[cig-commits] r13910 - doc/cigma/manual

luis at geodynamics.org luis at geodynamics.org
Wed Jan 21 13:34:32 PST 2009 

Author: luis
Date: 2009-01-21 13:34:32 -0800 (Wed, 21 Jan 2009)
New Revision: 13910

Modified:
   doc/cigma/manual/main.lyx
Log:
Update main.lyx introduction.

Modified: doc/cigma/manual/main.lyx
===================================================================
--- doc/cigma/manual/main.lyx	2009-01-21 21:20:35 UTC (rev 13909)
+++ doc/cigma/manual/main.lyx	2009-01-21 21:34:32 UTC (rev 13910)
@@ -54,10 +54,919 @@
 \end_layout
 
 \begin_layout Section
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Error-Analysis"
+
+\end_inset
+
 Error Analysis
 \end_layout
 
+\begin_layout Standard
+When solving differential equations representing physical systems of interest,
+ we are often able to obtain a family of solutions by applying a variety
+ of solution techniques.
+ Sometimes an analytic method can be found, but most of the time we end
+ up resorting to a numerical algorithm, such as the finite element method.
+ Assessing the quality of these solutions is an important task, so we would
+ like to develop a quantitative measure for indicating just how close our
+ solutions approach the exact answer.
+\end_layout
+
+\begin_layout Standard
+The simplest possible quantitative measure of the difference between two
+ distinct functions consists of taking the pointwise difference at a common
+ set of points.
+ While no finite sample of points can perfectly represent a continuum of
+ values, valuable information can be inferred from the statistics of the
+ resulting set of differences.
+ However, the functions we want to compare may not be defined at a common
+ set of points.
+ Unless we are able to interpolate the two functions on an intermediate
+ set of points, this simple pointwise measure becomes inapplicable.
+\end_layout
+
+\begin_layout Standard
+A very useful distance measure we can use is the 
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm, defined by the following integral
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\varepsilon=||u-v||_{L_{2}}=\sqrt{\int_{\Omega}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}}\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $u$
+\end_inset
+
+ and 
+\begin_inset Formula $v$
+\end_inset
+
+ are the two functions defined on a global coordinate system.
+ This gives us a single global estimate 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ representing the distance between the two functions 
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ and 
+\begin_inset Formula $v(\vec{x})$
+\end_inset
+
+.
+ Alternatively, you may think of this as the size, or norm, of the 
+\series bold
+\emph on
+residual field
+\series default
+\emph default
+ 
+\begin_inset Formula $\rho(\vec{x})=u(\vec{x})-v(\vec{x})$
+\end_inset
+
+.
+ This measure is useful because it is well known that solutions obtained
+ with the finite element method converge only in a weak sense, as an average
+ over a geometric region.
+ In the rest of this chapter we will discuss the details involved in evaluating
+ the above integral.
+\end_layout
+
+\begin_layout Standard
+Another quantitative measure, which we won't discuss in this version of
+ Cigma, is the energy norm.
+ This norm is typically employed in 
+\emph on
+a posteriori
+\emph default
+ error analysis and is problem dependent.
+ In general, the residual field used under this norm is defined in terms
+ of the linearized equation from which the solution was obtained.
+ Yet another interesting possibility would be to 
+\end_layout
+
 \begin_layout Section
+Functions
+\end_layout
+
+\begin_layout Standard
+The functions that we discuss in this manual are assumed to be defined in
+ a common region of space denoted by 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Elements of 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ are written as 
+\begin_inset Formula $\vec{x}$
+\end_inset
+
+, and the number of components in the corresponding value of 
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+, is said to be the 
+\series bold
+\emph on
+rank
+\series default
+\emph default
+ of the function 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Thus, 
+\emph on
+scalar functions
+\emph default
+ have a rank of 1, 
+\emph on
+vector functions
+\emph default
+ would have a rank of 2 or 3, and 
+\emph on
+tensor functions
+\emph default
+ would have a rank of 6 or 9.
