FEM basis polynomial order and the differential equation order

[chowdhury]

Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved
$\frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x} }{\partial x} \right ) + \omega^2 u_{x} = 0$

This is second order equation in $u_{x}$. Which order polynomial basis shall I choose?

 

[bigfooted]

The degree of the polynomial basis function is connected to the local differentiability of your solution, this makes the degree difficult to determine beforehand.
Suppose your solution is locally linear. Then using a linear basis function would capture this behavior. You could still use basis functions of higher order, but they will not give you additional accuracy. Your Helmholtz equation above is infinitely differentiable, so you could use very high order basis functions and still see a benefit from the increase in polynomial degree. Increasing the degree of the basis function leads to faster convergence than increasing the number of cells (as function of degrees of freedom). So in general, the best strategy is to increase locally the degree of the basis function until the degree of the local solution (so the degree of the Hilbert space of the local solution). In case of shock waves, you can locally use a low degree basis function, and away from the shock a higher degree basis function.

A very practical book explaining this in detail is "Spectral h/p methods for Computational Fluid Dynamics" by Sherwin and Karniadakis. A more mathematical book is "The Finite Element Method and its Reliability" by Babuska and Strouboulis.