From: Wolfgang Bangerth Date: Thu, 16 Sep 2021 21:40:30 +0000 (-0600) Subject: Add a few references to the FE_Q_iso_Q1 element documentation. X-Git-Tag: v9.4.0-rc1~982^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F12765%2Fhead;p=dealii.git 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 b425869a1d..0635cfc949 100644 --- 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 244b46e5cf..4d8bcdf426 100644 --- 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 pdim 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 pdim 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 h/p. 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 hp+1 in the L2 norm, + * achieve interpolation orders of hp+1 in the $L_2$ norm, * whereas these elements reach only (h/p)2. 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. * - *
  • 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.
    • + *
  • 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.
    • * *
  • Preconditioning systems of FE_Q systems of higher order @p p with a * preconditioner based on @p Qp-iso-Q1 elements: Some preconditioners like