Reference documentation for deal.II version 9.4.0

## Detailed Description

This module contains the base class Quadrature as well as the quadrature formulas provided by deal.II. Quadrature formulas provide two essential pieces of data: the locations of quadrature points on the unit cell [0,1]^d, and the weight of each quadrature point.

Since deal.II uses quadrilaterals and hexahedra, almost all quadrature formulas are generated as tensor products of 1-dimensional quadrature formulas defined on the unit interval [0,1], which makes their definition for the higher-dimensional case almost trivial. However, the library also allows anisotropic tensor products (more quadrature points in one coordinate direction than in another) through the QAnisotropic class, as well as the definition of quadrature formulas that are not tensor products.

In the grand scheme of things, the classes of this module interact with a variety of other parts of the library:

### Use

Quadrature formulas are used, among other uses, when integrating matrix entries and the components of the right hand side vector. To this end, the quadrature point defined on the unit cell have to be mapped to the respective locations on a real cell, and the weights have to be multiplied by the determinant of the Jacobian. This step is done by classes derived from the Mapping base class, although this is often hidden since many parts of the library fall back to using an object of type MappingQ1 if no particular mapping is provided.

The next step is to evaluate shape functions and their gradients at these locations. While the classes derived from the FiniteElement base class provide a description of the shape functions on the unit cell, the actual evaluation at quadrature points and joining this with the information gotten from the mapping is done by the FEValues class and its associates. In essence, the FEValues class is therefore a view to the finite element space (defined by the FiniteElement classes) evaluated at quadrature points (provided by the Quadrature classes) mapped to locations inside cells in real, as opposed to unit, space (with the mapping provided by the Mapping classes).

The FEValues class provides, as a side product, the location of the quadrature points as mapped to a real cell, for other uses as well. This can then be used, for example, to evaluate a right hand side function at these points.

### QIterated

The class QIterated is used to construct an iterated quadrature formula out of an existing one, thereby increasing the accuracy of the formula without increasing the order. For example, by iterating the trapezoidal rule with points at 0 and 1 and weights 1/2 and 1/2 twice, we get a quadrature formula with points at 0, 1/2, and 1 with weights 1/4, 1/2, and 1/4, respectively. This formula is obtained by projecting the quadrature formula onto the subintervals [0,1/2] and [1/2,1], respectively, and then merging the right endpoint of the left interval with the left endpoint of the right interval. In the same way, all one-dimensional quadrature formulas can be iterated. Higher dimensional iterated formulas are generated as tensor products of one-dimensional iterated formulas.

### QAnisotropic

While the usual quadrature formulas of higher dimensions generate tensor products which are equal in each direction, the class QAnisotropic generates tensor products of possibly different formulas in each direction.

### QProjector

The class QProjector is not actually a quadrature rule by itself, but it provides functions for computing quadrature formulas on the surfaces of higher dimensional cells.

All other classes in this module actually implement quadrature rules of different order and other characteristics.