From a6418290ad26cc4382ec158f7a4421b6ea777646 Mon Sep 17 00:00:00 2001 From: Martin Kronbichler Date: Fri, 15 Dec 2017 16:44:30 +0100 Subject: [PATCH] Extend discussion on what happens if manifolds are blended in a bad way. --- doc/doxygen/headers/manifold.h | 22 ++++++++++++++++++---- 1 file changed, 18 insertions(+), 4 deletions(-) diff --git a/doc/doxygen/headers/manifold.h b/doc/doxygen/headers/manifold.h index 92b6f54e74..a08a7804cc 100644 --- a/doc/doxygen/headers/manifold.h +++ b/doc/doxygen/headers/manifold.h @@ -350,13 +350,27 @@ * version 9.0 deal.II used something called Laplace smoothing where the * weights that are applied to the nodes on the circumference to get the * position of the interior nodes are determined by solving a Laplace equation - * on the unit element. However, this did lead to boundary layers close to the + * on the unit element. However, this lead to boundary layers close to the * curved description, i.e., singularities in the higher derivatives of the * mapping from unit to real cell. * - * For example, the above case with only 3 circumferential cells leads to the - * following mesh with Laplace smoothing rather than the interpolation from - * the boundary: + * If the transition from a curved boundary description to a straight + * description in the interior is done wrong, it is typically impossible to + * achieve high order convergence rates. For example, the Laplace smoothing + * inside a single layer lead to a singularity in the fourth derivative of the + * mapping from the reference to the real cell, limiting the convergence rate + * to 3 in the cells at the boundary (and 3.5 if global L2 errors were + * measured in 2D). Other more crude strategies, like completely ignoring the + * manifold for the additional points of a high-order mapping, could lead to + * even worse convergence rates. The current implementation in deal.II, on the + * other hand, has been extensively verified in this respect and should behave + * optimally. + * + * A bad strategy for blending a curved boundary representation with flat + * interior representations obviously also reflects mesh quality. For example, + * the above case with only 3 circumferential cells leads to the following + * mesh with Laplace smoothing rather than the interpolation from the + * boundary: * * @image html hypershell-boundary-only-3-old.png "" * -- 2.39.5