A_{ij}=\mu_{{\boldsymbol{m}}(i)}^{-1/2}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\int_{\Gamma_{{\boldsymbol{m}}(j)}}\Phi(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x),\qquad
i,j\in{1,\ldots,N},$ (93)

where ${\boldsymbol{m}}(1),\ldots,{\boldsymbol{m}}(N)$ is an ordering of the
elements of $L_{h}$ (corresponding to the order of $f^{1},\ldots,f^{N}$ in
§5.3), and for brevity we have written
$\mu_{{\boldsymbol{m}}}:=\mathcal{H}^{d}(\Gamma_{{\boldsymbol{m}}})$ for
${\boldsymbol{m}}\in I_{\mathbb{N}_{0}}$. Recall from §3 that
${\boldsymbol{m}}\in I_{\mathbb{N}_{0}}$ means that either
${\boldsymbol{m}}=0$, in which case
$\Gamma_{{\boldsymbol{m}}}=\Gamma_{0}=\Gamma$ and
$\mu_{\boldsymbol{m}}=\mathcal{H}^{d}(\Gamma)$, or
${\boldsymbol{m}}=(m_{1},\ldots,m_{\ell})$ for some $\ell\in\mathbb{N}$, in
which case
$\mu_{{\boldsymbol{m}}}=\rho_{m_{1}}^{d}\cdots\rho_{m_{\ell}}^{d}\mathcal{H}^{d}(\Gamma)$.

For $i\neq j$ the integrand in (93) is continuous, because, by assumption, the
components of $\Gamma$ are disjoint. In that case, to evaluate (93) we use the
composite barycentre rule of [25, Defn 3.5]. In our context, this means
partitioning the BEM elements $\Gamma_{{\boldsymbol{m}}(i)}$ and
$\Gamma_{{\boldsymbol{m}}(j)}$, which have diameter approximately equal to
$h$, into a union of self-similar subsets of a possibly smaller approximate
diameter $0<h_{Q}\leq h$, writing the integral in (93) as a sum of integrals
over the Cartesian products of these subsets, and approximating these
integrals by a one-point barycentre rule. Specifically, we approximate

$\displaystyle A_{ij}\approx
A_{ij}^{Q}:=\mu_{{\boldsymbol{m}}(i)}^{-1/2}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(i)}}\sum_{\boldsymbol{n}^{\prime}\in
L_{h_{Q}}^{{\boldsymbol{m}}(j)}}\mu_{\boldsymbol{n}}\mu_{\boldsymbol{n}^{\prime}}\Phi(x_{\boldsymbol{n}},x_{\boldsymbol{n}^{\prime}}),\qquad
i,j\in{1,\ldots,N},\,i\neq j,$ (94)

where, for $i\in{1,\ldots,N}$,

$\displaystyle
L_{h_{Q}}^{{\boldsymbol{m}}(i)}:=\begin{cases}\\{{\boldsymbol{m}}(i)\\},&h_{Q}\geq\operatorname{diam}(\Gamma_{{\boldsymbol{m}}(i)}),\\\
\big{\\{}\boldsymbol{n}\in
L_{h_{Q}}:\,\Gamma_{\boldsymbol{n}}\subset\Gamma_{{\boldsymbol{m}}(i)}\\},&h_{Q}<\operatorname{diam}(\Gamma_{{\boldsymbol{m}}(i)}),\end{cases}$

describes the partitioning of $\Gamma_{{\boldsymbol{m}}(i)}$ and, for
$\boldsymbol{n}\in I_{\mathbb{N}_{0}}$,

