From: bangerth Date: Mon, 28 Aug 2006 19:48:29 +0000 (+0000) Subject: Update comments X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=970221effd5eec180ac112a3e475877eccd64be3;p=dealii-svn.git Update comments git-svn-id: https://svn.dealii.org/trunk@13769 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 da97d77997..bf38c7d0cc 100644 --- a/deal.II/deal.II/include/fe/fe_dgp.h +++ b/deal.II/deal.II/include/fe/fe_dgp.h @@ -26,9 +26,18 @@ template class MappingQ; * Discontinuous finite elements based on Legendre polynomials. * * This finite element implements complete polynomial spaces, that is, - * dim-dimensional polynomials of degree p. The underlying - * polynomials form a Legendre basis on the unit square. Thus, the - * mass matrix is diagonal, if the grid cells are parallelograms. + * dim-dimensional polynomials of degree p. For example, in 2d the + * element FE_DGP(1) would represent the span of the functions + * $\{1,\hat x,\hat y\}$, which is in contrast to the element FE_DGQ(1) + * that is formed by the span of $\{1,\hat x,\hat y,\hat x\hat y\}$. Since the + * DGP space has only three unknowns for each quadrilateral, it is + * immediately clear that this element can not be continuous. + * + * The basis functions for this element are chosen to form a Legendre + * basis on the unit square. Thus, the mass matrix is diagonal, if the + * grid cells are parallelograms. Note that this is in contrast to the + * FE_DGPMonomial class that actually uses the monomial basis listed + * above as basis functions. * * The shape functions are defined in the class PolynomialSpace. The * polynomials used inside PolynomialSpace are Polynomials::Legendre @@ -36,6 +45,16 @@ template class MappingQ; * basis functions, refer to PolynomialSpace, remebering that the * Legendre polynomials are ordered by ascending degree. * + * + *

Transformation properties

+ * + * It is worth noting that under a (bi-, tri-)linear mapping, the + * space described by this element does not contain $P(k)$, even if we + * use a basis of polynomials of degree $k$. Consequently, for + * 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). + * * @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 6e041c822d..114c87a22f 100644 --- a/deal.II/deal.II/include/fe/fe_dgp_monomial.h +++ b/deal.II/deal.II/include/fe/fe_dgp_monomial.h @@ -24,14 +24,37 @@ template class MappingQ; /*@{*/ /** - * Discontinuous finite elements based on monomials of degree up to - * p. + * Discontinuous finite elements based on monomials. * - * This finite element makes use of the PolynomialsP class which - * implements dim-dimensional polynomials of degree - * p based the Polynomials::Polynomial and the - * PolynomialSpace classes. + * This finite element implements complete polynomial spaces, that is, + * dim-dimensional polynomials of degree p. For example, in 2d the + * element FE_DGP(1) would represent the span of the functions + * $\{1,\hat x,\hat y\}$, which is in contrast to the element FE_DGQ(1) + * that is formed by the span of $\{1,\hat x,\hat y,\hat x\hat y\}$. Since the + * DGP space has only three unknowns for each quadrilateral, it is + * immediately clear that this element can not be continuous. * + * The basis functions for this element are chosen to be the monomials + * listed above. Note that this is the main difference to the FE_DGP + * class that uses a set of polynomials of complete degree + * p that form a Legendre basis on the unit square. Thus, + * there, the mass matrix is diagonal, if the grid cells are + * parallelograms. The basis here does not have this property; + * however, it is simpler to compute. On the other hand, this element + * has the additional disadvantage that the local cell matrices + * usually have a worse condition number than the ones originating + * from the FE_DGP element. + * + * + *

Transformation properties

+ * + * It is worth noting that under a (bi-, tri-)linear mapping, the + * space described by this element does not contain $P(k)$, even if we + * use a basis of polynomials of degree $k$. Consequently, for + * 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). + * * @author Ralf Hartmann, 2004 */ template