Starting in the spring 2013, I videotaped the lectures
for my *MATH 676: Finite element methods in scientific
computing* course
at the KAMU TV studio at
Texas A&M. These are lectures
on many aspects of scientific computing, software,
and the practical aspects of the finite element method, as
well as their implementation in the
deal.II software library. Support
for creating these videos was also provided by the
National Science Foundation and
the Computational
Infrastructure in Geodynamics.

**Note 1:** In some of the videos, I demonstrate code or user
interfaces. If you can't read the text, change the
video quality by clicking on the "gear" symbol at the
bottom right of the YouTube player.

**Note 2:**
deal.II is an
actively developed library, and in the course of this
development we occasionally deprecate and remove
functionality. In some cases, this implies that we also
change tutorial programs, but the nature of videos is that
this is not reflected in something that may have been
recorded years ago. If in doubt, consult
the *current* version of the tutorial.

**Lecture 17: step-6: Adaptive meshes**

This lecture demonstrates how one uses adaptive mesh refinement and the constraints discussed in the previous lecture to solve the Laplace equation. We show the 10 or 15 lines of additional code necessary to deal with adaptive meshes. The remainder of the program is unchanged despite the radically different way of viewing meshes in this situation.

In the code, I refine 30% of the cells and coarsen 3%. These "magic"
numbers have a purpose: In 2d, if you refine a fraction
*x* of your *N* cells, then you replace each of these cells
by their four children, so you will get a total of
*(1-x)N+4xN=(1+3x)N* cells in the next round. *x=30%*
therefore ensures that we get roughly twice as many cells the next
time around. The fraction of 3% for coarsening is small enough so that
not many cells that were previously refined will be coarsened again,
but large enough that we can undo our own mistakes.

Of course, this then leads to the question of why it is useful to
double the number of cells in each refinement. To understand this,
let's assume that we have *M* cells on the final mesh. Then we
had *M/2*, *M/4*, *M/8*, ... cells on the meshes in
previous iterations. If we have a perfect solver in which it costs us
*O(M)* operations to solve a linear system with *M*
unknowns, then solving on a sequence of meshes that always double will
cost us a total compute time of
*...+O(M/8)+O(M/4)+O(M/2)+O(M)=O(2M)* operations, or about twice
as many as on the finest mesh -- not too large a penalty to pay, given
that we have likely constructed a good mesh by adaptively refining. On
the other hand, if we have chosen a fraction *x* much smaller
than 30% (i.e., we are very careful in refining the mesh) then we may
have a slightly better mesh (i.e., *M* is smaller), but the sum
above adds up to something larger. On the other hand, if we chose
*x=1* (i.e., global refinement), then we will likely have a mesh
that doesn't suit the solution at all (i.e., we need a very large
*M* to get any kind of accuracy at all) but the sum for the work
would be *...+O(M/32)+O(M/8)+O(M/4)+O(M)=O(4M/3)*: even in the
best of cases (if a globally refined mesh is what is needed) only one
third faster than the adaptive refinement with *x=30%*.

It is worth noting that the factor *x* that leads to a doubling
of the number of cells is different in 3d.

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