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6.0.1 Convergence of a Sequence of Approximate Solutions     continued...

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.