*
* Thus for the P1 nonconforming element, the function values at midpoints on edges of a cell are important.
* The first attempt to define (local) degrees of freedom (DOFs) on a quadrilateral
- * is by using midpoint values of a function as usual nonconforming finite elements.
+ * is by using midpoint values of a function.
*
* However, these 4 functionals are not linearly independent
* because a linear function on 2D is uniquely determined by only 3 independent values.
* and it is same as the dimension of the linear polynomial space on a cell in 2D.
*
* <h3>Shape functions</h3>
- * Before introducing the DOFs, we present 4 local shape functions on a cell.
+ * Before introduction of the DOFs, we present 4 local shape functions on a cell.
* Due to the dice rule, we need a special construction for shape functions.
* Although the following 4 shape functions are not linearly independent within a cell,
* they are helpful to define the global basis functions which are linearly independent on whole domain.
* We denote such a function associated with vertex $v_j$ by $\phi_j$.
* Then the set of 4 shape functions is a partition of unity on a cell: $\sum_{j=0}^{3} \phi_j = 1$.
*
- * The following figures represent $\phi_j$, $j=0,\cdots,3$ with its values at midpoints.
+ * The following figures represent $\phi_j$ for $j=0,\cdots,3$ with its midpoint values.
*
* <ul>
* <li> shape function $\phi_0$:
*
* We want to emphasize that the shape functions are constructed on each cell, not on the reference cell only.
* Usual finite elements are defined based on a 'parametric' concept.
- * That means, a function space for a finite element is defined on one reference cell, and it is transfomed
+ * It means that a function space for a finite element is defined on one reference cell, and it is transfomed
* into each cell via a mapping from the reference cell.
* However the P1 nonconforming element does not follow such concept. It defines a function space with
* linear shape functions on each cell without any help of a function space on the reference cell.
*
* When the Neumann boundary condition is given,
* the global basis functions associated with all nodes (including boundary nodes)
- * are actually not linearly independent. There exists 1 redundancy.
+ * are actually not linearly independent. There exists one redundancy.
* Thus in this case, the number of DOFs is equal to the number of all nodes minus 1.
*
* <h3>Unit support points</h3>
* For a smooth function, we construct a piecewise linear function which belongs to the element space by
* using its nodal values as DOF values.
- * This interpolation is implemented by using appropriate @p unit_support_points.
*
- * Note that two nodal values of a smooth function and its interpolant do not coincide in general,
- * contrast with ordinary Lagrange finite elements.
+ * Note that for the P1 nonconforming element two nodal values of a smooth function and its interpolant do not
+ * coincide in general, contrast with ordinary Lagrange finite elements.
* Of course, it is meaningless to refer 'nodal value' because the element space has nonconformity.
* But it is also true even though the single global basis function associated a node is considered
- * with a valid expression 'nodal value'.
+ * with the unique 'nodal value' at the node.
* For instance, consider the basis function associated with a node.
* Consider two lines representing the level sets for value 0.5 and 0, respectively, by connecting two midpoints.
* Then we cut the quad into two sub-triangles by the diagonal which is placed along those two lines.
* It gives another level set for value 0.25 which coincides with the cutting diagonal.
- * Therefore these three level sets are all parallel and it gives the value 0.75 at the base node, not value 1.
+ * Therefore these three level sets are all parallel in the quad and it gives the value 0.75 at the base node, not value 1.
* Even though a general quad is given, this is also true.
*
* <h3>References</h3>
- * You can find the paper about the P1NC element
- * Park & Sheen (2003). P1-nonconforming quadrilateral finite element methods for second-order elliptic problems.
- * SIAM Journal on Numerical Analysis, 41(2), 624-640,
- * available at http://epubs.siam.org/doi/abs/10.1137/S0036142902404923.
+ * You can find the original paper for the P1 nonconforming element which is
+ * accessible at http://epubs.siam.org/doi/abs/10.1137/S0036142902404923.
+ * @code(.bib)
+ * @article{park2003p,
+ * title = {P 1-nonconforming quadrilateral finite element methods for second-order elliptic problems},
+ * author = {Park, Chunjae and Sheen, Dongwoo},
+ * journal = {SIAM Journal on Numerical Analysis},
+ * volume = {41},
+ * number = {2},
+ * pages = {624--640},
+ * year = {2003},
+ * publisher = {SIAM}
+ * }
+ * @endcode
*
*/