+\end_layout
+
+\begin_layout Section
+Integral Approximations
+\end_layout
+
+\begin_layout Standard
+In order to evaluate the integral for our 
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm, we will need to define an integration mesh which partitions the common
+ domain 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ on which our two functions are defined.
+ We can then use a numerical approximation, or 
+\series bold
+\emph on
+integration rule
+\series default
+\emph default
+, on each of the discrete cell elements 
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+ of our partition.
+ Each of these cell integrals of the continuous residual norm 
+\begin_inset Formula $||\rho(\vec{x})||^{2}$
+\end_inset
+
+ can then be replaced by a weighed sum evaluated at a finite set of points.
+ This integral will be valid up to a certain accuracy.
+ In Chapter 5, we discuss in more detail how choose the location and values
+ for the integration points and weights which give optimal accuracy.
+\end_layout
+
+\begin_layout Standard
+Thus, in general, an integration rule with 
+\begin_inset Formula $Q$
+\end_inset
+
+ points and weights that allows us to integrate the scalar function 
+\begin_inset Formula $F(\vec{x})$
+\end_inset
+
+ is given by the equation
+\begin_inset Formula \begin{equation}
+\int_{\Omega_{i}}\ F(\vec{x})\ d\vec{x}\approx\sum_{q=1}^{Q}F(\vec{X_{q}})W_{q}\end{equation}
+
+\end_inset
+
+where the integration points 
+\begin_inset Formula $\vec{X}_{q}$
+\end_inset
+
+ and integration weights 
+\begin_inset Formula $W_{q}$
+\end_inset
+
+ depend on the integration region 
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+.
+ As we will see in Chapter 7, one possibility is to pre-calculate these
+ points and weights explicitly on a specific discretization.
+ Thus, as discussed in more detail in Chapters 4 and 5, we will generally
+ want to define a reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ which acts as a template for cells with the same geometric shape.
+ We can achieve this by defining a 
+\series bold
+\emph on
+reference map
+\series default
+\emph default
+ 
+\begin_inset Formula $\chi_{i}:\hat{\Omega}\to\Omega_{i}$
+\end_inset
+
+ that describes how points in a specific cell 
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+ originate from the reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ In this sense, the reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ is said to define a 
+\series bold
+\emph on
+local coordinate system
+\series default
+\emph default
+ for the 
+\begin_inset Formula $i$
+\end_inset
+
+-th cell 
+\begin_inset Formula $\Omega_{i}.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Note that a given discretization may contain cells of different shapes,
+ which would require us to define an appropriate number of reference cells
+ for each type of cell.
+ However, for the purposes of this manual we may restrict the discussions
+ in this manuals, without any loss of generality, to discretizations consisting
+ of a single geometric shape.
+ (XXX: reword this last sentence)
+\end_layout
+
+\begin_layout Standard
+The advantage of this approach is readily apparent when one realizes that
+ the integration points and weights can be calculated once and for all on
+ the reference cell, and then reused for the other cells through the application
+ of the corresponding reference map.
+ In other words,
+\begin_inset Formula \begin{eqnarray*}
+\vec{X}_{q} & = & \chi_{i}(\vec{\xi}_{q})\\
+W_{q} & = & w_{q}J_{i}(\vec{\xi}_{q})\end{eqnarray*}
+
+\end_inset
+
+where 
+\begin_inset Formula $\vec{\xi}_{q}$
+\end_inset
+
+ and 
+\begin_inset Formula $w_{q}$
+\end_inset
+
+ are the optimal integration points and weights for the integration rule
+ defined on the reference domain 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ Here, the factor 
+\begin_inset Formula $J_{i}(\vec{\xi})=\det\ \left|\frac{d\chi_{i}}{d\vec{\xi}}\right|$
+\end_inset
+
+ is the Jacobian determinant of the transformation 
+\begin_inset Formula $\chi_{i}$
+\end_inset
+
+.
