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

Abstract

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

Appendix: Proof of Lemma 5.1

For the most part, Stein’s proof carries over without change, and hence, we shall mostly be concerned with some necessary modifications.

The first step consists in reducing the problem to smooth functions. It is easily seen that the set $C^\infty (D)$ is dense in ${\mathcal {K}}^m_{p,a}(D)$ for all parameters: given a function $u\in {\mathcal {K}}^m_{p,a}(D)$, we can multiply it with a smooth cut-off function, hence we may assume $u$ to have compact support. With standard mollification arguments we see that such a function (and simultaneously its partial derivatives) can be approximated in the $L_2$-sense by $C^\infty$-functions. Clearly, this immediately extends also to approximation w.r.t. the ${\mathcal {K}}^m_{p,a}(D)$-norm.

Stein then shows first that the extension operator applied to a smooth function $u\in C^\infty (D)$ yields again a smooth function $\mathfrak {E}u\in C^\infty (\mathbb {R}^d)$. It now remains to check the corresponding norm estimates. Below we shall keep close to Stein’s notation.

Step 1: Preparations First, let $D$ be a special Lipschitz domain, i.e. $D=\{(x,y)\in \mathbb {R}^{d+1}:y>\varphi (x)\}$ with $\varphi :\mathbb {R}^d\longrightarrow \mathbb {R}$ being Lipschitz continuous with Lipschitz constant $M$. Let $\delta (\xi )=d(\xi ,\partial D)$ be the distance of the point $x\in \overline{D}^c$ to the boundary, and let $\Delta (\xi )$ be the regularized version as constructed by Stein [22, Section VI.2.1]. For this distance function we have the estimates $C_1\delta (\xi )\le \Delta (\xi )\le C_2\delta (x)$, as well as

$$\begin{aligned} \biggl |\frac{\partial ^\alpha }{\partial \xi ^\alpha }\Delta (\xi )\biggr | \le B_\alpha \bigl (\delta (\xi )\bigr )^{1-|\alpha |},\quad \xi \in \overline{D}^c, \end{aligned}$$

for constants $C_1,\,C_2$ and $B_\alpha$ independent of $D$.

Now let $\xi _0=(x_0,\varphi (x_0))\in \partial D$ and put

$$\begin{aligned} \varGamma _{\xi _0}=\left\{ \xi =(x,y):y<\varphi (x_0),|y-\varphi (x_0)|>M|x-x_0|\right\} , \end{aligned}$$

the lower cone with vertex at $\xi _0$. Then we clearly have $\varGamma _{\xi _0}\cap \overline{D}=\{\xi _0\}$, and elementary geometric calculations for $(x,y)\in \overline{D}^c$ and the cone $\varGamma _{(x,\varphi (x))}$ yield

$$\begin{aligned} \delta (x,y)\ge \frac{M^{-1}}{(1+M^{-2})^{1/2}}\bigl (\varphi (x)-y\bigr ) =\frac{1}{(1+M^2)^{1/2}}\bigl (\varphi (x)-y\bigr ). \end{aligned}$$

Thus, it follows

$$\begin{aligned} \varphi (\xi )-y\le (1+M^2)^{1/2}\delta (x) \le \frac{1}{C_1}(1+M^2)^{1/2}\Delta (x)\equiv C_3\Delta (x). \end{aligned}$$

Now we further put $\delta ^*(\xi )=2C_3\Delta (\xi )$, and hence obtain the estimate

$$\begin{aligned} \delta ^*(x,y)\ge 2\bigl (\varphi (x)-y\bigr ). \end{aligned}$$

From the definition of $\delta$, we obtain

$$\begin{aligned} \delta (x,y)\le d\bigl ((x,y),(x,\varphi (x))\bigr ) =|y-\varphi (x)|=\varphi (x)-y \end{aligned}$$

for all points $(x,y)\in \overline{D}^c$. It further follows

$$\begin{aligned} y+\lambda \delta ^*(x,y)\ge y+\delta ^*(x,y) \ge y+2\big (\varphi (x)-y\bigr )=2\varphi (x)-y \end{aligned}$$

for all $\lambda >1$. Finally, we also have

$$\begin{aligned} \delta ^*(x,y)=2C_3\Delta (x,y)\le 2C_3C_2\delta (x,y)\le 2C_3C_2\bigl (\varphi (x)-y\bigr ). \end{aligned}$$

