[web] Added docu for quadrature schemes

web/content/docs/userguide/basics/quadrature.md

++++
+date = "2022-03-02T09:30:00+01:00"
+title = "Quadrature schemes and extrapolation"
+author = "Christoph Lehmann"
+weight = 7
+
+
+[menu]
+  [menu.userguide]
+    parent = "basics"
++++
+
+OpenGeoSys has implemented certain numerical integration/quadrature rules for
+each mesh element type. Currently, all of the implemented quadrature rules are
+Gauss-Legendre quadrature or schemes in that spirit. I.e., integration points
+are located inside the elements, not at its faces, edges or corners.
+
+The table below gives an overview over the implemented schemes and which
+polynomials can be integrated exactly with the respective schemes. The data are
+[unit tested](https://gitlab.opengeosys.org/ogs/ogs/-/tree/master/Tests/MathLib/TestGaussLegendreIntegration.cpp).
+
+Integration methods are implemented up to *integration order* 4.
+For each integration order *n* there is a maximum polynomial degree *P* that can be
+integrated exactly with the respective integration method. For the classical
+Gauss-Legendre integration the following holds: *P = 2 · n - 1*. Methods that
+fulfill this relation are marked **bold** in column **P** in the table, methods
+that are deficient are marked *italic*.
+
+The columns contain the following data:
+
+* #IP: The number of integration points of the integration method.
+* P: The maximum polynomial degree that the integration method can integrate
+  exactly. The monomials in the polynomial are *x*<sup>i</sup> *y*<sup>j</sup>
+  *z*<sup>k</sup> with *i + j + k ≤ P*
+* Q: The maximum polynomial degree that the integration method can integrate
+  exactly. The monomials in the polynomial are *x*<sup>i</sup> *y*<sup>j</sup>
+  *z*<sup>k</sup> with *i ≤ Q*, *j ≤ Q* and *k ≤ Q*. I.e., maximum monomial
+  degrees are higher in the Q column than in the P column.
+
+
+| Integration order → | 1       | 1     | 1     | 2       | 2     | 2     | 3       | 3     | 3     | 4       | 4     | 4     |
+| ------------------- | ------: | ----: | ----: | ------: | ----: | ----: | ------: | ----: | ----: | ------: | ----: | ----: |
+| **Mesh element ↓**  | **#IP** | **Q** | **P** | **#IP** | **Q** | **P** | **#IP** | **Q** | **P** | **#IP** | **Q** | **P** |
+| Point               | 1       | \*    | \*    | 1       | \*    | \*    | 1       | \*    | \*    | 1       | \*    | \*    |
+| Line                | 1       | 1     | **1** | 2       | 3     | **3** | 3       | 5     | **5** | 4       | 7     | **7** |
+| Quad                | 1       | 1     | **1** | 4       | 3     | **3** | 9       | 5     | **5** | 16      | 7     | **7** |
+| Hexahedron          | 1       | 1     | **1** | 8       | 3     | **3** | 27      | 5     | **5** | 64      | 7     | **7** |
+| Triangle            | 1       | 0     | **1** | 3       | 1     | **3** | 4       | 1     | *3*   | 7       | 2     | *5*   |
+| Tetrahedron         | 1       | 0     | **1** | 5       | 1     | **3** | 14      | 1     | **5** | 20      | 1     | *5*   |
+| 3-sided prism       | 1       | 0     | **1** | 6       | 1     | **3** | 21      | 2     | *3*   | 28      | 2     | *5*   |
+| Pyramid             | 1       | 1     | **1** | 5       | 1     | **3** | 13      | 3     | *3*   | 13      | 3     | *3*   |
+
+Note, that on pyramids det(*J*) varies over the mesh element, even for linear
+elements. Therefore, for pyramids we are actually integrating a polynomial of
+higher degree than the degrees P and Q given in the table above.
+
+## Extrapolation of integration point data to mesh nodes
+
+OpenGeoSys can extrapolate integration point data to mesh nodes for easy
+postprocessing. **Note, however, that the extrapolation procedures is not exact
+and can lead to more or less subtle errors that are hard to find!**
+
+Since OpenGeoSys extrapolates integration point data element-wise, the number of
+integration points of each mesh element must be greater or equal to the number of nodes of the element.
+Therefore, the ability to extrapolate data is linked to the chosen integration
+order, i.e., the number of integration points must be greater or equal to the
+number of nodes. The relation is presented in the table below.
+
+The columns contain the following data:
+
+* #IP: The number of integration points of the integration method.
+* L:
+  Whether extrapolation can be performed on the *linear* elements (e.g. Quad4) of the
+  respective type with the given integration order.
+* Q:
+  Whether extrapolation can be performed on the *quadratic* elements (e.g.
+  Hex20) of the respective type with the given integration order.
+
+As you can see, for integration order ≥ 2 extrapolation works for all linear
+elements and for integration order ≥ 3 extrapolation works for all element types
+implemented in OpenGeoSys.
+
+| Integration order → | 1       | 1     | 1     | 2       | 2     | 2     | 3       | 3     | 3     | 4       | 4     | 4     |
+| ------------------- | ------: | :---: | :---: | ------: | :---: | :---: | ------: | :---: | :---: | ------: | :---: | :---: |
+| **Mesh element ↓**  | **#IP** | **L** | **Q** | **#IP** | **L** | **Q** | **#IP** | **L** | **Q** | **#IP** | **L** | **Q** |
+| Point               | 1       |   ✓   |   ✓   | 1       |   ✓   |   ✓   | 1       |   ✓   |   ✓   | 1       |   ✓   |   ✓   |
+| Line                | 1       |   –   |   –   | 2       |   ✓   |   –   | 3       |   ✓   |   ✓   | 4       |   ✓   |   ✓   |
+| Quad                | 1       |   –   |   –   | 4       |   ✓   |   –   | 9       |   ✓   |   ✓   | 16      |   ✓   |   ✓   |
+| Hexahedron          | 1       |   –   |   –   | 8       |   ✓   |   –   | 27      |   ✓   |   ✓   | 64      |   ✓   |   ✓   |
+| Triangle            | 1       |   –   |   –   | 3       |   ✓   |   –   | 4       |   ✓   |   ✓   | 7       |   ✓   |   ✓   |
+| Tetrahedron         | 1       |   –   |   –   | 5       |   ✓   |   –   | 14      |   ✓   |   ✓   | 20      |   ✓   |   ✓   |
+| 3–sided prism       | 1       |   –   |   –   | 6       |   ✓   |   –   | 21      |   ✓   |   ✓   | 28      |   ✓   |   ✓   |
+| Pyramid             | 1       |   –   |   –   | 5       |   ✓   |   –   | 13      |   ✓   |   ✓   | 13      |   ✓   |   ✓   |
+