+ Recall that the index 
+\begin_inset Formula $i$
+\end_inset
+
+ corresponds to the 
+\begin_inset Formula $i$
+\end_inset
+
+-th cell 
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+, and the index 
+\begin_inset Formula $q$
+\end_inset
+
+ ranges over the usual 
+\begin_inset Formula $q=1,\ldots,Q$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Interpolation Functions
+\end_layout
+
+\begin_layout Standard
+Our two functions 
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ and 
+\begin_inset Formula $v(\vec{x})$
+\end_inset
+
+ are given in terms of the point
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+, and are said to be defined on a 
+\series bold
+\emph on
+global coordinate system
+\series default
+\emph default
+.
+ Based on our discussion in Section 3.1, we see that we only need to know
+ the values of 
+\begin_inset Formula $u$
+\end_inset
+
+ and 
+\begin_inset Formula $v$
+\end_inset
+
+ at a finite set of global points in order to calculate an approximation
+ to the 
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm of the residual field 
+\begin_inset Formula $\rho=u-v$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Specifically, given a function 
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ we must be able to calculate its values at exactly those integration points.
+ If we know a formula or algorithm for 
+\begin_inset Formula $f$
+\end_inset
+
+, then our task is easy.
+ Alternatively, these values can be given explicitly as a finite list.
+ In all other cases, we will need an interpolation scheme that allows us
+ to calculate 
+\begin_inset Formula $f$
+\end_inset
+
+ on any intermediate points.
+ This ability is also important because we may want to increase the accuracy
+ of our norm by using more integration points, and thus we will need to
+ be able to evaluate our function 
+\begin_inset Formula $f$
+\end_inset
+
+ at the new points.
+\end_layout
+
+\begin_layout Standard
+In general, an interpolation scheme involves a set of known functions 
+\begin_inset Formula $\phi_{j}(\vec{x})$
+\end_inset
+
+ that we can compute anywhere and a set of parameters 
+\begin_inset Formula $c_{j}$
+\end_inset
+
+ that define how these known functions are combined.
+ Usually, the relation between the parameters and the known functions is
+ a linear combination, 
+\begin_inset Formula \begin{equation}
+f(\vec{x})=d_{1}\phi_{1}(\vec{x})+d_{2}\phi_{2}(\vec{x})+\cdots+d_{m}\phi_{m}(\vec{x})\end{equation}
+
+\end_inset
+
+where the parameters 
+\begin_inset Formula $d_{j}$
+\end_inset
+
+, with 
+\begin_inset Formula $j=1,\ldots,m$
+\end_inset
+
+, are also sometimes referred to as the 
+\series bold
+\emph on
+degrees of freedom
+\series default
+\emph default
+ of the function 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ As described in Chapter 4, the functions 
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+, also known as 
+\series bold
+\emph on
+shape functions
+\series default
+\emph default
+, must satisfy a number of conditions.
+ Because the choice of shape functions affects how well the true solution
+ is represented, the convergence of the numerical method used to obtain
+ the solution, the optimal choice of shape functions is very problem dependent.
+\end_layout
+
+\begin_layout Standard
+If we know the shape functions 
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+ directly in terms of the global points 
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+, they are said to form a 
+\series bold
+\emph on
+global interpolation scheme
+\series default
+\emph default
+, and we may use Eq X directly to find the values of 
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+, and we may refer to the 
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+as 
+\emph on
+global shape functions
+\emph default
+.
+ Note that since we are working in the global coordinate system, we don't
+ need the discretization of the domain 
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Examples of global shape functions include spherical harmonics, and ...
+\end_layout
+
+\begin_layout Standard
+Perhaps a more typical case is when 
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ is defined piecewise, on each discretization element 
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+.
+ Because these cells partition the original domain by definition, a given
+ point in our domain will be found in one and only one cell, which will
+ have a local definition.
