# Approximation of functions in fractional Sobolev spaces

First, some background and motivation: Convergence estimates in finite elements are often of the form $\|u-u_h \|_{L^2} \leq Ch^m \|u\|_{H^m}$, where $h$ is the mesh norm, $u_h$ is some discrete solution from a Galerkin method, and $H^m$ is the Sobolev space of order $m$, $W^m_2$. Interpolation by polynomials is usually the key step to deriving this error estimate. First, an estimate of the form $\|u-I_h u\|_{H^{1}} \leq Ch^{m-1} \|u\|_{H^m}$ is found, where $I_h u$ is some piecewise polynomial interpolant. Then Cea's lemma is invoked to arrive at the convergence estimate.

I am curious about the following question: Consider $\Omega \subset \mathbb{R}^2$ and some mesh with mesh norm $h$. Let $u \in H^s$ for $0 < s < 1$. Note that $H^s$ does not embed into the space of continuous functions, so these spaces can include discontinuous functions. Is there a comparable error estimate for piecewise polynomials that says something like $\|u-I_h u\|_{L^2(\Omega)} \leq C h^{s} \|u\|_{H^s}$? It's unclear to me if interpolation necessarily makes sense, as $u$ might be discontinuous along an element boundary. Perhaps the correct approach is to replace $I_h u$ with a least squares type of approximation to have optimal $L^2$ convergence. Can anyone point me to a textbook or paper that derives estimates of these types?

Yes, there are interpolation estimates in those fractional order Sobolev spaces. As you already indicated, one crucial point is the regularity required for the interpolation operator. Nodal interpolation is not an option as point evaluations make no sense for $s \leq \frac{d}{2}$, with $d=2$ in your case. The Scott-Zhang interpolation operator relies on the continuity of trace operations and hence is only applicable for $s > \frac12$. For $s > \frac12$ the result you've been looking for has been proven in a recent paper by Patrick Ciarlet, Jr. ("Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces", DOI: 10.1515/jnum-2013-0007). For $s \leq \frac12$ some Clèment-type interpolation operator (of least squares type) might be the only choice. For that case however, I don't know of any literature proving your estimate.