这种加速格式具有三阶的收敛速度.     

Hermite subdivision on manifolds via parallel transport

We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C ¹ smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from.