]> https://gitweb.dealii.org/ - dealii.git/commitdiff
Clarify an issue in the introduction of step-61. 12038/head
authorWolfgang Bangerth <bangerth@colostate.edu>
Mon, 12 Apr 2021 17:24:33 +0000 (11:24 -0600)
committerWolfgang Bangerth <bangerth@colostate.edu>
Mon, 12 Apr 2021 17:24:33 +0000 (11:24 -0600)
examples/step-61/doc/intro.dox

index f2609ee1720c28eb5511bd972e9de286109844b5..701912a96055fde8955a321d14a97f78b3fdc832 100644 (file)
@@ -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

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