+ That is, for a particular 
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+ we will be able to find a unique cell 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, for some index 
+\begin_inset Formula $e$
+\end_inset
+
+.
+ We can refer to this as a 
+\series bold
+\emph on
+local interpolation scheme
+\series default
+\emph default
+.
+\end_layout
+
+\begin_layout Standard
+See Chapter 5, for more details on the specific interpolation schemes available
+ in Cigma.
+\end_layout
+
+\begin_layout Section
+Global Error Measure
+\end_layout
+
+\begin_layout Standard
+In this section we show the formula used by Cigma to compute the 
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm of the residual field 
+\begin_inset Formula $\rho$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Suppose the domain 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ into an appropriate set of cells 
+\begin_inset Formula $\Omega_{1},\Omega_{2},\ldots,\Omega_{n_{el}}$
+\end_inset
+
+, we can compute the global error 
+\begin_inset Formula $\varepsilon^{2}$
+\end_inset
+
+ as a sum over localized cell contributions, 
+\begin_inset Formula $\varepsilon^{2}=\sum_{e}\varepsilon_{e}^{2}$
+\end_inset
+
+, where each cell residual 
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ is given by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In general, we won't be able to integrate each cell residual 
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ exactly since either of the functions 
+\begin_inset Formula $u$
+\end_inset
+
+ and 
+\begin_inset Formula $v$
+\end_inset
+
+ may have an incompatible representation relative to the finite element
+ space on each domain 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+.
+ However, we can still calculate an approximation of each cell residual
+ by applying an appropriate quadrature rule with a tolerable truncation
+ error 
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Encyclopaedia of Cubature Formulas 2005"
+
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+To obtain an approximation to the integral of a function 
+\begin_inset Formula $F(\vec{x})$
+\end_inset
+
+ over a cell 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, we simply apply the quadrature rule with weights 
+\begin_inset Formula $w_{e,1},w_{e,2},\ldots,w_{e,n_{Q}}$
+\end_inset
+
+ and integration points 
+\begin_inset Formula $\vec{x}_{e,1},\vec{x}_{e,2},\ldots,\vec{x}_{e,n_{Q}}$
+\end_inset
+
+ appropriate for the physical element 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Applying this quadrature rule directly over the entire physical domain 
+\begin_inset Formula $\Omega=\cup\Omega_{e}$
+\end_inset
+
+ gives us 
+\begin_inset Formula \[
+\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{e=1}^{n_{el}}\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For efficiency reasons, it is undesirable in finite element applications
+ to perform calculations in a global coordinate system.
+ To avoid duplication of work, shape function evaluations may be performed
+ once on a reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ and then transformed back into the corresponding physical cell 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ as needed.
+\end_layout
+
+\begin_layout Standard
+To compute integrals of 
+\begin_inset Formula $F$
+\end_inset
+
+ in a reference coordinate system, we need to apply a change of variables:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}\]
+
+\end_inset
+
+where 
+\color none
+the additional factor 
+\begin_inset Formula $J_{e}(\vec{\xi})=\det\left[\frac{d\vec{x}_{e}}{d\vec{\xi}}\right]$
+\end_inset
+
+ is the Jacobian determinant of the reference map 
+\begin_inset Formula $\vec{x}_{e}(\vec{\xi}):\hat{\Omega}\to\Omega_{e}$
+\end_inset
+
+.
+ This map describes how the physical points 
+\begin_inset Formula $\vec{x}\in\Omega_{e}$
+\end_inset
+
+ are transformed from the reference points 
+\begin_inset Formula $\vec{\xi}\in\hat{\Omega}$
+\end_inset
+
+.
