Characterisation of approximation properties of finite elements

We extend some earlier results on the approximation properties of a large class of operators that approximates a function into an integer-shift invariant space in L² by projection of the function. Concretely we work out a Fourier analysis method that applies to the L² norm of the derivative of the approximation error. We apply this result the approximation of functions by B-splines in function of a given sampling parameter, and we obtain a series of asymptotic constants for these approximations.

These results can be applied an analysis of the accuracy of the approximation using finite elements, as is shown in the second half of the paper. In particular we obtain asymptotic constants for Legendre and Hermite finite elements and show that spline approximation yelds better results in this respect.