Add a few references to the FE_Q_iso_Q1 element documentation.

diff --git a/doc/doxygen/references.bib b/doc/doxygen/references.bib
index b425869a1d0e0f333e7724be60741e68036ed57e..0635cfc9498fd65ae303922cbd6349d579d76c86 100644 (file)
--- a/doc/doxygen/references.bib
+++ b/doc/doxygen/references.bib
@@ -1478,3 +1478,40 @@
   year={2019},
   publisher={ACM New York, NY, USA}
 }
+
+@article{Boffi2011,
+  doi = {10.1007/s10915-011-9549-4},
+  url = {https://doi.org/10.1007/s10915-011-9549-4},
+  year = {2011},
+  month = nov,
+  publisher = {Springer Science and Business Media {LLC}},
+  volume = {52},
+  number = {2},
+  pages = {383--400},
+  author = {D. Boffi and N. Cavallini and F. Gardini and L. Gastaldi},
+  title = {Local Mass Conservation of Stokes Finite Elements},
+  journal = {Journal of Scientific Computing}
+}
+
+@article{Taylor73,
+title={{A numerical solution of the Navier-Stokes equations using the finite element technique}},
+author={C. Taylor and P. Hood},
+journal={Comput. Fluids},
+volume={1},
+number={},
+pages={73--100},
+year={1973}}
+
+@article{Bercovier1979,
+  doi = {10.1007/bf01399555},
+  url = {https://doi.org/10.1007/bf01399555},
+  year = {1979},
+  month = jun,
+  publisher = {Springer Science and Business Media {LLC}},
+  volume = {33},
+  number = {2},
+  pages = {211--224},
+  author = {M. Bercovier and O. Pironneau},
+  title = {Error estimates for finite element method solution of the Stokes problem in the primitive variables},
+  journal = {Numerische Mathematik}
+}
\ No newline at end of file

diff --git a/include/deal.II/fe/fe_q_iso_q1.h b/include/deal.II/fe/fe_q_iso_q1.h
index 244b46e5cf3eb55adc72da3ea2296089ee412566..4d8bcdf426e57299b85e8c6429bd8e0cf8def36f 100644 (file)
--- a/include/deal.II/fe/fe_q_iso_q1.h
+++ b/include/deal.II/fe/fe_q_iso_q1.h
@@ -30,20 +30,24 @@ DEAL_II_NAMESPACE_OPEN
 /*@{*/
 
 /**
- * Implementation of a scalar Lagrange finite element @p Qp-iso-Q1 that
- * defines the finite element space of continuous, piecewise linear elements
- * with @p p subdivisions in each coordinate direction. It yields an element
- * with the same number of degrees of freedom as the @p Qp elements but using
- * linear interpolation instead of higher order one. This type of element is
- * also called macro element in the literature as it really consists of
- * several smaller elements, namely <i>p</i><tt><sup>dim</sup></tt> such
- * sub-cells.
+ * Implementation of a scalar Lagrange finite element @p Qp-iso-Q1
+ * that defines the finite element space of continuous, piecewise
+ * linear elements with @p p subdivisions in each coordinate
+ * direction. It yields an element with the same number of degrees of
+ * freedom as the @p Qp elements but using linear interpolation
+ * instead of higher order one. In other words, on every cell, the
+ * shape functions are not of higher order polynomial degree
+ * interpolating a set of node points, but are piecewise (bi-,
+ * tri-)linear *within* the cell and interpolating the same set of
+ * node points. This type of element is also called *macro element* in
+ * the literature as it can be seen as consisting of several smaller
+ * elements, namely <i>p</i><tt><sup>dim</sup></tt> such sub-cells.
  *
  * The numbering of degrees of freedom is done in exactly the same way as in
  * FE_Q of degree @p p. See there for a detailed description on how degrees of
  * freedom are numbered within one element.
  *
- * This element represents a Q-linear finite element space on a reduced mesh
+ * This element represents a Q-linear finite element space on a reduced mesh of
  * size <i>h/p</i>. Its effect is equivalent to using FE_Q of degree one on a
  * finer mesh by a factor @p p if an equivalent quadrature is used. However,
  * this element reduces the flexibility in the choice of (adaptive) mesh size
@@ -51,7 +55,7 @@ DEAL_II_NAMESPACE_OPEN
  * other hand, comparing this element with @p p subdivisions to the FE_Q
  * element of degree @p p on the same mesh shows that the convergence is
  * typically much worse for smooth problems. In particular, @p Qp elements
- * achieve interpolation orders of <i>h<sup>p+1</sup></i> in the L2 norm,
+ * achieve interpolation orders of <i>h<sup>p+1</sup></i> in the $L_2$ norm,
  * whereas these elements reach only <i>(h/p)<sup>2</sup></i>. For these two
  * reasons, this element is usually not very useful as a standalone. In
  * addition, any evaluation of face terms on the boundaries within the
@@ -71,11 +75,19 @@ DEAL_II_NAMESPACE_OPEN
  * solution and stabilization techniques are used that work for linears but
  * not higher order elements. </li>
  *
- * <li> Stokes/Navier Stokes systems such as the one discussed in step-22 could be
- * solved with Q2-iso-Q1 elements for velocities instead of Q2 elements.
- * Combined with Q1 pressures they give a stable mixed element pair. However,
- * they perform worse than the standard (Taylor-Hood $Q_2\times Q_1$)
- * approach in most situations.  </li>
+ * <li> Stokes/Navier Stokes systems such as the one discussed in
+ * step-22 could be solved with Q2-iso-Q1 elements for velocities
+ * instead of $Q_2$ elements.  Combined with $Q_1$ pressures they give
+ * a stable mixed element pair. However, they perform worse than the
+ * standard (Taylor-Hood $Q_2\times Q_1$) approach in most
+ * situations. (See, for example, @cite Boffi2011 .)  This combination
+ * of subdivided elements for the velocity and non-subdivided elements
+ * for the pressure is sometimes called the "Bercovier-Pironneau
+ * element" and dates back to around the same time as the Taylor-Hood
+ * element (namely, the mid-1970s). For more information, see the
+ * paper by Bercovier and Pironneau from 1979 @cite Bercovier1979, and
+ * for the origins of the comparable Taylor-Hood element see
+ * @cite Taylor73 from 1973.</li>
  *
  * <li> Preconditioning systems of FE_Q systems of higher order @p p with a
  * preconditioner based on @p Qp-iso-Q1 elements: Some preconditioners like