zero or not depends on the equation under consideration, but the
indicated positions in the matrix tell us which shape functions can
and which can't couple, if the equation is a local, i.e. differential
-one):
+one):
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renumbering. Although this might not be apparent, the number of
nonzero entries is the same in both pictures, of course.
-A common observation is that the more refined the grid is, the better
-the clustering around the diagonal will get.
+<h3> Possible extensions </h3>
+
+Just as with step-1, you may want to play with the program a bit to
+familiarize yourself with deal.II. For example, in the
+<code>distribute_dofs</code> function, we use linear finite elements
+(that's what the argument "1" to the FE_Q object is). Explore how the
+sparsity pattern changes if you use higher order elements, for example
+cubic or quintic ones (by using 3 and 5 as the respective arguments).
+
+Or, you could see how the sparsity pattern changes with more
+refinements. You will see that not only the size of the matrix
+changes, but also its bandwidth (the distance from the diagonal of
+those nonzero elements of the matrix that are farthest away from the
+diagonal), though the ratio of bandwidth to size typically shrinks,
+i.e. the matrix clusters more around the diagonal.
+
+Another idea of experiments would be to try other renumbering
+strategies than Cuthill-McKee from the DoFRenumbering namespace.