To randomly add to this discussion...
In general higher order polynomials are good for approximating smooth
functions, such as the tail of an exponential decay. Interestingly, a
denser grid with lower order polynomials sometimes works better for
approximating a function that oscillates rapidly, such as the peak of a
tightly packed sinosoidal wave.
Think what would happen if you try to represent a zero valued function on
one element. The order of the interpolation does not affect the accuracy
its approximation! Then try to map an exponentially decaying function with
increasing polynomial order and see what happens... That is a good exercise
for a rainy day or a sunday afternoon. :-)
Maybe try to approximate the functions, x, x^2, ..., x^n???
The result is, that if you want to save dofs, and your solution is mixed,
hp-adaptivity is definately worth exploring - and it is alot of fun!
That's my penny worth. :-)
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