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decision is the absence of closure cells. Treatment of hanging nodes
is much better done in the numerical than in the geometrical context
(cf. Section <A HREF="node3.html#sub:hnodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>).
+ SRC="cross_ref_motif.gif"></A>).
<P>
Two other important criteria in the design of the library, unfortunately not
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quadrilaterals in two, and hexahedra in three space dimensions. A
triangulation is therefore made up of the objects in Figure
<A HREF="node2.html#fig:topology"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>.
+ SRC="cross_ref_motif.gif"></A>.
<BR>
<DIV ALIGN="CENTER"><A NAME="fig:topology"> </A><A NAME="262"> </A>
<TABLE WIDTH="50%">
Lagrangian multiplier (which would lead to saddle point problems), but
can be inserted into the matrix and right hand side, thus retaining
the properties of the matrices (cf. Section <A HREF="node3.html#sub:hnodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>). In
+ SRC="cross_ref_motif.gif"></A>). In
particular, symmetric and positive definite matrices retain these properties.
<P>
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example the interpolation points of Lagrangian elements or face integrals for
some nonconforming elements, may be associated with any of the basic topological
objects of a mesh, as shown in Figure <A HREF="node2.html#fig:topology"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>. The <TT>FiniteElement</TT>
+ SRC="cross_ref_motif.gif"></A>. The <TT>FiniteElement</TT>
class describing a certain finite element states the objects on which these
functionals are defined. For neighboring cells, these nodes may coincide, in
which case they are identified.
carry different node values, which we call <EM>hanging nodes</EM>. Then,
additional compatibility conditions have to be imposed numerically, as we will
explain below in Section <A HREF="node3.html#sub:hnodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>.
+ SRC="cross_ref_motif.gif"></A>.
<P>
<A NAME="tex2html3"
HREF="footnode.html#foot400"><SUP>1</SUP></A>
see Figure <A HREF="node3.html#fig:hanging-nodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>. The
+ SRC="cross_ref_motif.gif"></A>. The
degrees of freedom on the refined side of a face (for example the left side of
the face with end points <I>P</I><SUB>1</SUB> and <I>P</I><SUB>2</SUB> in the figure) are not all matched by
degrees of freedom on the coarse side, so in order to guarantee continuity
SRC="img14.gif"
ALT="$\varphi_1$">
associated with node <I>P</I><SUB>1</SUB>in Figure <A HREF="node3.html#fig:hanging-nodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A> can be written as
+ SRC="cross_ref_motif.gif"></A> can be written as
<!-- MATH: $\varphi_1=\tilde\varphi_1+\tfrac 12\tilde\varphi_{10}$ -->
<IMG
where in the latter representation we
identified the function <I>u</I> by the vector of its nodal
values. For the grid in Figure <A HREF="node3.html#fig:hanging-nodes"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A> and bilinear elements,
+ SRC="cross_ref_motif.gif"></A> and bilinear elements,
the matrix <I>C</I> has the following form:
<BR><P></P>
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<P>
The examples in Figure <A HREF="node5.html#fig:conservation"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A> of computations with locally
+ SRC="cross_ref_motif.gif"></A> of computations with locally
refined grids for these equations were done by R. Hartmann. The left and
middle picture show solutions of the nonlinear Burgers equation with two
merging shocks on a coarse grid and a refined one; bilinear discontinuous
<BR>
<P>
In Figure <A HREF="node5.html#fig:parameter"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A> we show two examples of estimated parameters
+ SRC="cross_ref_motif.gif"></A> we show two examples of estimated parameters
along with the exact values. In order to reduce the degree of ill-posedness
and to reduce the number of unknowns, the discretization of the parameter <I>q</I>was performed on a coarser grid than the state variable <I>u</I>. In fact, the
discretization was done on the same grid but we posed additional constraints
HREF="node6.html#Ban98">1</A>,<A
HREF="node6.html#BR99b">3</A>]. In Figure
<A HREF="node5.html#fig:wave3d"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A>, a wave traveling outward from the lower left corner of the
+ SRC="cross_ref_motif.gif"></A>, a wave traveling outward from the lower left corner of the
domain is shown. We used a simple error indicator to track the spreading of
the wave to adjust the computational meshes accordingly; it can be seen that
rather coarse cells were used where the solution is smooth.
<P>
Figure <A HREF="node5.html#fig:helmholtz"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
- SRC="/data/allarch/usr/latex2html/icons.gif/cross_ref_motif.gif"></A> shows examples of solutions to Poisson's
+ SRC="cross_ref_motif.gif"></A> shows examples of solutions to Poisson's
equation using Neumann boundary conditions on an embedded boundary
curve. The interior of the circle is not part of the domain on which
the problem is posed and the solution is set to zero there.
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