$x_{\boldsymbol{n}}:=\frac{\int_{\Gamma_{\boldsymbol{n}}}x\,\mathrm{d}\mathcal{H}^{d}(x)}{\int_{\Gamma_{\boldsymbol{n}}}\,\mathrm{d}\mathcal{H}^{d}(x)}=\mu_{\boldsymbol{n}}^{-1}\int_{\Gamma_{\boldsymbol{n}}}x\,\mathrm{d}\mathcal{H}^{d}(x)$

is the barycentre of $\Gamma_{\boldsymbol{n}}$ with respect to the measure
$\mathcal{H}^{d}$. (Note that $x_{\boldsymbol{n}}$ is not necessarily an
element of $\Gamma_{\boldsymbol{n}}$.) The similarities can be written as
$s_{m}(x)=\rho_{m}A_{m}x+v_{m}$, for some $v_{m}\in\mathbb{R}^{n}$ and some
orthogonal $A_{m}\in\mathbb{R}^{n\times n}$ ($A_{m}=\pm 1$ in the case $n=1$),
and then, writing $\boldsymbol{n}=(n_{1},\ldots,n_{\ell})$, we find (see [25,
Prop. 3.3]) that

$x_{\boldsymbol{n}}=s_{n_{1}}\circ s_{n_{2}}\circ\ldots\circ
s_{n_{\ell}}\Bigg{(}\bigg{[}I-\sum_{m=1}^{M}\rho_{m}^{d+1}A_{m}\bigg{]}^{-1}\bigg{(}\sum_{m=1}^{M}\rho_{m}^{d}v_{m}\bigg{)}\Bigg{)},$

the formula that we use for calculation of $x_{\boldsymbol{n}}$ in §6.

For $i=j$ the integral (93) is singular. To evaluate it we adopt the
singularity subtraction approach of [25, §5], writing

$\displaystyle
A_{ii}=\mu_{{\boldsymbol{m}}(i)}^{-1}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\left(\Phi_{\rm
sing}(x,y)+\Phi_{\rm
reg}(x,y)\right)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x),$
(95)

where

$\Phi_{\rm sing}(x,y):=\begin{cases}-\frac{1}{2\pi}\log{|x-y|},&\quad n=1,\\\
\frac{1}{4\pi|x-y|},&\quad n=2,\end{cases}$

and $\Phi_{\rm reg}:=\Phi-\Phi_{\rm sing}$. The integral of $\Phi_{\rm reg}$
has a continuous integrand and can be evaluated using the composite barycentre
rule, viz.

$\displaystyle\int_{\Gamma_{{\boldsymbol{m}}(i)}}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\Phi_{\rm
reg}(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x)\approx\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(i)}}\sum_{\boldsymbol{n}^{\prime}\in
L_{h_{Q}}^{{\boldsymbol{m}}(i)}}\mu_{\boldsymbol{n}}\mu_{\boldsymbol{n}^{\prime}}\Phi_{\rm
reg}(x_{\boldsymbol{n}},x_{\boldsymbol{n}^{\prime}}).$ (96)

The integral of $\Phi_{\rm sing}$ over
$\Gamma_{{\boldsymbol{m}}(i)}\times\Gamma_{{\boldsymbol{m}}(i)}$ is singular,
but, using the self-similarity of $\Gamma$ and the symmetry and homogeneity
properties of $\Phi_{\rm sing}$, namely the fact that, for $m=1,\ldots,M$,

$\Phi_{\rm sing}(s_{m}(x),s_{m}(y))=\left\\{\begin{array}[]{cc}\Phi_{\rm
sing}(x,y)-\frac{1}{2\pi}\log\rho_{m},&n=1,\\\ \rho_{m}^{-1}\Phi_{\rm
sing}(x,y),&n=2,\end{array}\right.$

it can be written (see [25, Thm 4.6]) as

$\displaystyle\int_{\Gamma_{{\boldsymbol{m}}(i)}}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\Phi_{\rm
sing}(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x)$
$\displaystyle=\begin{cases}\displaystyle\bigg{(}1-\sum_{m=1}^{M}\rho_{m}^{2d}\bigg{)}^{-1}\sum_{m=1}^{M}\Bigg{(}-\frac{1}{2\pi}\mu_{{\boldsymbol{m}}(i)}^{2}\rho_{m}^{2d}\log(\rho_{m})&\\\
\displaystyle\hskip 99.58464pt+\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}\int_{\Gamma_{({\boldsymbol{m}}(i),m)}}\int_{\Gamma_{({\boldsymbol{m}}(i),m^{\prime})}}\\!\\!\\!\\!\\!\\!\Phi_{\rm
sing}(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x)\Bigg{)},&n=1,\\\
\displaystyle\bigg{(}1-\sum_{m=1}^{M}\rho_{m}^{2d-1}\bigg{)}^{-1}\sum_{m=1}^{M}\sum_{\begin{subarray}{c}m^{\prime}=1\\\
m^{\prime}\neq
m\end{subarray}}^{M}\int_{\Gamma_{({\boldsymbol{m}}(i),m)}}\int_{\Gamma_{({\boldsymbol{m}}(i),m^{\prime})}}\Phi_{\rm
sing}(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x),&n=2.\end{cases}$
(97)

