![]() Quadrature objects are used in a number of places within deal.II where integration is performed, most notably via the FEValues and related classes. For example, the QGauss formulae include N dim quadrature points in dim dimensions, where \(N\) is the constructor parameter of QGauss. There is a special constructor to generate a quadrature formula from two others. Tensor product quadratureĪt least for hypercube reference cells (i.e., squares and cubes), 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. The tensor product formulae are exact on tensor product polynomials of degree m in each space direction, but they are still only of (m+1)st order. For the optimal formulae QGauss we have \(m = 2N-1\), where \(N\) is the constructor parameter to QGauss. The number m is to be found in the documentation of each concrete formula. The order of the integration error is m+1, that is, the error is the size of the cell to the m+1 by the Bramble-Hilbert Lemma. This number is given in the documentation of each formula. Mathematical backgroundįor each quadrature formula we denote by m, the maximal degree of polynomials integrated exactly on the reference cell the quadrature formula corresponds to. The main purpose of these formulae is their use in QProjector, which will create a useful formula of dimension one out of them. Access to the weight is possible, while access to the quadrature point is not permitted, since a Point of dimension zero contains no information. Since an integral over zero dimensions is the evaluation at a single point, any constructor of such a formula initializes to a single quadrature point with weight one. In order to allow for dimension independent programming, a quadrature formula of dimension zero exists. Refer to the list of derived classes for more details.Īt least for quadrilaterals and hexahedra (or, more precisely, since we work on reference cells: for the unit square and the unit cube), quadrature formulas are typically tensor products of one-dimensional formulas (see also the section on implementation detail below). There are a number of derived classes, denoting concrete integration formulae. Integration over concrete cells is done by coordinate transformation to the reference cell represented by the current class. That is, points and weights are expressed in the coordinate system of a reference cell (see the ReferenceCell class) and as such serves to represent quadrature points and weights on the unit line segment \(\) in 1d, on the unit square or unit triangle in 2d, as well as the unit tetrahedron, cube, pyramid, and wedge reference cells in 3d. This class stores quadrature points \(\mathbf x_q\) and weights \(w_q\) for concrete "quadrature formulas" when \(K\) (the domain we integrate over) is a reference cell. Quadrature is a means to approximate an integral by evaluating the integrand at specific points \(\mathbf x_q\) and summing the point values with specific weights \(w_q\) that is, quadrature computes Map_iterator = decltype( counter_map)::iteratorīase class for quadrature formulae in arbitrary dimensions. Map_value_type = decltype( counter_map)::value_type List_subscribers (StreamType &stream) constĮxcInUse ( int arg1, std::string arg2, std::string arg3)ĮxcNoSubscriber (std::string arg1, std::string arg2) Unsubscribe (std::atomic *const validity, const std::string &identifier="") const Subscribe (std::atomic *const validity, const std::string &identifier="") const This is mostly used by the SmartPointer class. Serialize (Archive &ar, const unsigned int version)Ĭlasses derived from Subscriptor provide a facility to subscribe to this object. Initialize (const std::vector > &points, const std::vector & weights) Quadrature (std::vector > &points, std::vector & weights) Quadrature (const std::vector > &points, const std::vector & weights) Quadrature ( Quadrature &) noexcept=default Quadrature (const Quadrature &quadrature_1d) Quadrature (const SubQuadrature &, const Quadrature &) ![]() Quadrature (const unsigned int n_quadrature_points=0) This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.
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