Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels

We study univariate integration with the Gaussian weight for a positive variance α. This is done for the reproducing kernel Hilbert space with the Gaussian kernel for a positive shape parameter γ. We study Gauss-Hermite quadratures, although this choice of quadratures may be questionable since polynomials do not belong to this space of functions. Nevertheless, we provide the explicit formula for the error of the Gauss-Hermite quadrature using n function values. In particular, for 2αγ ²<1 we have an exponential rate of convergence, and for 2αγ ²=1 we have no convergence, whereas for 2αγ ²>1 we have an exponential divergence.     

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.     

Computing Discrepancies of Smolyak Quadrature Rules

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as possible competition to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing theirL₂-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing theL₂-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules and Hammersley and Monte Carlo sequences.     

A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Gaussian product-type quadratures

Ordinary N-term integral quadratures require the evaluation of the entire integrand at N points. However, m-by-n product type quadratures involve the evaluation of one factor of the integrand at m points and the reamaining factor at n points. The principal results of this paper include the generalization of the product-type quadrature concept to arbitrary weight functions and to infinite as well as finite intervals, the calculation of the mn coefficients of this product quadrature formula from the LU decoposition of one n-byn, and the extension of the precision of the formula. Nuerical examples are included to illustrate the application of Gaussian product-type quadratures and to compare them with the ordinary Gaussian quadratures.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

A fast Fourier-collocation method for second boundary integral equations

In this paper we develop a fast collocation method for second boundary integral equations by the trigonometric polynomials. We propose a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis and the corresponding collocation functionals. The compression leads to a sparse matrix with only O(nlog²n) number of nonzero entries, where 2n+1 denotes the order of the matrix. Thus we develop a fast Fourier-collocation method. We prove that the fast Fourier-collocation method gives the optimal convergence order up to a logarithmic factor. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We establish that this algorithm preserves the quasi-optimal convergence of the approximate solution with requiring a number of O(nlog³n) multiplications.     

High‐order numerical methods for one‐dimensional parabolic singularly perturbed problems with regular layers

In this work we construct and analyze some finite difference schemes used to solve a class of time‐dependent one‐dimensional convection‐diffusion problems, which present only regular layers in their solution. We use the implicit Euler or the Crank‐Nicolson method to discretize the time variable and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize the spatial variable. In both cases we prove that the numerical method is uniformly convergent with respect to the diffusion parameter, having order near two in space and order one or 3/2, depending on the method used, in time. We show some numerical examples which illustrate the theoretical results, in the case of using the Euler implicit method, and give better numerical behaviour than that predicted theoretically, showing order two in time and order N^?2log²N in space, if the Crank‐Nicolson scheme is used to discretize the time variable. Finally, we construct a numerical algorithm by combining a third order A‐stable SDIRK with two stages and a third‐order HODIE difference scheme, showing its uniformly convergent behavior, reaching order three, up to a logarithmic factor. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Curve denoising by multiscale singularity detection and geometric shrinkage

We propose a method for denoising piecewise smooth curves, given a number of noisy sample points. Using geometric variants of wavelet shrinkage methods, our algorithm preserves corners while enforcing that the smoothed arcs lie in an L² Sobolev space Hα of order α chosen by the operator. The reconstruction is scale-invariant when using the Sobolev space H3/2, adapts to the local noise level, and is essentially free of tuning parameters. In particular, our noise-adaptivity ensures that there is no arbitrarily-chosen “diffusion time” parameter in the denoising. Further, in cases where the distinction between signal and noise is unclear, we show how statistics gathered from the curve can be used to identify a finite number of “good” choices for the denoising.     

On polynomial collocation for second kind integral equations with fixed singularities of Mellin type

Summary.  We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asymptotics of the solution, we derive the rates of convergence. Finally, we describe an algorithm to compute the stiffness matrix based on simple Gau? quadratures and an alternative algorithm based on a recursion in the spirit of Monegato and Palamara Orsi. All together an almost best approximation to the solution of the integral equation can be computed with 𝒪(n ²[log n]²) resp. 𝒪(n ²) operations, where n is the dimension of the polynomial trial space. Received February 18, 2002 / Revised version received May 15, 2002 / Published online October 29, 2002 RID="⋆" ID="⋆" Correspondence to: A. Rathsfeld Mathematics Subject Classification (1991): 65R20     

Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature ² × ¹ of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on ² and ¹, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on ² and ¹ lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.     

The effect of numerical integration on the finite element approximation of linear functionals

In this paper, we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k, in the context of approximating linear functionals, which are also known as “quantities of interest”. We have obtained the optimal order of convergence, O(h^2k){\mathcal{O}(h^{2k})}, of the error in the computed functional, when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k − 1. However, this result was obtained under an increased regularity assumption on the data, which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature. We have obtained a lower bound of the error in the computed functional for a particular problem, which indicates the necessity of the increased regularity requirement of the data. Numerical experiments have been presented indicating that over-integration may be necessary to accurately approximate the functional, when the data lack the increased regularity.     

An alternating direction implicit method for orthogonal spline collocation linear systems

Summary An Alternating Direction Implicit method is analyzed for the solution of linear systems arising in high-order, tensor-product orthogonal spline collocation applied to some separable, second order, linear, elliptic partial differential equations in rectangles. On anNxN partition, with Jordan's selection of the acceleration parameters, the method requiresO(N ² ln ² N) arithmetic operations to produce an approximation whose accuracy, in theH ¹-norm, is that of the collocation solution.     

Computational parametric Willmore flow

We propose a new algorithm for the computation of Willmore flow. This is the L ²-gradient flow for the Willmore functional, which is the classical bending energy of a surface. Willmore flow is described by a highly nonlinear system of PDEs of fourth order for the parametrization of the surface. The spatially discrete numerical scheme is stable and consistent. The discretization relies on an adequate calculation of the first variation of the Willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements. The algorithm uses finite elements on surfaces. We give numerical examples and tests for piecewise linear finite elements. A convergence proof for the full algorithm remains an open question.     

Stability of collocation methods for delay differential equations with vanishing delays

We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.     

On ϵ-Collocation for Pseudodifferential Equations on a Closed Curve

This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L²-convergence for operators with smooth and piecewise continuous coefficients.     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

AC 0-collocation-like method for two-point boundary value problems

Summary A method of a collocation type based onC ⁰-piecewise polynomial spaces is presented for a two-point boundary value problem of the second order. The method has an optimal order of convergence under smoothness requirements on the exact solution which are weaker than forC ¹-collocation methods. If the differential operator is symmetric, a modification of this method leads to a symmetric system of linear equations. It is shown that if the collocation solution is a piecewise polynomial of degree not greater thanr, the method is stable and convergent with orderh ^r inH ¹-norm. A similar symmetric modification forC ⁰-colloction-finite element method [7] is also obtained. Superconvergence at the nodes is established.