Quadrature Points

Numerical quadrature (or, quadrature rules) are primary components of finite element methods. Numerical quadrature is used for evaluating integration, which is necessary for computing the finite element matrices and vectors.

Numerical quadrature are evaluated on reference elements. Every numerical quadrature rule has an order of accuracy.

A quadrature rule is denoted by weights $w_{i}$ and quadrature points $\xi_{i}$ .

After getting $\{\xi, w\}$ we can compute an integration by using following expression:

\int_{\Omega^{*}} f(\mathbf{x}) d\Omega = \sum_{i=1}^{n}{f(\mathbf{x}_{i})}w_{i}

where, $\Omega^{*}$ denotes the reference element domain.

easifemBase

defines a user-defined data type called QuadraturePoint_ to handle the quadrature points.

note

Quadrature points $\xi$ are located in the reference element.

Creating an instance of `QuadraturePoint_`

Initiate

Use Initiate method for constructing an instance of QuadraturePoint.
To use this subroutine we need to create ReferenceElement first.

The following example shows how to construct GaussLegendre quadrature points on line.

Gauss Legendre points on line elements
PROGRAM main
USE easifemBase
IMPLICIT NONE
TYPE(QuadraturePoint_) :: obj
TYPE(ReferenceLine_) :: refelem
INTEGER(I4B) :: order

refelem = ReferenceLine(nsd=1_I4B)
CALL display(refelem, "refelem: ")
order = 4_I4B

CALL initiate(obj=obj, &
  & refelem=refelem, &
  & order=order,  &
  & quadratureType=GaussLegendre)
CALL display(obj, "ans: ")
END PROGRAM main

we can use following values for quadratureType, which are declared in easifemBase's GlobalData:

GaussLegendre
GaussLegendreLobatto
GaussLegendreRadau, GaussLegendreRadauLeft
GaussLegendreRadauRight
GaussChebyshev
GaussChebyshevLobatto
GaussChebyshevRadau, GaussChebyshevRadauLeft
GaussChebyshevRadauRight
GaussJacobi
GaussJacobiLobatto
GaussJacobiRadau, GaussJacobiRadauLeft
GaussJacobiRadauRight
GaussUltraspherical
GaussUltrasphericalLobatto
GaussUltrasphericalRadau, GaussUltraspericalRadauLeft
GaussUltrasphericalRadauRight

info

In the case of GaussJacobi***, alpha and beta should be given, and in the case of GaussUltraspherical**, lambda should be given.

caution

In the case of triangle, tetrahedron, pyramid, and prism, quadratureType is ineffective, that is, we do not use quadratureType for these elements.