add note on discontinuous coefficient

diff --git a/examples/step-6/doc/results.dox b/examples/step-6/doc/results.dox
index b41f3cf9f263d564d1934c5e6b04e8b20ace5a1d..890eaa33e6cde8b6048d48b52720a1632cabf10d 100644 (file)
--- a/examples/step-6/doc/results.dox
+++ b/examples/step-6/doc/results.dox
@@ -31,7 +31,7 @@ Cycle 7:
 
 
 
-As intended, the number of cells roughly doubles in each cycle.  The
+As intended, the number of cells roughly doubles in each cycle. The
 number of degrees is slightly more than four times the number of
 cells; one would expect a factor of exactly four in two spatial
 dimensions on an infinite grid (since the spacing between the degrees
@@ -484,7 +484,43 @@ step-49, step-53, and step-54, among other tutorial programs that cover
 the issue. Information on curved domains can also be found in the
 documentation module on @ref manifold "Manifold descriptions".
 
-
+Why does it make sense to choose a mesh that tracks the internal
+interface? There are a number of reasons, but the most essential one
+comes down to what we actually integrate in our bilinear
+form. Conceptually, we want to integrate the term $A_{ij}^K=\int_K
+a(\mathbf x) \nabla \varphi_i(\mathbf x) \nabla \varphi_j(\mathbf x) ; dx$ as the
+contribution of cell $K$ to the matrix entry $A_{ij}$. We can not
+compute it exactly and have to resort to quadrature. We know that
+quadrature is accurate if the integrand is smooth. That is because
+quadrature in essence computes a polynomial approximation to the
+integrand that coincides with the integrand in the quadrature points,
+and then computes the volume under this polynomial as an approximation
+to the volume under the original integrand. This polynomial
+interpolant is accurate if the integrand is smooth on a cell, but it
+is usually rather inaccurate if the integrand is discontinuous on a
+cell.
+
+Consequently, it is worthwhile to align cells in such a way that the
+interfaces across which the coefficient is discontinuous are aligned
+with cell interfaces. This way, the coefficient is constant on each
+cell, following which the integrand will be smooth, and its polynomial
+approximation and the quadrature approximation of the integral will
+both be accurate. Note that such an alignment is common in many
+practical cases, so deal.II provides a number of functions (such as
+@ref GlossMaterialId "material_id") to help manage such a scenario.
+Refer to step-28 and step-46 for examples of how material id's can be
+applied.
+
+Finally, let us consider the case of a coefficient that has a smooth
+and non-uniform distribution in space. We can repeat once again all of
+the above discussion on the representation of such a function with the
+quadrature. So, to simulate it accurately there are a few readily
+available options: you could reduce the cell size, increase the order
+of the polynomial used in the quadrature formula, select a more
+appropriate quadrature formula, or perform a combination of these
+steps. The key is that providing the best fit of the coefficient's
+spatial dependence with the quadrature polynomial will lead to a more
+accurate finite element solution of the PDE.
 
 <h4>Playing with the regularity of the solution</h4>