Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

A spline collocation method for parabolic pseudodifferential equations

The purpose of this paper is to examine a boundary element collocation method for some parabolic pseudodifferential equations. The basic model problem for our investigation is the two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition. Certain choices of the representation formula for the heat potential yield boundary integral equations of the first kind, namely the single layer and the hypersingular heat operator equations. Both of these operators, in particular, are covered by the class of parabolic pseudodifferential operators under consideration. Moreover, the spatial domain is allowed to have a general smooth boundary curve. As trial functions the tensor products of the smoothest spline functions of odd degree (space) and continuous piecewise linear splines (time) are used. Stability and convergence of the method is proved in some appropriate anisotropic Sobolev spaces.     

A quadrature-based approach to improving the collocation method

Summary The collocation method is a popular method for the approximate solution of boundary integral equations, but typically does not achieve the high order of convergence reached by the Galerkin method in appropriate negative norms. In this paper a quadrature-based method for improving upon the collocation method is proposed, and developed in detail for a particular case. That case involves operators with even symbol (such as the logarithmic potential) operating on 1-periodic functions; a smooth-spline trial space of odd degree, with constant mesh spacingh=1/n; and a quadrature rule with 2n points (where ann-point quadrature rule would be equivalent to the collocation method). In this setting it is shown that a special quadrature rule (which depends on the degree of the splines and the order of the operator) can yield a maximum order of convergence two powers ofh higher than the collocation method.     

High-order discrete-time orthogonal spline collocation methods for singularly perturbed 1D parabolic reaction–diffusion problems

Quasi-optimal error estimates are derived for the continuous-time orthogonal spline collocation (OSC) method and also two discrete-time OSC methods for approximating the solution of 1D parabolic singularly perturbed reaction–diffusion problems. OSC with C¹ splines of degree r ≥ 3 on a Shishkin mesh is employed for the spatial discretization while the Crank–Nicolson method and the BDF2 scheme are considered for the time-stepping. The results of numerical experiments validate the theoretical analysis and also exhibit additional quasi-optimal results, in particular, superconvergence phenomena.     

The convergence of spline collocation for strongly elliptic equations on curves

Summary Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthdayThis work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the Stiftung Volkswagenwerk     

Least Squares Solution of Bivariate Surface Fitting Problems Using Tensor Product Splines

The paper treats bivariate surface fitting problems, where the data points lie on lines parallel to one of the axes. The associated bivariate collocation matrix is investigated as a block Kronecker product of univariate collocation matrices. Based on various properties of this block Kronecker product, such scattered data are characterized where the associated interpolation problem using tensor product splines admits a unique solution.     

Collocation with WEB–Splines

We describe a collocation method with weighted extended B–splines (WEB–splines) for arbitrary bounded multidimensional domains, considering Poisson’s equation as a typical model problem. By slightly modifying the B–spline classification for the WEB–basis, the centers of the supports of inner B–splines can be used as collocation points. This resolves the mismatch between the number of basis functions and interpolation conditions, already present in classical univariate schemes, in a simple fashion. Collocation with WEB–splines is particularly easy to implement when the domain boundary can be represented as zero set of a weight function; sample programs are provided on the website http://www.web-spline.de. In contrast to standard finite element methods, no mesh generation and numerical integration is required, regardless of the geometric shape of the domain. As a consequence, the system equations can be compiled very efficiently. Moreover, numerical tests confirm that increasing the B–spline degree yields highly accurate approximations already on relatively coarse grids. Compared with Ritz-Galerkin methods, the observed convergence rates are decreased by 1 or 2 when using splines of odd or even order, respectively. This drawback, however, is outweighed by a substantially smaller bandwidth of collocation matrices.     

Quantitative Inheritance Properties for Simultaneous Approximation by Tensor Product Operators

Our results describe how quantitative properties of univariate operators are inherited by the tensor product of their parametric extensions. This includes statements concerning simultaneous approximation. The estimates are in terms of partial and total moduli of smoothness of higher order. Applications are given for cubic interpolatory splines and Bernstein operators. Further applications are possible.     

Maximum‐norm a posteriori error bounds for a collocation method applied to a singularly perturbed reaction–diffusion problem in three dimensions

Collocation with triquadratic C¹‐splines for a singularly perturbed reaction–diffusion problem in three dimension is studied. A posteriori error bound in the maximum norm is derived for the collocation method on arbitrary tensor‐product meshes which is robust in the perturbation parameter. Numerical results are presented that support our theoretical estimate.     

