# Polynomial Approximation Theory in Sobolev Spaces

Chapter

We will now develop the approximation theory appropriate for the finite elements developed in Chapter 3. We take a constructive approach, defining an averaged version of the Taylor polynomial familiar from calculus. The key estimates are provided by some simple lemmas from the theory of Riesz potentials, which we derive. As a corollary, we provide a proof of Sobolev's inequality, much in the spirit given originally by Sobolev.

## Keywords

Sobolev Space Interpolation Error Reference Element Nodal Variable Interpolation Operator
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© Springer Science+Business Media, LLC 2008