SRC="img8.gif"
ALT="$\hat K$">
to the element <I>K</I> in real space, see Figure
-<A HREF="mapping.html#fig:mapping">1</A>.
+<A HREF="index.html#fig:mapping">1</A>.
<BR>
<DIV ALIGN="CENTER"><A NAME="fig:mapping"> </A><A NAME="183"> </A>
<TABLE WIDTH="50%">
The definition of <IMG
WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img1.gif"
- ALT="$\sigma $">(<A HREF="mapping.html#eq:def-mapping-q">1</A>) ensures that each of the unit support points
+ ALT="$\sigma $">(<A HREF="index.html#eq:def-mapping-q">1</A>) ensures that each of the unit support points
<IMG
WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$\hat K$">
based on a tensor
product mesh. In the following we only consider the two-dimensional
-case, <I>d</I>=2. For that case, Figure <A HREF="mapping.html#fig:unit-mapping-points">2</A>
+case, <I>d</I>=2. For that case, Figure <A HREF="index.html#fig:unit-mapping-points">2</A>
shows the distributions of the unit support points <IMG
WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
Let the ordering and numbering of the unit support points be as
follows: first the corners, then the points on the edges and finally
the inner support points, see also Figure
-<A HREF="mapping.html#fig:unit-mapping-points">2</A>. Thus the first 4<I>p</I> points are placed
+<A HREF="index.html#fig:unit-mapping-points">2</A>. Thus the first 4<I>p</I> points are placed
on the boundary
<!-- MATH: $\partial\hat K$ -->
<IMG
</DIV>
<BR CLEAR="ALL">
<P></P>
-According to (<A HREF="mapping.html#eq:point-mappings">2</A>) these points are mapped to the
+According to (<A HREF="index.html#eq:point-mappings">2</A>) these points are mapped to the
mapping support points <I>p</I><SUB><I>k</I></SUB>,
<!-- MATH: $k=0,\ldots,4p-1$ -->
<IMG
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
-into the Laplace problem (<A HREF="mapping.html#eq:local-laplace-problem">3</A>) yields the
+into the Laplace problem (<A HREF="index.html#eq:local-laplace-problem">3</A>) yields the
zero boundary value problem,
<BR><P></P>
<DIV ALIGN="CENTER">
</DIV>
<BR CLEAR="ALL">
<P></P>
-and recalling definitions (<A HREF="mapping.html#eq:def-mapping-q">1</A>), (<A HREF="mapping.html#eq:def-tilde-sigma">5</A>) and (<A HREF="mapping.html#eq:discrete-boundary-function">4</A>) gives
+and recalling definitions (<A HREF="index.html#eq:def-mapping-q">1</A>), (<A HREF="index.html#eq:def-tilde-sigma">5</A>) and (<A HREF="index.html#eq:discrete-boundary-function">4</A>) gives
<BR><P></P>
<DIV ALIGN="CENTER">
</DIV>
<BR CLEAR="ALL">
<P></P>
-The solutions to problem (<A HREF="mapping.html#eq:discrete-local-laplace">7</A>) for <I>l</I>=1,2 are
+The solutions to problem (<A HREF="index.html#eq:discrete-local-laplace">7</A>) for <I>l</I>=1,2 are
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
-of the linear combination (<A HREF="mapping.html#eq:linear-combination-laplace">8</A>), that
+of the linear combination (<A HREF="index.html#eq:linear-combination-laplace">8</A>), that
represents the dependency of the <I>j</I>th inner mapping support point
<I>p</I><SUB>4<I>p</I>+<I>j</I></SUB> on the support points <I>p</I><SUB><I>k</I></SUB>,
<!-- MATH: $k=0,\ldots,4p-1$ -->
</DIV>
<BR CLEAR="ALL">
<P></P>
-see also Figure <A HREF="mapping.html#fig:coefficients-q2-q3">3</A>, left.
+see also Figure <A HREF="index.html#fig:coefficients-q2-q3">3</A>, left.
<BR>
<DIV ALIGN="CENTER"><A NAME="fig:coefficients-q2-q3"> </A><A NAME="185"> </A>
<TABLE WIDTH="50%">
</TABLE>
</DIV>
<BR>
-For the case that <I>p</I>=3, Figure <A HREF="mapping.html#fig:coefficients-q2-q3">3</A>, right, shows the coefficients <I>c</I><SUB>12,<I>k</I></SUB> of the linear combination for the inner mapping support point <I>p</I><SUB>12</SUB>. The coefficents for the points <I>p</I><SUB>13</SUB>, <I>p</I><SUB>14</SUB> and <I>p</I><SUB>15</SUB> can be obtain by rotation of the coefficients.
+For the case that <I>p</I>=3, Figure <A HREF="index.html#fig:coefficients-q2-q3">3</A>, right, shows the coefficients <I>c</I><SUB>12,<I>k</I></SUB> of the linear combination for the inner mapping support point <I>p</I><SUB>12</SUB>. The coefficents for the points <I>p</I><SUB>13</SUB>, <I>p</I><SUB>14</SUB> and <I>p</I><SUB>15</SUB> can be obtain by rotation of the coefficients.
<P>
<B>Implementation in deal.II.</B> The coefficients <I>c</I><SUB><I>jk</I></SUB>, see
-(<A HREF="mapping.html#eq:coefficients">9</A>), are represented in the <I>MappingQ</I>
+(<A HREF="index.html#eq:coefficients">9</A>), are represented in the <I>MappingQ</I>
class by the <I>laplace_on_quad_vector</I> as follows
<BR><P></P>
<DIV ALIGN="CENTER">
the <I>MappingQ</I> class. This is done by calling the
<I>MappingQ::set_laplace_on_quad_vector</I> function that
includes the coefficients hardcoded for <I>p</I>=2 and <I>p</I>=3 in <I>d</I>=2dimensions, and a routine for computing the coefficients according to
-(<A HREF="mapping.html#eq:coefficients">9</A>) for all other cases. The mapping support
+(<A HREF="index.html#eq:coefficients">9</A>) for all other cases. The mapping support
points <IMG
WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img14.gif"
<I>MappingQ::add_line_support_points</I>), then by calling
<I>MappingQ::apply_laplace_vector</I> the remaining (<I>p</I>-1)<SUP>2</SUP>inner mapping supports points are computed, where
<I>MappingQ::apply_laplace_vector</I> just performes the linear
-combination given in (<A HREF="mapping.html#eq:linear-combination-laplace">8</A>).
+combination given in (<A HREF="index.html#eq:linear-combination-laplace">8</A>).
<BR><HR>
</FONT>
<ADDRESS>