Nonlinear Approximation Rates and Besov Regularity for Elliptic PDEs on Polyhedral Domains

We investigate the Besov regularity for solutions of elliptic PDEs. This is based on regularity results in Babuska-Kondratiev spaces. Following the argument of Dahlke and DeVore, we first prove an embedding of these spaces into the scale B τ,τ (D) of Besov spaces with 1 τ = r d + 1 p . This scale is known to be closely related to n-term approximation w.r.to wavelet systems, and also adaptive Finite element approximation. Ultimately this yields the rate n for u ∈ K p,a(D) ∩H s p(D) for r < r ∗ ≤ m. In order to improve this rate to n we leave the scale B τ,τ (D) and instead consider the spacesB τ,∞(D). We determine conditions under which the spaceK m p,a(D)∩H s p(D) is embedded into some space B τ,∞(D) for some m d + 1 p > 1 τ ≥ 1 p , which in turn indeed yields the desired n-term rate. As an intermediate step we also prove an extension theorem for Kondratiev spaces.