From 190bcab9a1aa753cef728e473cdaf7b6756b51fa Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Tue, 28 Jul 2015 12:00:30 -0500 Subject: [PATCH] Minor updates to step-10. In particular, reference step-53. --- examples/step-10/doc/intro.dox | 30 ++++++++++++++++++++---------- 1 file changed, 20 insertions(+), 10 deletions(-) diff --git a/examples/step-10/doc/intro.dox b/examples/step-10/doc/intro.dox index 2749945848..5236b0f708 100644 --- a/examples/step-10/doc/intro.dox +++ b/examples/step-10/doc/intro.dox @@ -22,11 +22,11 @@ approximation, and so on. For some differential equations, it is known that piecewise linear approximations of the boundary, i.e. $Q_1$ mappings, are not -sufficient if the boundary of the domain is curved. Examples are the +sufficient if the boundary of the exact domain is curved. Examples are the biharmonic equation using $C^1$ elements, or the Euler -equation on domains with curved reflective boundaries. In these cases, +equations of gas dynamics on domains with curved reflective boundaries. In these cases, it is necessary to compute the integrals using a higher order -mapping. The reason, of course, is that if we do not use a higher +mapping. If we do not use such a higher order mapping, the order of approximation of the boundary dominates the order of convergence of the entire numerical scheme, irrespective of the order of convergence of the discretization in the interior of @@ -42,20 +42,30 @@ different methods. The first method uses a triangulated approximation of the circle with -unit radius and integrates the unit function over it. Of course, if -the domain were the exact unit circle, then the area would be pi, but +unit radius and integrates the function that is constant one over it. Of course, if +the domain were the exact unit circle, then the area would be $\pi$, but since we only use an approximation by piecewise polynomial segments, -the value of the area is not exactly pi. However, it is known that as +the value of the area we integrate over is not exactly $\pi$. However, it is known that as we refine the triangulation, a $Q_p$ mapping approximates the boundary with an order $h^{p+1}$, where $h$ is the mesh -width. We will check the values of the computed area of the circle and -their convergence towards pi under mesh refinement for different +size. We will check the values of the computed area of the circle and +their convergence towards $\pi$ under mesh refinement for different mappings. We will also find a convergence behavior that is surprising at first, but has a good explanation. The second method works similarly, but this time does not use the area -of the triangulated unit circle, but rather its perimeter. Pi is then -approximated by half of the perimeter, as the radius is equal to one. +of the triangulated unit circle, but rather its perimeter. $\pi$ is then +approximated by half of the perimeter, as we choose the radius equal to one. + +@note This tutorial shows in essence how to choose a particular +mapping for integrals, by attaching a particular geometry to the +triangulation (as had already been done in step-1, for example) and +then passing a mapping argument to the FEValues class that is used for +all integrals in deal.II. The geometry we choose is a circle, for +which deal.II already has a class (SphericalManifold) that can be +used. If you want to define your own geometry, for example because it +is complicated and cannot be described by the classes already +available in deal.II, you will want to read through step-53. -- 2.39.5