From fea65d9ddb62942c3c4daaee272340c0ced00b58 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Mon, 4 Jan 2021 17:39:06 -0700 Subject: [PATCH] Better explain the purpose of the L-singularity function. --- include/deal.II/base/function_lib.h | 24 +++++++++++++++++++++--- 1 file changed, 21 insertions(+), 3 deletions(-) diff --git a/include/deal.II/base/function_lib.h b/include/deal.II/base/function_lib.h index e4b9a83c60..589a2fd654 100644 --- a/include/deal.II/base/function_lib.h +++ b/include/deal.II/base/function_lib.h @@ -381,11 +381,29 @@ namespace Functions /** - * Harmonic singularity on the L-shaped domain in 2D. + * A function that solves the Laplace equation (with specific + * boundary values but zero right hand side) and that has a + * singularity at the center of the L-shaped domain in 2D (i.e., + * at the location of the re-entrant corner of this non-convex + * domain). * * The function is given in polar coordinates by $r^{\frac{2}{3}} - * \sin(\frac{2}{3} \phi)$ with a singularity at the origin and should be - * used with GridGenerator::hyper_L(). + * \sin(\frac{2}{3} \phi)$ with a singularity at the origin and + * should be used with GridGenerator::hyper_L(). Here, $\phi$ is + * defined as the *clockwise* angle against the positive $x$-axis. + * + * This function is often used to illustrate that the solutions of the Laplace + * equation + * @f[ + * -\Delta u = 0 + * @f] + * can be singular even if the boundary values are smooth. (Here, if the + * domain is the L-shaped domain $(-1,1)^2 \backslash [0,1]^2$, the + * boundary values for $u$ are zero on the two line segments adjacent to the + * origin, and equal to $r^{\frac{2}{3}} \sin(\frac{2}{3} \phi)$ on the + * remaining parts of the boundary.) The function itself remains bounded on + * the domain, but its gradient is of the form $r^{-1/3}$ in the vicinity of + * the origin and consequently diverges as one approaches the origin. * * @ingroup functions */ -- 2.39.5