Step 2 Stein defined the operator $\mathfrak {E}$ on special Lipschitz domains by

$$\begin{aligned} \mathfrak {E}f(x,y)=\int _1^\infty f\bigl (x,y+\lambda \delta ^*(x,y)\bigr )\psi (\lambda )\hbox {d}\lambda , \qquad x\in \mathbb {R}^d, y\in \mathbb {R}, \end{aligned}$$

where $\psi :[1,\infty )\longrightarrow \mathbb {R}$ is a rapidly decaying continuous function such that $\int _1^\infty \psi (\lambda )\hbox {d}\lambda =1$ and $\int _1^\infty \lambda ^k\psi (\lambda )\hbox {d}\lambda =0$ for all $k\in \mathbb {N}$.

Now fix a point $(x_0,\varphi (x_0))\in \partial D$. The properties of the function $\psi$ particularly imply $|\psi (\lambda )|\le A\lambda ^{-2}$ for some constant $A$. Using this and the previous estimates for $\delta ^*$ we can estimate for $y<\varphi (x_0)$

$$\begin{aligned} \mathbf{} |\mathfrak {E}f(x_0,y)|&\le A\int _1^\infty |f(x_0,y+\lambda \delta ^*)|\frac{\hbox {d}\lambda }{\lambda ^2} \le A\delta ^*\int _{y+\delta ^*}^\infty |f(x_0,s)|(s-y)^{-2}\hbox {d}s\nonumber \\&\le A\delta ^*\int _{2\varphi (x_0)-y}^\infty |f(x_0,s)|(s-y)^{-2}\hbox {d}s\nonumber \\&\lesssim \bigl (\varphi (x_0)-y\bigr )\int _{2\varphi (x_0)-y}^\infty |f(x_0,s)|(s-y)^{-2}\hbox {d}s. \end{aligned}$$

(7.1)

This pointwise estimate now is the basis for proving estimates in weighted Sobolev norms.

Step 3 As a first case, consider the weight function $\rho (x,y)^2=|x|^2+|y|^2$, i.e. the distance to a fixed point for which we w.l.o.g. choose the origin. In particular, we now assume $\varphi (0)=0$, and start with the case $m=0$. We then obtain for arbitrary $\beta \in \mathbb {R}$

$$\begin{aligned} \int _{-\infty }^{\varphi (x_0)}&\rho (x_0,y)^{p\beta }|\mathfrak {E}f(x_0,y)|^p \hbox {d}y\\&\lesssim \int _{-\infty }^{\varphi (x_0)}\bigl (\varphi (x_0)-y\bigr )^p\rho (x_0,y)^{p\beta } \biggl (\int _{2\varphi (x_0)-y}^\infty |f(x_0,s)|(s-y)^{-2}\hbox {d}s\biggr )^p\hbox {d}y\\&=\int _0^\infty \tilde{y}^p\rho \bigl (x_0,\varphi (x_0)-\tilde{y}\bigr )^{p\beta } \biggl (\int _{\varphi (x_0)+\tilde{y}}^\infty |f(x_0,s)|(s-\varphi (x_0)\!+\!\tilde{y})^{-2}\hbox {d}s\biggr )^p\hbox {d}\tilde{y}\\&\le \int _0^\infty \tilde{y}^p\rho \bigl (x_0,\varphi (x_0)-\tilde{y}\bigr )^{p\beta } \biggl (\int _{\tilde{y}}^\infty |f(x_0,\tilde{s}+\varphi (x_0))|\tilde{s}^{-2}d\tilde{s}\biggr )^p\hbox {d}\tilde{y} \end{aligned}$$