The integrals appearing in (97) all have continous integrands and can be
evaluated using the composite barycentre rule (as in [25, Eqn (47)]), viz.

$\displaystyle\int_{\Gamma_{({\boldsymbol{m}}(i),m)}}\int_{\Gamma_{({\boldsymbol{m}}(i),m^{\prime})}}\Phi_{\rm
sing}(x,y)\,\mathrm{d}\mathcal{H}^{d}(y)\mathrm{d}\mathcal{H}^{d}(x)\approx\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{({\boldsymbol{m}}(i),m)}}\sum_{\boldsymbol{n}^{\prime}\in
L_{h_{Q}}^{({\boldsymbol{m}}(i),m^{\prime})}}\mu_{\boldsymbol{n}}\mu_{\boldsymbol{n}^{\prime}}\Phi_{\rm
sing}(x_{\boldsymbol{n}},x_{\boldsymbol{n}^{\prime}}).$ (98)

Let $A_{ii}^{Q}$ denote the resulting approximation of $A_{ii}$, obtained by
combining (95)-(98).

Assuming the data $g$ is sufficiently smooth, the right-hand side entries (61)
are regular single integrals and can be evaluated by a single integral version
of the composite barycentre rules described for double integrals above, as in
[25, Defn 3.1]). Explicitly, we use the approximation

$\displaystyle
b_{i}=-\mu_{{\boldsymbol{m}}(i)}^{-1/2}\int_{\Gamma_{{\boldsymbol{m}}(i)}}\mathrm{tr}_{\Gamma}g(x)\,\mathrm{d}\mathcal{H}^{d}(x)\approx
b_{i}^{Q}:=-\mu_{{\boldsymbol{m}}(i)}^{-1/2}\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(i)}}\mu_{\boldsymbol{n}}\mathrm{tr}_{\Gamma}g(x_{\boldsymbol{n}}).$
(99)

Having computed $A^{Q}=[A_{ij}^{Q}]_{i,j=1}^{N}\approx A$ and
${\vec{b}}^{Q}=(b_{1}^{Q},\ldots,b_{N}^{Q})^{T}\approx{\vec{b}}$, we solve,
instead of (57), the perturbed linear system
$A^{Q}{\vec{c}}^{\,Q}={\vec{b}}^{Q}$. Provided this is uniquely solvable (for
which see Corollary 5.17 below), our fully discrete approximation to $\phi$ is
$\phi_{N}^{Q}:=\sum_{j=1}^{N}c_{j}^{Q}e^{j}$.

Provided $\varphi\in(\widetilde{H}{}^{1/2}(\Gamma^{c}))^{\perp}$ is
sufficiently smooth, we approximate $J(\phi)$, given by (62), by an
approximation $J^{Q}(\phi_{N}^{Q})$ defined in a similar way to (99). Using
(65), we have that

$J(\phi)\approx
J(\phi_{N})=\sum_{j=1}^{N}c_{j}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\int_{\Gamma_{{\boldsymbol{m}}(j)}}\mathrm{tr}_{\Gamma}\varphi\,\mathrm{d}\mathcal{H}^{d}\approx
J^{Q}(\phi^{Q}_{N})=\sum_{j=1}^{N}c_{j}^{Q}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(j)}}\mu_{\boldsymbol{n}}\mathrm{tr}_{\Gamma}\varphi(x_{\boldsymbol{n}}),$
(100)

where, for every $\psi_{N}=\sum_{j=1}^{N}d_{j}e^{j}\in V_{N}$,

$J^{Q}(\psi_{N}):=\sum_{j=1}^{N}d_{j}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(j)}}\mu_{\boldsymbol{n}}\mathrm{tr}_{\Gamma}\varphi(x_{\boldsymbol{n}}).$
(101)