+ Put another way, the inverse reference map 
+\color inherit
+
+\begin_inset Formula $\vec{\xi}=\vec{x}_{e}^{-1}(\vec{x})$
+\end_inset
+
+ tells us how the physical domain 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ maps into the reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+At this point, we can assume without loss of generality that every physical
+ cell 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ can be derived from a single reference cell 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+, so that our quadrature rule becomes simply a set of weights 
+\begin_inset Formula $w_{1},\ldots,w_{n_{Q}}$
+\end_inset
+
+ and points 
+\begin_inset Formula $\vec{\xi}_{1},\ldots,\vec{\xi}_{n_{Q}}$
+\end_inset
+
+ over 
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ After changing variables, we end up with the final form of the quadrature
+ rule that we can use to integrate the global residual field:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}=\sum_{q=1}^{n_{Q}}w_{q}F(\vec{x}_{e}(\vec{\xi}_{q}))J_{e}(\vec{\xi}_{q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+If we let 
+\begin_inset Formula $F(\vec{x})=||u(\vec{x})-v(\vec{x})||^{2}$
+\end_inset
+
+ in the above expression, we can find the cell residual contribution over
+ 
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ from
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}\ ||u(\vec{x})-v(\vec{x})||^{2}\ d\vec{x}\\
+ & = & \int_{\hat{\Omega}}\ ||u(\vec{x}_{e}(\vec{\xi}))-v(\vec{x}_{e}(\vec{\xi}))||^{2}J_{e}(\vec{\xi})\ d\vec{\xi}\\
+ & = & \sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The global error 
+\begin_inset Formula $\varepsilon=\sqrt{\sum_{e}\varepsilon_{e}^{2}}$
+\end_inset
+
+ is then approximated by the expression
+\end_layout
+
+\begin_layout Standard
+\begin_inset Box Boxed
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100text%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+We use this final form to calculate the global and localized errors on arbitrary
+ discretizations.
+\end_layout
+
+\begin_layout Section
+Comparing Residuals
+\end_layout
+
+\begin_layout Standard
+For comparing errors of different solutions, you can normalize the global
+ error by the norm of the exact solution.
+ For a family of solutions 
+\begin_inset Formula $u_{h}(\vec{x})$
+\end_inset
+
+, parametrized by the the maximum element size 
+\begin_inset Formula $h$
+\end_inset
+
+ of the underlying discretization, and an exact solution 
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+, this normalized error is given by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{rel} & = & \frac{||u-u_{h}||_{L_{2}}}{||u||_{L_{2}}}\\
+ & = & \frac{\int_{\Omega}||u(\vec{x})-u_{h}(\vec{x})||^{2}d\vec{x}}{\int_{\Omega}||u(\vec{x})||^{2}d\vec{x}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This normalized error can be interpreted as the average error in the physical
+ quantity being evaluated, so that a value of 0.01 corresponds to a 1% averaged
+ error.
+\end_layout
+
+\begin_layout Standard
+Even if the exact solution is not currently known, this normalized error
+ may be used to test the accuracy between two or more numerical solutions,
+ defined on successively refined meshes.
+\end_layout
+
+\begin_layout Section
+Convergence Rates
+\end_layout
+
+\begin_layout Standard
+Even if the exact solution 
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ is not known, but we have calculated a family of solutions 
+\begin_inset Formula $u_{h}(\vec{x})$
+\end_inset
+
+ on discrete meshes, we may calculate a convergence rate by using the standard
+ error estimate 
+\begin_inset Formula $||u-u_{h}||_{p}\leq Ch^{\alpha},$
+\end_inset
+
+where 
+\begin_inset Formula $C$
+\end_inset
+
+ is independent of both 
+\begin_inset Formula $h$
+\end_inset
+
+ and 
+\begin_inset Formula $u$
+\end_inset
+
+.
+ This error bound holds problems
+\end_layout
+
+\begin_layout Section
+Uses in Benchmarking
+\end_layout
+
+\begin_layout Standard
+As we saw in Section 3.5, it is easier to calculate the convergence rate
+ on a sequence of meshes if we have access to the exact solution.
+ 
+\end_layout
+
+\begin_layout Section
 Citation
 \end_layout

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