Measures of convergence often depend on how the * closeness* of
measuring the distance between functions is defined. Another common
description of measuring convergence is, * uniform convergence*, which
requires that the maximum value of in the
domain vanish as . This is stronger than pointwise
convergence as it requires a unform rate of convergence at every point in
the domain. Two other commonly used measures are * convergence in
energy* and * convergence in mean*, which involve measuring an *
average* of a function of the pointwise error over the domain [45].

In general, proving pointwise convergence is very difficult except in the simplest cases, while proving the convergence of an averged value, such as energy, is often easier. Of course, scientists and engineers are often much more interested in assuring that their answers are accurate in a pointwise sense than an energy sense because they typically want to know values of the solution and graidents, at specific places.