The following quadrature error estimates follow from results in [25],
specifically [25, Prop. 5.2, Thms 3.6(iii), 5.7 & 5.11]. Following the
terminology of [25], we say an IFS attractor $\Gamma$ is hull-disjoint if the
convex hulls $\mathrm{Hull}(\Gamma_{1}),\ldots,\mathrm{Hull}(\Gamma_{M})$ are
disjoint, which holds if and only if the OSC holds for some open set
$O\supset\mathrm{Hull}(\Gamma)$ (cf. Lemma 2.5, the proof of which works in
the same way for hull-disjointness as for disjointness). The assumption of
hull-disjointness permits analysis of the singular quadrature rules in [25] by
Taylor expansion. (The point here is that while the barycentre
$x_{{\boldsymbol{m}}}$ of a component $\Gamma_{{\boldsymbol{m}}}$ may not lie
in $\Gamma_{{\boldsymbol{m}}}$, it must lie in
$\mathrm{Hull}(\Gamma_{{\boldsymbol{m}}}$).) However, numerical experiments
suggest that hull-disjointness is not essential for the applicability of the
quadrature rules in [25] (see [25, §6]). We also suspect that the estimates
(105) and (104) may not be sharp in their $h$-dependence — see [25, Rem. 5.10]
and the discussion around [25, Fig. 7]. In Remark 5.19 we describe
modifications to our quadrature rule which may improve its efficiency.

######  Theorem 5.14.

Let $\Gamma$ be an IFS attractor satisfying the OSC with $n-1<d={\rm
dim_{H}}(\Gamma)<n$. Suppose that $0<h_{Q}\leq h$, and let $A_{ij}$,
$A_{ij}^{Q}$, $b_{i}$ and $b_{i}^{Q}$ be as in (93)-(99), and $J(\cdot)$ and
$J^{Q}(\cdot)$ be as defined by (62) and (101), respectively, for some
$\varphi\in(\widetilde{H}{}^{1/2}(\Gamma^{c}))^{\perp}$.

  1. (i)

Suppose that $\mathrm{tr}_{\Gamma}g=G|_{\Gamma}$, for some $G$ that is twice
boundedly differentiable in some open neighbourhood of
$\mathrm{Hull}(\Gamma)$. (For the scattering problem this holds with
$G=-u^{i}$, if $u^{i}$ satisfies the Helmholtz equation in a neighbourhood of
$\mathrm{Hull}(\Gamma)$.) Then, for $i=1,\ldots,N$,

$\displaystyle|b_{i}-b^{Q}_{i}|$ $\displaystyle\leq
h_{Q}^{2}|G|_{2,\mathrm{Hull}(\Gamma)}\mu_{{\boldsymbol{m}}(i)}^{1/2}\qquad\text{
and hence }\qquad\|{\vec{b}}-{\vec{b}}^{Q}\|_{2}\leq
h_{Q}^{2}|G|_{2,\mathrm{Hull}(\Gamma)}\mathcal{H}^{d}(\Gamma)^{1/2},$ (102)

where
$|G|_{2,\mathrm{Hull}(\Gamma)}:=\max_{x\in\mathrm{Hull}(\Gamma)}\max_{\begin{subarray}{c}\alpha\in\mathbb{N}_{0}^{n}\\\
|\alpha|=2\end{subarray}}|D^{\alpha}G(x)|$.

  2. (ii)

Suppose that $\mathrm{tr}_{\Gamma}\varphi=V|_{\Gamma}$, where $V$ is twice
boundedly differentiable in some open neighbourhood of
$\mathrm{Hull}(\Gamma)$. (For $\varphi$ given by (63) this holds with $V=-v$
if $v$ is $C^{\infty}$ in a neighbourhood of $\mathrm{Hull}(\Gamma)$.) