From: hartmann Date: Mon, 19 Jul 1999 13:55:34 +0000 (+0000) Subject: additional documentation X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e563d40e9a1e79e90d1d5378e09cb14c87b467e6;p=dealii-svn.git additional documentation git-svn-id: https://svn.dealii.org/trunk@1604 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/base/include/base/quadrature.h b/deal.II/base/include/base/quadrature.h index 79bed6d693..85fc8b68f8 100644 --- a/deal.II/base/include/base/quadrature.h +++ b/deal.II/base/include/base/quadrature.h @@ -20,18 +20,36 @@ * * There are a number of derived classes, denoting concrete integration * formulae. These are named by a prefixed #Q#, the name of the formula - * (e.g. #Gauss#) and finally the order of integration. For example, - * #QGauss2# denotes a second order Gauss integration formula in - * any dimension. Second order means that it integrates polynomials of - * third order exact. In general, a formula of order #n# exactly - * integrates polynomials of order #2n-1#. + * (e.g. #Midpoint# or #Gauss#) and finally (only for Gauss integration + * formulae) the number of quadrature points. + * + * For each quadrature formula + * there exists a number #m#, that denotes the maximal degree of polynomials + * the formula of integration is exact for. This number is given + * in the documentation of each formula. The order of integration is then + * given by #m+1#, that means that the error representation of the quadrature + * formula includes the $(m+1).$ derivative of the function to be integrated. + * As the $(m+1).$ derivate of polynomials of degree #m# is 0, the order of + * integration is always one larger than the degree of polynomials the + * quadrature formula is exact for. For example, the #Midpoint# quadrature + * formula is exact for polynomials of degree 1 (linear polynomials) and its order + * of integration is 2, i.e. `The midpoint-formula is of order 2'. + * + * Note the special case of Gauss integration formulae: The number #n# in the + * n-Point-Gauss Quadrature formula #QGaussn# denotes the number of quadrature + * points of the formula in one dimension. This formula is exact for polynomials + * of degree #2n-1# and its order of integration is #2n#. For example, + * #QGauss2<1># denotes the 2-Point-Gauss quadrature formula in 1 dimension. + * It is exact for polynomials of degree 3 and its order of integration is 4. * * Most integration formulae in more than one space dimension are tensor * products of quadrature formulae in one space dimension, or more * generally the tensor product of a formula in #(dim-1)# dimensions and * one in one dimension. There is a special constructor to generate a * quadrature formula from two others. - * + * For example, the #QGauss2# formulae includes $2^dim$ quadrature points + * in #dim# dimensions but is still exact for polynomials of degree 3 and its + * order of integration is 4. * * For some programs it is necessary to have a quadrature object for faces. * These programs fail to link if compiled for only one space dimension, @@ -44,7 +62,7 @@ * necessary to provide a class #Point<0># to make the compiler happy. This * class also does nothing. * - * @author Wolfgang Bangerth, 1998 + * @author Wolfgang Bangerth, 1998, documentation: Ralf Hartmann, 1999 */ template class Quadrature { diff --git a/deal.II/base/include/base/quadrature_lib.h b/deal.II/base/include/base/quadrature_lib.h index 6982cd08aa..455ddb9ca0 100644 --- a/deal.II/base/include/base/quadrature_lib.h +++ b/deal.II/base/include/base/quadrature_lib.h @@ -10,7 +10,7 @@ /** - * 2-Point-Gauss quadrature formula. + * 2-Point-Gauss quadrature formula, exact for polynomials of degree 3. * * Reference: Ward Cheney, David Kincaid: Numerical Mathematics and Computing. */ @@ -24,7 +24,7 @@ class QGauss2 : public Quadrature /** - * 3-Point-Gauss quadrature formula. + * 3-Point-Gauss quadrature formula, exact for polynomials of degree 5. * * Reference: Ward Cheney, David Kincaid: Numerical Mathematics and Computing. */ @@ -38,7 +38,7 @@ class QGauss3 : public Quadrature /** - * 4-Point-Gauss quadrature formula. + * 4-Point-Gauss quadrature formula, exact for polynomials of degree 7. * * Reference: Ward Cheney, David Kincaid: Numerical Mathematics and Computing. */ @@ -53,7 +53,7 @@ class QGauss4 : public Quadrature /** - * 5-Point-Gauss quadrature formula. + * 5-Point-Gauss quadrature formula, exact for polynomials of degree 9. * * Reference: Ward Cheney, David Kincaid: Numerical Mathematics and Computing. */ @@ -67,7 +67,8 @@ class QGauss5 : public Quadrature /** - * 6-Point-Gauss quadrature formula. I have not found explicite + * 6-Point-Gauss quadrature formula, exact for polynomials of degree 11. + * I have not found explicite * representations of the zeros of the Legendre functions of sixth * and higher degree. If anyone finds them, please replace the existing * numbers by these expressions. @@ -85,7 +86,8 @@ class QGauss6 : public Quadrature /** - * 7-Point-Gauss quadrature formula. I have not found explicite + * 7-Point-Gauss quadrature formula, exact for polynomials of degree 13. + * I have not found explicite * representations of the zeros of the Legendre functions of sixth * and higher degree. If anyone finds them, please replace the existing * numbers by these expressions. @@ -103,7 +105,8 @@ class QGauss7 : public Quadrature /** - * 8-Point-Gauss quadrature formula. I have not found explicite + * 8-Point-Gauss quadrature formula, exact for polynomials of degree 15. + * I have not found explicite * representations of the zeros of the Legendre functions of sixth * and higher degree. If anyone finds them, please replace the existing * numbers by these expressions. @@ -123,7 +126,7 @@ class QGauss8 : public Quadrature /** - * First order midpoint quadrature rule. + * Midpoint quadrature rule, exact for linear polynomials. * For compatibility, this rule may be accessed as #QGauss1#, too. */ template @@ -136,7 +139,7 @@ class QMidpoint : public Quadrature #define QGauss1 QMidpoint /** - * Simpson quadrature rule. + * Simpson quadrature rule, exact for polynomials of degree 3. */ template class QSimpson : public Quadrature @@ -148,7 +151,7 @@ class QSimpson : public Quadrature /** - * Trapezoidal quadrature rule. + * Trapezoidal quadrature rule, exact for linear polynomials. */ template class QTrapez : public Quadrature