From: Wolfgang Bangerth Date: Wed, 10 Sep 2014 00:46:14 +0000 (-0500) Subject: Add a section discussing consequences. X-Git-Tag: v8.2.0-rc1~155^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F140%2Fhead;p=dealii.git Add a section discussing consequences. This pertains in particular to the question of getting discontinuous geometries when using manifolds and a discussion of what that means, if it is important, and if it could be fixed. --- diff --git a/examples/step-53/doc/results.dox b/examples/step-53/doc/results.dox index 6f7dc62860..beaaf746e8 100644 --- a/examples/step-53/doc/results.dox +++ b/examples/step-53/doc/results.dox @@ -120,3 +120,163 @@ values, we just go down this list and set these elevations in the their original zero value). This vector is then output using DataOut as usual and can be visualized as shown above. + +

Issues with adaptively refined meshes generated this way

+ +If you zoomed in on the mesh shown above and looked closely enough, you would +find that at hanging nodes, the two small edges connecting to the hanging +nodes are not in exactly the same location as the large edge of the +neighboring cell. This can be shown more clearly by using a different surface +description in which we enlarge the vertical topography to enhance the effect +(courtesy of Alexander Grayver): + + + +So what is happening here? Partly, this is only a result of visualization, but +there is an underlying real cause as well: + + + +The situation is slightly more complicated if you use a higher order mapping +using the MappingQ class, but not fundamentally different. Let's take a +quadratic mapping for the moment (nothing fundamental changes with even higher +order mappings). Then you need to imagine each edge of the cells you integrate +on as a quadratic curve despite the fact that you will never actually see it +plotted that way by a visualization program. But imagine it that way for a +second. So which quadratic curve does MappingQ take? It is the quadratic curve +that goes through the two vertices at the end of the edge as well as a point +in the middle that it queries from the manifold. In the case of the long edge +on the unrefined side, that's of course exactly the location of the hanging +node, so the quadratic curve describing the long edge does go through the +hanging node, unlike in the case of the linear mapping. But the two small +edges are also quadratic curves; for example, the left small edge will go +through the left vertex of the long edge and the hanging node, plus a point it +queries halfway in between from the manifold. Because, as before, the point +the manifold returns halfway along the left small edge is rarely exactly on +the quadratic curve describing the long edge, the quadratic short edge will +typically not coincide with the left half of the quadratic long edge, and the +same is true for the right short edge. In other words, again, the geometries +of the large cell and its smaller neighbors at hanging nodes do not touch +snuggly. + +This all begs two questions: first, does it matter, and second, could this be +fixed. Let us discuss these in the following: + +