scholarly journals Stable High Order Quadrature Rules for Scattered Data and General Weight Functions

2020 ◽  
Vol 58 (4) ◽  
pp. 2144-2164
Author(s):  
Jan Glaubitz
Keyword(s):  
Scattered Data ◽  
High Order ◽  
Quadrature Rules ◽  
Weight Functions ◽  
General Weight

scholarly journals Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

1970 ◽  
Vol 29 ◽  
pp. 117-125
Author(s):  
HT Rathod ◽  
RD Sathish ◽  
Md Shafiqul Islam ◽  
Arun Kumar Gali
Keyword(s):  
Gaussian Quadrature ◽  
High Order ◽  
Quadrature Rules ◽  
Zeros Of Polynomials ◽  
Matlab Program ◽  
Present Context ◽  
First Time ◽  
Legendre Quadrature ◽  
Weight Coefficients ◽  
Integration Rules

Gauss Legendre Quadrature rules are extremely accurate and they should be considered seriously when many integrals of similar nature are to be evaluated. This paper is concerned with the derivation and computation of numerical integration rules for the three integrals: (See text for formulae) which are dependent on the zeros and the squares of the zeros of Legendre Polynomial and is quite well known in the Gaussian Quadrature theory. We have developed the necessary MATLAB programs based on symbolic maths which can compute the sampling points and the weight coefficients and are reported here upto 32 – digits accuracy and we believe that they are reported to this accuracy for the first time. The MATLAB programs appended here are based on symbolic maths. They are very sophisticated and they can compute Quadrature rules of high order, whereas one of the recent MATLAB program appearing in reference [21] can compute Gauss Legendre Quadrature rules upto order twenty, because the zeros of Legendre polynomials cannot be computed to desired accuracy by MATLAB routine roots (……..). Whereas we have used the MATLAB routine solve (……..) to find zeros of polynomials which is very efficient. This is worth noting in the present context. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 117-125  DOI: http://dx.doi.org/10.3329/ganit.v29i0.8521 

A high order multivariate approximation scheme for scattered data sets

2010 ◽  
Vol 229 (18) ◽  
pp. 6343-6361 ◽  
Author(s):  
Qiqi Wang ◽  
Parviz Moin ◽  
Gianluca Iaccarino
Keyword(s):  
Approximation Scheme ◽  
Scattered Data ◽  
High Order ◽  
Multivariate Approximation ◽  
Data Sets

A Residual Based Variational Method for Reducing Dispersion Error in Finite Element Methods

2005 ◽  
Author(s):  
Lonny L. Thompson ◽  
Prapot Kunthong
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Least Squares ◽  
High Order ◽  
High Order Accuracy ◽  
Quadrature Rules ◽  
Source Terms ◽  
Order Accuracy ◽  
Dispersion Error ◽  
Variational Framework

A difficulty of the standard Galerkin finite element method has been the ability to accurately resolve oscillating wave solutions at higher frequencies. Many alternative methods have been developed including high-order methods, stabilized Galerkin methods, multi-scale variational methods, and other wave-based discretization methods. In this work, consistent residuals, both in the form of least-squares and gradient least-squares are linearly combined and added to the Galerkin variational Helmholtz equation to form a new generalized Galerkin least-squares method (GGLS). By allowing the stabilization parameters to vary spatially within each element, we are able to select optimal parameters which reduce dispersion error for all wave directions from second-order to fourth-order in nondimensional wavenumber; a substantial improvement over standard Galerkin elements. Furthermore, the stabilization parameters are frequency independent, and thus can be used for both time-harmonic solutions to the Helmholtz equation as well as direct time-integration of the wave equation, and eigenfrequency/eigenmodes analysis. Since the variational framework preserves consistency, high-order accuracy is maintained in the presence of source terms. In the case of homogeneous source terms, we show that our consistent variational framework is equivalent to integrating the underlying stiffness and mass matrices with optimally selected numerical quadrature rules. Optimal GGLS stabilization parameters and equivalent quadrature rules are determined for several element types including: bilinear quadrilateral, linear triangle, and linear tetrahedral elements. Numerical examples on unstructured meshes validate the expected high-order accuracy.

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