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 $$B^r_{\tau ,\tau }(D)$$Bτ,τr(D) of Besov spaces with $$\frac{1}{\tau }=\frac{r}{d}+\frac{1}{p}$$1τ=rd+1p. This scale is known to be closely related to $$n$$n-term approximation w.r.t. wavelet systems, and also adaptive finite element approximation. Ultimately, this yields the rate $$n^{-r/d}$$n-r/d for $$u\in {\mathcal {K}}^m_{p,a}(D)\cap H^s_p(D)$$u∈Kp,am(D)∩Hps(D) for $$r<r^*\le m$$r<r∗≤m. In order to improve this rate to $$n^{-m/d}$$n-m/d, we leave the scale $$B^r_{\tau ,\tau }(D)$$Bτ,τr(D) and instead consider the spaces $$B^m_{\tau ,\infty }(D)$$Bτ,∞m(D). We determine conditions under which the space $${\mathcal {K}}^m_{p,a}(D)\cap H^s_p(D)$$Kp,am(D)∩Hps(D) is embedded into some space $$B^m_{\tau ,\infty }(D)$$Bτ,∞m(D) for some $$\frac{m}{d}+\frac{1}{p}>\frac{1}{\tau }\ge \frac{1}{p}$$md+1p>1τ≥1p, which in turn indeed yields the desired $$n$$n-term rate. As an intermediate step, we also prove an extension theorem for Kondratiev spaces.