The essential step now lies in applying the following version of Hardy’s inequality with weights (see [19]),

$$\begin{aligned} \int _0^\infty \biggl (u(x)\int _x^\infty g(y)\hbox {d}y\biggr )^q \hbox {d}x \lesssim C(u,v)\int _0^\infty \bigl (v(y)g(y)\bigr )^q \hbox {d}y, \end{aligned}$$

which holds for $g$ non-negative and $1\le q\le \infty$ if, and only if

$$\begin{aligned} C(u,v)=\sup _{r>0}\biggl (\int _0^r |u(t)|^q\hbox {d}t\biggr )^{1/q}\biggl (\int _r^\infty |v(t)|^{-q'}\hbox {d}t\biggr )^{1/q'}<\infty . \end{aligned}$$

This shall be applied with $g(\tilde{s})=|f(x_0,\tilde{s}+\varphi (x_0))|\tilde{s}^{-2},\,u(t)=t\rho (x_0,\varphi (x_0)-t)^\beta$ and $v(t)=t^2\rho (x_0,t+\varphi (x_0))^\beta$. Upon $C(u,v)$ being finite, we obtain

$$\begin{aligned} \int _{-\infty }^{\varphi (x_0)}\rho (x_0,y)^{p\beta }|\mathfrak {E}f(x_0,y)|^p \hbox {d}y&\lesssim \int _0^\infty |f(x_0,\tilde{s}+\varphi (x_0))|^p \rho (x_0,\tilde{s}+\varphi (x_0))^{p\beta }\hbox {d}\tilde{s}\\&=\int _{\varphi (x_0)}^\infty |f(x_0,s)|^p\rho (x_0,s)^{p\beta }\hbox {d}s. \end{aligned}$$

Since for $y>\varphi (x_0)$ we have $\mathfrak {E}f(x_0,y)=f(x_0,y)$, we finally arrive at

$$\begin{aligned} \int _{-\infty }^\infty \rho (x_0,y)^{p\beta }|\mathfrak {E}f(x_0,y)|^p \mathrm{d}y \lesssim \int _{\varphi (x_0)}^\infty |f(x_0,s)|^p\rho (x_0,s)^{p\beta }\hbox {d}s, \end{aligned}$$

from which we easily obtain the claim for $m=0$.

Step 4 It remains to check the condition $C(u,v)<\infty$ for Hardy’s inequality. Note that we have to find a bound independent of $x_0$ or $\varphi (x_0)$. First we note that due to the assumption $\varphi (0)=0$, we also have $|\varphi (x_0)|\le M|x_0|$. For simplification we then assume $\beta \ge 0$, the case $\beta <0$ can be handled with similar arguments.

We start by noting

$$\begin{aligned} |x_0|^2&\le \rho \bigl (x_0,\varphi (x_0)\pm t\bigr )^2\le |x_0|^2+\bigl (|\varphi (x_0)|+t\bigr )^2\\&\le |x_0|^2+\bigl (M|x_0|+t\bigr )^2\le C^2_M\max (|x_0|,t)^2. \end{aligned}$$

We then find

$$\begin{aligned} \int _0^r u(t)^p\hbox {d}t \le \int _0^r t^pC_M^{\beta p}\max \bigl (|x_0|,t\bigr )^{\beta p}\hbox {d}t \lesssim C_M^{\beta p}r^{p+1}\max \bigl (|x_0|,r\bigr )^{\beta p}. \end{aligned}$$

Moreover, we obtain for $r<(2M+1)|x_0|$ (note $-2p'+1<0$)

$$\begin{aligned} \int _r^\infty v(t)^{-p'}\hbox {d}t \le \int _r^\infty t^{-2p'}|x_0|^{-\beta p'}\hbox {d}t \lesssim |x_0|^{-\beta p'}r^{-2p'+1}. \end{aligned}$$

For $r\ge (2M+1)|x_0|$ we have to take a little more care. In this case we have $t\ge r\ge 2M|x_0|\ge 2|\varphi (x_0)|$ and hence

$$\begin{aligned} \rho (x_0,\varphi (x_0)\pm t)\ge |\varphi (x_0)\pm t|\ge \bigl |t-|\varphi (x_0)|\bigr |\ge t/2. \end{aligned}$$

