higher order mappings. By <it>mapping</it> we mean the transformation
between the unit cell (i.e. the unit line, square, or cube) to the
cells in real space. In all the previous examples, we have implicitly
-used linear or d-linear mappings; you will not have notices this at
+used linear or d-linear mappings; you will not have noticed this at
all, since this is what happens if you do not do anything
special. However, if your domain has curved boundaries, there are
cases where the piecewise linear approximation of the boundary
your computational domain is an approximation to the real domain using
curved boundaries as well. If the boundary approximation uses
piecewise quadratic parabolas to approximate the true boundary, then
-we say that this is a quadratic or <em>Q2</em> approximation. If we
-use piecewise graphs of cubic polynomials, then this is a <em>Q3</em>
+we say that this is a quadratic or <em>Q<sub>2</sub></em> approximation. If we
+use piecewise graphs of cubic polynomials, then this is a <em>Q<sub>3</sub></em>
approximation, and so on.
</p>
<p>
For some differential equations, it is known that piecewise linear
-approximations of the boundary, i.e. Q1 mappings, are not
+approximations of the boundary, i.e. Q<sub>1</sub> mappings, are not
sufficient if the boundary of the domain is curved. Examples are the
biharmonic equation using C<sup>1</sup> elements, or the Euler
-equation. In these cases, it is necessary to compute the integrals
-using a higher order mapping. The reason, of course, is that if we do
-not use a higher order mapping, the order of approximation of the
-boundary dominates the order of convergence of the entire numerical
-scheme, irrespective of the order of convergence of the discretization
-in the interior of the domain.
+equation on domains with curved reflective boundaries. In these cases,
+it is necessary to compute the integrals using a higher order
+mapping. The reason, of course, is that if we do not use a higher
+order mapping, the order of approximation of the boundary dominates
+the order of convergence of the entire numerical scheme, irrespective
+of the order of convergence of the discretization in the interior of
+the domain.
</p>
<p>
Rather than demonstrating the use of higher order mappings with one of
-these more complicated mappings, we do only a brief computation:
+these more complicated examples, we do only a brief computation:
calculating the value of pi=3.141592653589793238462643... by two
different methods.
</p>
the domain were the exact unit circle, then the area would be pi, but
since we only use an approximation by piecewise polynomial segments,
the value of the area is not exactly pi. However, it is known that as
-we refine the triangulation, a Qp mapping approximates the boundary
+we refine the triangulation, a Q<sub>p</sub> mapping approximates the boundary
with an order <it>h<sup>p+1</sup></it>, where <it>h</it> is the mesh
width. We will check the values of the computed area of the circle and
their convergence towards pi under mesh refinement for different
</td>
</tr>
</table>
-The quadratic mapping obviously quite well approximates the
-boundary, while for the cubic mapping the difference between
-approximated domain and true one is hardly visible already for the
-coarse grid.
+Obviously the quadratic mapping approximates the boundary quite well,
+while for the cubic mapping the difference between approximated domain
+and true one is hardly visible already for the coarse grid.
</p>
<p>
<p>
One of the immediate observations from the output is that in all cases
the values converge quickly to the true value of
-pi=3.141592653589793238462643. Note that for for Q4 mapping, the last
+pi=3.141592653589793238462643. Note that for the Q<sub>4</sub> mapping, the last
number is correct to 13 digits in both computations, which is already
-quite a lot. However, also note that for the Q1 mapping, even on the
+quite a lot. However, also note that for the Q<sub>1</sub> mapping, even on the
finest grid the accuracy is significantly worse than on the coarse
-grid for a Q4 mapping!
+grid for a Q<sub>4</sub> mapping!
</p>
<p>
The last column of the output shows the convergence order, in powers
of the mesh width <it>h</it>. In the introduction, we had stated that
-the convergence order for a <it>Qp</it> mapping should be
-<it>h<sup>p+1</sup></it>. However, in the example shown, the Q2 and Q4
+the convergence order for a <it>Q<sub>p</sub></it> mapping should be
+<it>h<sup>p+1</sup></it>. However, in the example shown, the Q<sub>2</sub> and Q<sub>4</sub>
mappings show a convergence order of <it>h<sup>p+2</sup></it>! This at
first surprising fact is readily explained by the particular boundary
-we have here. In fact, the circle is described by the function
+we have chosen in this example. In fact, the circle is described by the function
<it>sqrt(1-x<sup>2</sup>)</it>, which has the series expansion
<it>1-x<sup>2</sup>/2-x<sup>4</sup>/8-x<sup>6</sup>/16+...</it>
around <it>x=0</it>. Thus, for the quadratic mapping where the
truncation error of the quadratic approximation should be cubic, there
is no such term but only a quartic one, which raises the convergence
-order to 4, instead of 3. The same happens for the Q4 mapping.
+order to 4, instead of 3. The same happens for the Q<sub>4</sub> mapping.
</p>
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