Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients

In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k?≥?1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green’s formula, which is not problem dependent, i.e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green’s formula.     

Double integral inequalities and application in numerical integration

Upper and lower error bounds for an optimal 2-point quadrature rule of open type are derived. These error bounds are sharp. Applications in numerical integration are given This revised version was published online in August 2006 with corrections to the Cover Date.     

On Birkhoff-Young quadrature of analytical functions

A simple method is given for constructing quadrature rules for the numerical integration of an analytic function over a line segment in the complex plane. The Birkhoff-Young 5-point, degree 5 rule is obtained as a special case. An error analysis is used to show how rules preferable to the Birkhoff-Young rule are easily developed.     

Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

Novel ideas in harmonic analysis are used to analyze the trapezoidal rule integration for two spheres. Sampling in spherical coordinates links three levels of harmonic analysis. Eigenfunctions of a nonstandard manifold Laplacian descend by restriction, first to a differential graph Laplacian, and then to difference operators. Trapezoidal rule integration with appropriate sampling is exact on eigenspaces of the manifold Laplacian, a fact which leads to trapezoidal rule error estimates on Sobolev-style spaces of functions. Singular functions with accurate trapezoidal rule integrals are identified, and a simplified analysis of smooth function numerical integration is provided.     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.     

A Note on Asymptotic Minimal Weights for an Integration Rule

Consider an integration rule with abscissas in the intervalof integration and which is exact for all polynomials up toa certain degree. The error of the rule applied to a Hilbertspace of functions analytic within an ellipse containing intervalof integration is a bounded linear functional. In this paperwe the asymptotic weights of the rule which minimize the normof the error functional as the underlying ellipse shrinks tothe interval of integration.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

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.     

Error inequalities for an optimal quadrature formula

An optimal 3-point quadrature formula of closed type is derived. It is shown that the optimal quadrature formula has a better error bound than the well-known Simpson’s rule. A corrected formula is also considered. Various error inequalities for these formulas are established. Applications in numerical integration are given.     

Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

A family of quadrature rules for integration over tetrahedral volumes is developed. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement using 1, 4, 10, 20, 35, and 56 quadrature points. The rules are characterized by rapid convergence, positive weights, and symmetry. Each rule is an optimal approximation in the sense that lower-order terms have zero contribution to the truncation error and the leading-order error term is minimized. Quadrature formulas up to order 9 are presented with relevant numerical examples.     

Evaluation of the local error of fractional-rational multistep methods with variable integration step

To ensure the proper qualitative characteristic of approximate numerical solution of the Cauchy problem for a system of ordinary differential equations, it is necessary to formulate certain conditions that have to be satisfied by numerical methods. The efficiency of a numerical method is determined by constructing the algorithm of integration step changing and the choice of the order of the method. The construction of such a method requires one to determine preliminarily the admissible error of the method in each integration step. A theorem on the evaluation of the local error of multistep numerical p th-order methods with variable integration step without taking into account the round-off error is formulated. This theorem enables one to construct an efficient algorithm for the step change and the choice of the corresponding order of the method.     

Verallgemeinerung der Quadraturfehlerabschätzung vonDavis auf mehrfache Integrale

Summary The error which occurs when the trapezoidal rule is applied to the numerical integration of a real analytic and periodic function over its period may be estimated according to a result due toDavis by means of a certain analytic continuation into the complex plane. This result is extended to multiple integrals. The error bound obtained is proportional to the multiplicity of the computed integral.     

Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.     

High-order quadratures for integral operators with singular kernels

A numerical integration method that has rapid convergence for integrands with known singularities is presented. Based on endpoint corrections to the trapezoidal rule, the quadratures are suited for the discretization of a variety of integral equations encountered in mathematical physics. The quadratures are based on a technique introduced by Rokhlin (1990). The present modification controls the growth of the quadrature weights and permits higher-order rules in practice. Several numerical examples are included.     

Parameter tuning and repeated application of the IMT-type transformation in numerical quadrature

Summary The IMT rule, which is especially suited for the integration of functions with end-point singularities, is generalized by introducing parameters and also by repeatedly applying the parametrized IMT transformation. The quadrature formulas thus obtained are improved considerably both in efficiency and in robustness against end-point singularities. Asymptotic error estimates and numerical results are also given.     

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

We present a high‐order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second‐order one‐dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well‐conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 307–349, 2016     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

Blending product-type quadrature rules

The idea of blending which was originally used for bivariate approximation is utilized for the numerical integration of the product of two functions. The combination of three product-type quadrature rules results in a rule with a lower error than each of the original rules. Rules of different exactness degrees as well as compounded rules of different step sizes can be taken for such a combination. Two explicit rules are constructed for demonstration; numerical examples confirm the asymptotic rates of convergence of these rules.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

二维带边界偏导数值的复化数值积分公式

构建了一类二维带边界偏导数值的复化数值积分公式,给出了所建立的两种数值积分公式的稳定性分析、误差分析和代数精度.与二维复化四点高斯数值积分公式相对比,所建立的带边界偏导数值的复化梯形、复化辛普森求积公式在达到相同精度时所需积分节点大大减少,积分的时间复杂度也随之大大减少,实例验证结果良好.