<h1>Introduction</h1>
<P>
-In this example program, we will mainly consider two aspects:
+In this program, we will mainly consider two aspects:
<UL>
<LI>Verification of correctness of the program;
<LI>Non-homogeneous Neumann boundary conditions for the Helmholtz equation.
<P>
The function <TT>integrate_difference</TT> evaluates the desired norm on each
cell <I>K</I> of the triangulation and returns a vector which holds these
-values. From the local values, we can then obtain the global error. For
+values for each cell. From the local values, we can then obtain the global error. For
example, if the vector (<I>e</I><SUB><I>i</I></SUB>) contains the local <I>L</I><SUB>2</SUB> norms, then
<BR><P></P>
<DIV ALIGN="CENTER">
WIDTH="61" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
src="step-7.data/intro/img19.gif"
ALT="${(g_2,v)}_{\Gamma_2}$">
-has appeared by partial integration and
+has appeared by integration by parts and
using
<!-- MATH: $\partial_n u=g$ -->
<IMG
faces of cells. One provides it with a quadrature formula for a manifold with
dimension one less than the dimension of the domain is, and the cell and the
number of its face on which we want to perform the integration. The class will
-then compute the values, gradients, normal vectors, weights, etc at the
+then compute the values, gradients, normal vectors, weights, etc. at the
quadrature points on this face, which we can then use in the same way as for
the domain integrals. The details of how this is done are shown in the
following program.
\begin{document}
-In this example program, we will mainly consider two aspects:
+In this program, we will mainly consider two aspects:
\begin{itemize}
\item Verification of correctness of the program;
\item Non-homogeneous Neumann boundary conditions for the Helmholtz equation.
The function \texttt{integrate\_difference} evaluates the desired norm on each
cell $K$ of the triangulation and returns a vector which holds these
-values. From the local values, we can then obtain the global error. For
+values for each cell. From the local values, we can then obtain the global error. For
example, if the vector $(e_i)$ contains the local $L_2$ norms, then
$$
E = \| {\mathbf e} \| = \left( \sum_i e_i^2 \right)^{1/2}
{(f,v)}_\Omega + {(g_2,v)}_{\Gamma_2}
$$
for all test functions $v\in H^1_0=\{v\in H^1: v|_{\Gamma_1}=0\}$. The
-boundary term ${(g_2,v)}_{\Gamma_2}$ has appeared by partial integration and
+boundary term ${(g_2,v)}_{\Gamma_2}$ has appeared by integration by parts and
using $\partial_n u=g$ on $\Gamma_2$ and $v=0$ on $\Gamma_1$. The cell
matrices and vectors which we use to build the global matrices and right hand
side vectors in the discrete formulation therefore look like this:
faces of cells. One provides it with a quadrature formula for a manifold with
dimension one less than the dimension of the domain is, and the cell and the
number of its face on which we want to perform the integration. The class will
-then compute the values, gradients, normal vectors, weights, etc at the
+then compute the values, gradients, normal vectors, weights, etc. at the
quadrature points on this face, which we can then use in the same way as for
the domain integrals. The details of how this is done are shown in the
following program.