From: Martin Kronbichler Date: Mon, 8 Jul 2019 16:02:06 +0000 (+0200) Subject: Fix documentation of HermiteLikeInterpolation X-Git-Tag: v9.2.0-rc1~1405^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F8355%2Fhead;p=dealii.git Fix documentation of HermiteLikeInterpolation --- diff --git a/include/deal.II/base/polynomial.h b/include/deal.II/base/polynomial.h index 93de2a51e6..4ba120336b 100644 --- a/include/deal.II/base/polynomial.h +++ b/include/deal.II/base/polynomial.h @@ -602,13 +602,13 @@ namespace Polynomials * Polynomials for a variant of Hermite polynomials with better condition * number in the interpolation than the basis from HermiteInterpolation. * - * In analogy to the actual Hermite polynomials this basis evaluates the + * In analogy to the proper Hermite polynomials, this basis evaluates the * first polynomial $p_0$ to 1 at $x=0$ and has both a zero value and zero * derivative at $x=1$. Likewise, the last polynomial $p_n$ evaluates to 1 - * at $x=1$ but has zero value and zero derivative at $x=0$. The second + * at $x=1$ with a zero value and zero derivative at $x=0$. The second * polynomial $p_1$ and the second to last polynomial $p_{n-1}$ represent - * the derivative degree of freedom at $x=0$ and $x=1$, respectively. As - * such, they are zero at both the end points $x=0, x=1$ and have zero + * the derivative degree of freedom at $x=0$ and $x=1$, respectively. + * They are zero at both the end points $x=0, x=1$ and have zero * derivative at the opposite end, $p_1'(1)=0$ and $p_{n-1}'(0)=0$. As * opposed to the original Hermite polynomials, $p_0$ does not have zero * derivative at $x=0$. The additional degree of freedom is used to make @@ -617,7 +617,7 @@ namespace Polynomials * respectively. Furthermore, the extension of these polynomials to higher * degrees $n>3$ is constructed by adding additional nodes inside the unit * interval, again ensuring better conditioning. The nodes are computed as - * the roots of the Jacobi polynomials for $\alpha=\beta=2$ which are + * the roots of the Jacobi polynomials for $\alpha=\beta=2$, which are * orthogonal against the generating function $x^2(1-x)^2$ with the Hermite * property. Then, these polynomials are constructed in the usual way as * Lagrange polynomials with double roots at $x=0$ and $x=1$. For example at @@ -625,12 +625,14 @@ namespace Polynomials * through the factor $(x-0.5)$. In summary, this basis is dominated by * nodal contributions, but it is not a nodal one because the second and * second to last polynomials that are non-nodal, and due to the presence of - * double nodes in $x=0$ and $x=1$. + * double nodes in $x=0$ and $x=1$. The weights of the basis functions are + * set such that the sum of all polynomials with unit weight represents the + * constant function 1, similarly to Lagrange polynomials. * * The basis only contains Hermite information at degree>=3, * but it is also implemented for degrees between 0 and two. For the linear * case, the usual hat functions are implemented, whereas the polynomials - * for degree=2 are $p_0(x)=(1-x)^2$, $p_1(x)=4x(x-1)$, and + * for degree=2 are $p_0(x)=(1-x)^2$, $p_1(x)=2x(x-1)$, and * $p_2(x)=x^2$, in accordance with the construction principle for degree 3 * that allows a non-zero of $p_0$ and $p_2$. * @@ -692,9 +694,10 @@ namespace Polynomials * * This polynomial inherits the advantageous property of Hermite polynomials * where only two functions have value and/or derivative nonzero on a face + * advantageous for discontinuous Galerkin methods * but gives better condition numbers of interpolation, which improves the * performance of some iterative schemes like conjugate gradients with - * point-Jacobi. + * point-Jacobi. This polynomial is used in FE_DGQHermite. * * @author Martin Kronbichler * @date 2018