Then we find

$$\begin{aligned} \int _r^\infty v(t)^{-p'}\hbox {d}t \le \int _r^\infty t^{-(2+\beta )p'}\hbox {d}t\lesssim r^{-(\beta +2)p'+1}. \end{aligned}$$

Combined this gives (recall $\frac{1}{p}+\frac{1}{p'}=1$)

$$\begin{aligned} \sup _{r>0}&\biggl (\int _0^r |u(t)|^p\hbox {d}t\biggr )^{1/p}\biggl (\int _r^\infty |v(t)|^{-p'}\hbox {d}t\biggr )^{1/p'}\\&\le c(M,p,\beta )\sup _{r>0}\Bigl (\max \bigl (|x_0|,r\bigr )^\beta r^{1+1/p} \max \bigl (|x_0|,r\bigr )^{-\beta }r^{-2+1/p'}\Bigr )=c(M,p,\beta ) \end{aligned}$$

for some constant $c(M,p,\beta )$ independent of $x_0$.

Step 5 For partial derivatives of $f$ (i.e. the case $m>0$) similar arguments can be used (as explained in [22]). Exemplary we show it for some second-order partial derivative, w.l.o.g. $\partial _j^2\mathfrak {E}f$. It holds

$$\begin{aligned} \partial _j\mathfrak {E}f =\int _1^\infty \partial _j f(\ldots )\psi (\lambda )\,\hbox {d}\lambda +\int _1^\infty \partial _y f(\ldots ) \lambda \partial _j\delta ^*\psi (\lambda )\,\hbox {d}\lambda \end{aligned}$$

and hence

$$\begin{aligned} \partial _j^2\mathfrak {E}f&=\int _1^\infty \partial _j^2 f(\ldots )\psi (\lambda )\,\hbox {d}\lambda +2\int _1^\infty \partial _j\partial _y f(\ldots ) \lambda \partial _j\delta ^*\psi (\lambda )\,\hbox {d}\lambda \nonumber \\&\qquad +\int _1^\infty \partial _y^2 f(\ldots )(\lambda \partial _j\delta ^*)^2\psi (\lambda )\,\hbox {d}\lambda +\int _1^\infty \partial _y f(\ldots )\lambda \partial _j^2\delta ^*\psi (\lambda )\,\hbox {d}\lambda . \end{aligned}$$

(7.2)

We first note $\partial ^\alpha \delta ^*\le c_\alpha (\delta ^*)^{1-|\alpha |}$ for all multiindices $\alpha$ and $|\psi (\lambda )|\le A_k\lambda ^{-k}$. For the first term we then find as above with $y<\varphi (x_0)$

$$\begin{aligned} \biggl |\int _1^\infty \partial _j^2 f(\ldots )\psi (\lambda )\,\hbox {d}\lambda \biggr |&\le \int _1^\infty \bigl |\partial _j^2 f(\ldots )\psi (\lambda )\bigr |\,\hbox {d}\lambda \le A_2\int _1^\infty \bigl |\partial _j^2 f(\ldots )\bigr |\,\frac{\hbox {d}\lambda }{\lambda ^2}\nonumber \\&\lesssim \bigl (\varphi (x_0)-y\bigr )\int _{2\varphi (x_0)-y}^\infty |\partial _j^2 f(x_0,s)|(s-y)^{-2}\,\hbox {d}s, \end{aligned}$$

(7.3)

and similarly

$$\begin{aligned}&\biggl |\int _1^\infty \partial _j\partial _y f(\ldots ) \lambda \partial _j\delta ^*\psi (\lambda )\,\hbox {d}\lambda \biggr |\nonumber \\&\quad \le c_jA_3\bigl (\varphi (x_0)-y\bigr )\int _{2\varphi (x_0)-y}^\infty |\partial _j\partial _y f(x_0,s)|(s-y)^{-2}\,\hbox {d}s \end{aligned}$$

