From: bangerth Date: Wed, 30 Aug 2006 02:42:25 +0000 (+0000) Subject: Add a bit of text about a puzzling observation X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=4f3f997ec786bcc4c0f6818fae137b7c88774af5;p=dealii-svn.git Add a bit of text about a puzzling observation git-svn-id: https://svn.dealii.org/trunk@13783 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/deal.II/include/fe/fe_dgp.h b/deal.II/deal.II/include/fe/fe_dgp.h index 0dec746f79..6c863c4032 100644 --- a/deal.II/deal.II/include/fe/fe_dgp.h +++ b/deal.II/deal.II/include/fe/fe_dgp.h @@ -55,6 +55,29 @@ template class MappingQ; * be exactly represented by elements of type FE_DGP(1) or * FE_DGPMonomial(1). * + * This can be understood by the following 2-d example: consider the + * cell with vertices at $(0,0),(1,0),(0,1),(s,s)$: + * @image html dgp_doesnt_contain_p.png + * + * For this cell, a bilinear transformation $F$ produces the relations + * $x=\hat x+\hat x\hat y$ and $y=\hat y+\hat x\hat y$ that correlate + * reference coordinates $\hat x,\hat y$ and coordinates in real space + * $x,y$. Under this mapping, the constant function is clearly mapped + * onto itself, but the two other shape functions of the $P_1$ space, + * namely $\phi_1(\hat x,\hat y)=\hat x$ and $\phi_2(\hat x,\hat + * y)=\hat y$ are mapped onto + * $\phi_1(x,y)=\frac{x-t}{t(s-1)},\phi_2(x,y)=t$ where + * $t=\frac{y}{s-x+sx+y-sy}$. + * + * For the simple case that $s=1$, i.e. if the real cell is the unit + * square, the expressions can be simplified to $t=y$ and + * $\phi_1(x,y)=x,\phi_2(x,y)=y$. However, for all other cases, the + * functions $\phi_1(x,y),\phi_2(x,y)$ are not linear any more, and + * neither is any linear combincation of them. Consequently, the + * linear functions are not within the range of the mapped $P_1$ + * polynomials. + * + * * @author Guido Kanschat, 2001, 2002, Ralf Hartmann 2004 */ template diff --git a/deal.II/deal.II/include/fe/fe_dgp_monomial.h b/deal.II/deal.II/include/fe/fe_dgp_monomial.h index 00e360abdd..8a0774f27a 100644 --- a/deal.II/deal.II/include/fe/fe_dgp_monomial.h +++ b/deal.II/deal.II/include/fe/fe_dgp_monomial.h @@ -54,6 +54,29 @@ template class MappingQ; * example, on meshes with non-affine cells, a linear function can not * be exactly represented by elements of type FE_DGP(1) or * FE_DGPMonomial(1). + * + * This can be understood by the following 2-d example: consider the + * cell with vertices at $(0,0),(1,0),(0,1),(s,s)$: + * @image html dgp_doesnt_contain_p.png + * + * For this cell, a bilinear transformation $F$ produces the relations + * $x=\hat x+\hat x\hat y$ and $y=\hat y+\hat x\hat y$ that correlate + * reference coordinates $\hat x,\hat y$ and coordinates in real space + * $x,y$. Under this mapping, the constant function is clearly mapped + * onto itself, but the two other shape functions of the $P_1$ space, + * namely $\phi_1(\hat x,\hat y)=\hat x$ and $\phi_2(\hat x,\hat + * y)=\hat y$ are mapped onto + * $\phi_1(x,y)=\frac{x-t}{t(s-1)},\phi_2(x,y)=t$ where + * $t=\frac{y}{s-x+sx+y-sy}$. + * + * For the simple case that $s=1$, i.e. if the real cell is the unit + * square, the expressions can be simplified to $t=y$ and + * $\phi_1(x,y)=x,\phi_2(x,y)=y$. However, for all other cases, the + * functions $\phi_1(x,y),\phi_2(x,y)$ are not linear any more, and + * neither is any linear combincation of them. Consequently, the + * linear functions are not within the range of the mapped $P_1$ + * polynomials. + * * * @author Ralf Hartmann, 2004 */