On the convergence of the spline collocation with discontinuous data

In this article we study the convergence of the collocation method in the case where the smoothest splines are used as trial functions. The given data is allowed to be piecewise continuous. Our model problem is stated by means of an explicit Fourier representation in the space of periodic functions. Thus the results are applicable e.g. to differential operators and to classical integral operators of the convolutional type. Error estimates are given for a class of Sobolev norms. An application to the single layer potential is discussed.     

A Generalization of the Arnold-Wendland Lemma to a Modified Collocation Method for Boundary Integral Equations in ℝ3

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic periodic pseudodifferential equations in two independent variables by a modified method of nodal collocation by odd degree polynomial splines. In the one-dimensional case, our method coincides with the method of nodal collocation when odd degree polynomial splines are employed for the trial functions. The convergence analysis is based on an equivalence which we establish between our method and a nonstandard Galerkin method for an operator closely related to the given operator. This equivalence is realized through a crucial intermediate result (which we now term the Arnold-Wendland lemma) to connect the solution of central finite difference equations and that of certain nonstandard Galerkin equations. The results of this paper are genuine two-dimensional generalizations of the results obtained by ARNOLD and WENDLAND in [2] for the one-dimensional equations.     

Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations

Collocation is based on the discretization of the strong form of the underlying partial differential equations, which requires basis functions of sufficient order and smoothness. Consequently, the use of isogeometric analysis (IGA) for collocation suggests itself, since splines can be readily adjusted to any order in polynomial degree and continuity required by the differential operators. In addition, they can be generated for domains of arbitrary geometric and topological complexity, directly linked to and fully supported by CAD technology. The major advantage of isogeometric collocation over Galerkin type IGA is the minimization of the computational effort for numerical quadrature. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Spline Collocation for Singular Integral Equations

This paper is devoted to the approximate solution of one-dimensional singular integral equations on a closed curve by spline collocation methods. As the main result we give conditions which are sufficient and in special cases also necessary for the convergence in SOBOLEV norms. The paper is organized as follows. In chapter 1 we indicate some definitions and some facts about projection methods. In chapter 2, we generalize a technique developed in [1] and study the convergence of collocations using splines of odd degree in periodic SOBOLEV spaces. In chapter 3, we apply our method to collocations by splines of even degree and consider the case of systems of equations. And in the last chapter, 4, the results are applied to singular integral equations and compared with known facts about piecewise linear spline collocation for such equations.     

二元三次样条空间S31,2(Δmn(2)) 的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2 型三角剖分上二元三次样条空间S₃^1,2(Δ_mn⁽²⁾)的若干样条拟插值算子. 这些变差缩减算子由样条函数B_ij¹支集上5 个网格点或中心和样条函数B_ij²支集上5 个网格点处函数值定义. 这些样条拟插值算子具有较好的逼近性,甚至算子V_mn（f） 能保持近最优的三次多项式性. 然后利用连续模,分析样条拟插值算子V_mn（f）一致逼近于充分光滑的实函数. 最后推导误差估计.     

Adaptive boundary element methods for strongly elliptic integral equations

Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.This work was carried out while Dr. De-hao Yu was an Alexander-von-Humboldt-Stiftung research fellow at the University of Stuttgart in 1987, 1988     

Spline collocation for singular integro-differential equations over (0.1)

Summary This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.     

Natural bicubic spline fractal interpolation

Fractal Interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied.     

The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane

Summary In this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions. It turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed+1. Some numerical examples which conform the theoretical results are presented. Present address: (1. 7. 1988–31. 12. 1988) Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Products of B-patches

Products and tensor products of multivariate polynomials in B-patch form are viewed as linear combinations of higher degree B-patches. Univariate B-spline segments and certain regions of simplex splines are examples of B-patches. A recursive scheme for transforming tensor product B-patch representations into B-patch representations of more variables is presented. The scheme can also be applied for transforming ann-fold product of B-patch expansions into a B-patch expansion of higher degree. Degree raising formulas are obtained as special cases. The scheme calculates the blossom of the (tensor) product surface and generalizes the pyramidal recursive scheme for B-patches.     

关于非等距的五次样条函数的缺插值

一f(x)是区间[0,1]上定义的函数,0=x_0相似文献