From: Wolfgang Bangerth Date: Thu, 8 Sep 2016 22:51:30 +0000 (-0600) Subject: Add a discussion about smoothness to step-6. X-Git-Tag: v8.5.0-rc1~671^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f9f8cb0fef4978c7e59028338649022e2bb6bfc2;p=dealii.git Add a discussion about smoothness to step-6. In particular, extend the 'Possibilities for extensions' section by a discussion of how to make the solution less smooth. --- diff --git a/examples/step-6/doc/results.dox b/examples/step-6/doc/results.dox index 0e154545a9..b41f3cf9f2 100644 --- a/examples/step-6/doc/results.dox +++ b/examples/step-6/doc/results.dox @@ -483,3 +483,102 @@ want, is a complex topic in itself. You can find much more on this in step-49, step-53, and step-54, among other tutorial programs that cover the issue. Information on curved domains can also be found in the documentation module on @ref manifold "Manifold descriptions". + + + +

Playing with the regularity of the solution

+ +From a mathematical perspective, solutions of the Laplace equation +@f[ + -\Delta u = f +@f] +on smoothly bounded, convex domains are known to be smooth themselves. The exact degree +of smoothness, i.e., the function space in which the solution lives, depends +on how smooth exactly the boundary of the domain is, and how smooth the right +hand side is. Some regularity of the solution may be lost at the boundary, but +one generally has that the solution is twice more differentiable in +compact subsets of the domain than the right hand side. +If, in particular, the right hand side satisfies $f\in C^\infty(\Omega)$, then +$u \in C^\infty(\Omega_i)$ where $\Omega_i$ is any compact subset of $\Omega$ +($\Omega$ is an open domain, so a compact subset needs to keep a positive distance +from $\partial\Omega$). + +The situation we chose for the current example is different, however: we look +at an equation with a non-constant coefficient $a(\mathbf x)$: +@f[ + -\nabla \cdot (a \nabla u) = f. +@f] +Here, if $a$ is not smooth, then the solution will not be smooth either, +regardless of $f$. In particular, we expect that wherever $a$ is discontinuous +along a line (or along a plane in 3d), +the solution will have a kink. This is easy to see: if for example $f$ +is continuous, then $f=-\nabla \cdot (a \nabla u)$ needs to be +continuous. This means that $a \nabla u$ must be continuously differentiable +(not have a kink). Consequently, if $a$ has a discontinuity, then $\nabla u$ +must have an opposite discontinuity so that the two exactly cancel and their +product yields a function without a discontinuity. But for $\nabla u$ to have +a discontinuity, $u$ must have a kink. This is of course exactly what is +happening in the current example, and easy to observe in the pictures of the +solution. + +In general, if the coefficient $a(\mathbf x)$ is discontinuous along a line in 2d, +or a plane in 3d, then the solution may have a kink, but the gradient of the +solution will not go to infinity. That means, that the solution is at least +still in the space $W^{1,\infty}$. On the other hand, we know that in the most +extreme cases -- i.e., where the domain has reentrant corners, the +right hand side only satisfies $f\in H^{-1}$, or the coefficient $a$ is only in +$L^\infty$ -- all we can expect is that $u\in H^1$, a much larger space than +$W^{1,\infty}$. It is not very difficult to create cases where +the solution is in a space $H^{1+s}$ where we can get $s$ to become as small +as we want. Such cases are often used to test adaptive finite element +methods because the mesh will have to resolve the singularity that causes +the solution to not be in $W^{1,\infty}$ any more. + +The typical example one uses for this is called the Kellogg problem +(referring to the paper "On the Poisson equation with intersecting interfaces" +by R. B. Kellogg, Applicable Analysis, vol. 4, pp. 101-129, 1974), which +in the commonly used form has a coefficient $a(\mathbf x)$ that has different values +in the four quadrants of the plane (or eight different values in the octants +of ${\mathbb R}^3$). The exact degree of regularity (the $s$ in the +index of the Sobolev space above) depends on the values of $a(\mathbf x)$ coming +together at the origin, and by choosing the jumps large enough, the +regularity of the solution can be made as close as desired to $H^1$. + +To implement something like this, one could replace the coefficient +function by the following (shown here only for the 2d case): +@code +template +double coefficient (const Point &p) +{ + if ((p[0] < 0) && (p[1] < 0)) // lower left quadrant + return 1; + else if ((p[0] >= 0) && (p[1] < 0)) // lower right quadrant + return 10; + else if ((p[0] < 0) && (p[1] >= 0)) // upper left quadrant + return 100; + else if ((p[0] >= 0) && (p[1] >= 0)) // upper right quadrant + return 1000; + else + { + Assert (false, ExcInternalError()); + return 0; + } +} +@endcode +(Adding the Assert at the end ensures that an exception +is thrown if we ever get to that point -- which of course we shouldn't, +but this is a good way to insure yourself: we all make mistakes by +sometimes not thinking of all cases, for example by checking +for p[0] to be less than and greater than zero, +rather than greater-or-equal to zero, and thereby forgetting +some cases that would otherwise lead to bugs that are awkward +to find. The return 0; at the end is only there to +avoid compiler warnings that the function does not end in a +return statement -- the compiler cannot see that the +function would never actually get to that point because of the +preceding Assert statement.) + +By playing with such cases where four or more sectors come +together and on which the coefficient has different values, one can +construct cases where the solution has singularities at the +origin. One can also see how the meshes are refined in such cases.