Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature
type: Item
instance of: scholarly article
title: Interpolation approximations based on Gauss-Lobatto-Legendre-Birkhoff quadrature
zbMATH Open document ID: 1179.47014
author: Lilian Wang
author: Ben-Yu Guo
published in: Journal of Approximation Theory
publication date: 2009-11-26T00:00:00Z
review text: This paper addresses Gauss-Lobatto-Legendre-Birkhoff (GLLB) quadratures \[ \int_{-1}^1 \phi(x) \,dx\approx \phi'(-1) \omega_{-} + \sum_{j=1}^{N-1} \phi(x_j) \omega_J +\phi'(1)\omega_{+}, \] which are exact for all polynomials of degree \(2N-1\) and whose interior nodes are zeros of the quasi-orthogonal polynomials formed by linear combination of Jacobi polynomials. GLLB quadratures are particularly important in spectral approximations of second-order elliptic problems with Neumann boundary conditions, as well as for many other important problems, such as numerical integration and integral equations with the data of derivatives at the endpoints. Its nodes can be a natural choice of the preassigned interpolation points. The authors continue studies of \textit{A.~Ezzirani} and \textit{A.~Guessab} [Math. Comput. 68, No. 225, 217--248 (1999; Zbl 0916.65020)] who proposed a fast algorithm for computing nodes and weights of Gauss-Birkhoff quadreature with mixed boundary conditions. The results are related to studies of asymptotic behaviors of the following nodes and weights: \( x_j \cong \cos{\frac{(2j+1/2)\pi}{2N+1}}; \,\, \omega_j \cong \frac{\pi}{N} \sin \theta_j,\, \theta_j=\arccos x_j,\, 1\leq j \leq N-1.\) It is to be noted that GLLB differentiation matrices are ill-conditioned. In the present paper, the authors construct finite-difference preconditioning for one-dimensional second-order elliptic equations. The user-oriented implementation and an error analysis of the GLLB pseudospectral methods based on variational formulations are also provided.
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 5639037
zbMATH Keywords: GLLB quadrature rule
zbMATH Keywords: collocation method
zbMATH Keywords: Neumann problems
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: interpolation errors
zbMATH Keywords: integral equations
reviewed by: Denis Sidorov
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: j.jat.2008.08.016
OpenAlex ID: W2071157981
cites work: Equivalent Norms for Sobolev Spaces
cites work: Techniques of scientific computing (Part 2)
cites work: Chebyshev and Fourier spectral methods.
cites work: Boundary Conditions in Chebyshev and Legendre Methods
cites work: Spectral Methods
cites work: Q4162607
cites work: Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems
cites work: A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations
cites work: A Practical Guide to Pseudospectral Methods
cites work: Q3140811
cites work: Calculation of Gauss Quadrature Rules
cites work: Spectral Methods and Their Applications
cites work: Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations
cites work: Jacobi interpolation approximations and their applications to singular differential equations
cites work: Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces
cites work: Spectral Methods for Time-Dependent Problems
cites work: The Pseudospectral Method for Third-Order Differential Equations
cites work: A New Class of Gaussian Quadrature Formulas Based on Birkhoff Type Data
cites work: Uniqueness of Gauss–Birkhoff Quadrature Formulas
cites work: Properties of collocation third-derivative operators
cites work: Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials
cites work: Spectral Methods in MATLAB
cites work: Quasi-orthogonal polynomials, quadrature, and interpolation
DOI: 10.1016/J.JAT.2008.08.016
description: scientific article; zbMATH DE number 5639037