(7.4)

as well as

$$\begin{aligned}&\biggl |\int _1^\infty \partial _y^2 f(\ldots ) (\lambda \partial _j\delta ^*)^2\psi (\lambda )\,\hbox {d}\lambda \biggr |\nonumber \\&\quad \le c_j^2A_4\bigl (\varphi (x_0)-y\bigr )\int _{2\varphi (x_0)-y}^\infty |\partial _y^2 f(x_0,s)|(s-y)^{-2}\,\hbox {d}s. \end{aligned}$$

(7.5)

It remains the last integral in (7.2). We re-write $\partial _y f$ as

$$\begin{aligned} \partial _y f(x^0,y+\lambda \delta ^*) =\partial _y f(x^0,y+\delta ^*) +\int _{y+\delta ^*}^{y+\lambda \delta ^*}\partial _y^2 f(x_0,t)\,\hbox {d}t. \end{aligned}$$

Due to the choice of $\psi$, i.e. $\int _1^\infty \lambda \psi (\lambda )\,\hbox {d}\lambda =0$, we then have

$$\begin{aligned} \int _1^\infty \partial _y^2 f(\ldots )\lambda \partial _j^2\delta ^*\psi (\lambda )\,\hbox {d}\lambda =\int _1^\infty \lambda \partial _j^2\delta ^*\psi (\lambda ) \int _{y+\delta ^*}^{y+\lambda \delta ^*}\partial _y^2 f(x_0,t)\,\hbox {d}t\,\hbox {d}\lambda . \end{aligned}$$

This can be estimated by

$$\begin{aligned} \biggl |\int _1^\infty \partial _y^2 f(\ldots )\lambda \partial _j^2\delta ^*\psi (\lambda )\,\hbox {d}\lambda \biggr |&\lesssim (\delta ^*)^{-1}A_4\int _1^\infty \biggl ( \int _{y+\delta ^*}^{y+\lambda \delta ^*}\bigl |\partial _y^2 f(x_0,t)\bigr |\,\hbox {d}t\biggr ) \lambda ^{-3}\,\hbox {d}\lambda \\&=(\delta ^*)^{-1}A_4\int _{y+\delta ^*}^\infty \biggl ( \int _{\frac{t-y}{\delta ^*}}^\infty \lambda ^{-3}\,\hbox {d}\lambda \biggr ) \bigl |\partial _y^2 f(x_0,t)\bigr |\,\hbox {d}t\\&\sim (\delta ^*)^{-1}\int _{y+\delta ^*}^\infty (\delta ^*)^2 \bigl |\partial _y^2 f(x_0,t)\bigr |\,\frac{\hbox {d}t}{(t-y)^2}\\&\lesssim \bigl (\varphi (x_0)-y\bigr )\int _{2\varphi (x_0)-y}^\infty |\partial _y^2 f(x_0,s)|(s-y)^{-2}\,\hbox {d}s. \end{aligned}$$

This last integral and those in (7.3)–(7.5) are immediate counterparts of (7.1). From there, the remaining estimates follow by analogous arguments.

Similarly for all other partial derivatives of $\mathfrak {E}f$, after differentiation under the integral, every term can be treated separately, and for terms involving lower order derivatives of $f$, we use Taylor expansion and the moment conditions for $\psi$.

Step 6 So far, all arguments were valid for arbitrary special Lipschitz domains in arbitrary dimension $d$. For Babuska–Kondratiev spaces on polyhedral domains in $\mathbb {R}^3$, however, we need to consider a second weight function. Hence, assume we are given a special Lipschitz domain with $\varphi (x_1,0)=0$ for all $x_1\in \mathbb {R}$, and consider the weight function $\tilde{\rho }(x_1,x_2,x_3)^2=x_2^2+x_3^2$ (i.e. the distance to a fixed straight, for which w.l.o.g. we chose the $x_1$-axis). Then, all previous arguments in Steps 3 and 5 carry over without any change, only the condition for Hardy’s inequality needs to be checked for the new weights $u(t)=t\tilde{\rho }(x_0,\varphi (x_0)-t)^\beta$ and $v(t)=t^2\tilde{\rho }(x_0,t+\varphi (x_0))^\beta$, but clearly also these calculations can be transferred, upon simply replacing $|x_0|$ by $|x_2|$.

