From: Wolfgang Bangerth Date: Mon, 12 Apr 2021 17:24:33 +0000 (-0600) Subject: Clarify an issue in the introduction of step-61. X-Git-Tag: v9.3.0-rc1~229^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=20ed4c0f4ccc29f10c0a05622f0e89cbe57e9ab2;p=dealii.git Clarify an issue in the introduction of step-61. --- diff --git a/examples/step-61/doc/intro.dox b/examples/step-61/doc/intro.dox index f2609ee172..701912a960 100644 --- a/examples/step-61/doc/intro.dox +++ b/examples/step-61/doc/intro.dox @@ -194,6 +194,15 @@ we need to think of $\nabla_{w,d} p_h|_K$ as that Raviart-Thomas function of degree $s$ for which the left hand side and right hand side are equal for all test functions. +A key point to make is then the following: While the usual gradient $\nabla$ is +a *local* operator that computes derivatives based simply on the value of +a function at a point and its (infinitesimal) neighborhood, the weak discrete gradient +$\nabla_{w,d}$ does not have this property: It depends on the values of the function +it is applied to on the entire cell, including the cell's boundary. Both are, +however, linear operators as is clear from the definition of $\nabla_{w,d}$ +above, and that will allow us to represent $\nabla_{w,d}$ via a matrix +in the discussion below. + @note It may be worth pointing out that while the weak discrete gradient is an element of the Raviart-Thomas space $RT_s(K)$ on each cell $K$, it is discontinuous between cells. On the other hand, the