Step 7 The results for special Lipschitz domains in Steps 1–6 now can be used to derive the estimate for general Lipschitz polyhedral domains. The idea is to consider a suitable covering of the singularity set by (finitely) man open sets $U_1,\ldots ,U_N\subset \mathbb {R}^d$. This cover of $S$ is chosen in such a way that in every set, $U_i$ the distance to the singularity set $S$ can be described (after rotation and translation) by either of the weight functions used in Steps 3 or 6, respectively. This cover of $S$ then is to be extended with additional finitely many open sets $U_{N+1},\ldots ,U_M$ to an open cover of $\overline{D}$. On these sets $U_{N+1},\ldots ,U_M$ the distance function $\eta$ shall be bounded from below. Finally, we assume that we can associate with every $U_i$ a special Lipschitz domain $D_i$ such that $U_i\cap D=U_i\cap D_i$. With these sets $U_i$ and $D_i$ in hand, we are back in the situation of [22, Section 3.3], where it is described how to glue together the extension operators $\mathfrak {E}_i$ (w.r.t. the domains $D_i$) to finally obtain $\mathfrak {E}$ (essentially it is a partition of unity argument for some partition adapted to the domains $D_i$ and the neighbourhoods $U_i$). Note that the operators $\mathfrak {E}_{N+1},\ldots ,\mathfrak {E}_M$ correspond to the unweighted case, i.e. the situation in Stein’s original work.

In particular, if $D\subset \mathbb {R}^2$ is a polygon (or a Lipschitz domain with polygonal structure), then $S$ consists of finitely many points, which trivially can be covered by $N=\# S$ many, pairwise disjoint open sets $U_i$. For these sets and the associated special Lipschitz domains $D_i$, we use the arguments in Steps 3–5 (the reference points being the respective vertices).

In case of a polyhedral domain $D\subset \mathbb {R}^3$, the situation is a little more diverse. The cover of $S$ then consists of three types of open sets: the first one covering the interior of exactly one edge each, but staying away from all vertices. This clearly corresponds to the setting of Step 6. To describe the other two types, let $A\in S$ be a vertex, and $\varGamma _1,\ldots ,\varGamma _n$ edges with endpoint in $A$. Then for every $j$, we can find a cone $C_{A,\varGamma _j}$ with vertex in $A$ and axis $\varGamma _j$ with sufficiently small height and opening angle, so that no two such cones intersect. Clearly, in any such cone the distance to $S$ is exactly the distance to the axis of the cone (the intersection with $S$ is just the edge $\varGamma _j$). Finally, let $\tilde{B}_A$ be a ball around $A$ with sufficiently small radius and denote by $\tilde{C}_{A,\varGamma _j}$ a cone with half the opening angle of $C_{A,\varGamma _j}$. As the last type of neighbourhoods, we define $B_A$ to be the interior of $\tilde{B}_A\setminus \bigcup _j\tilde{C}_{A,\varGamma _j}$. Then on $B_A$, the distance to $S$ is equivalent to the distance to $A$.

The norm estimates for $\mathfrak {E}$ carry over to our situation without change, we only note that the estimates in Steps 2–6 due to the assumptions on the $U_i$ exactly correspond to estimates for the $\Vert \cdot |\mathcal {K}^m_{p,a}(S)\Vert$-norm. $\square$

Remark 8

The definition of the neighbourhoods for vertices and edges is essentially taken from [1]. In that article also, similar extension arguments can be found (Lemmas 3.15, 3.16). However, their arguments seem to contain some slight gaps: they fixed a reference point (a reference axis) for the weight functions and assumed $\varphi (x_0)=0$, not noting that for $\varphi (x_0)\ne 0$ “a simple translation in $x_3$” also moves the reference point (axis) for the weight.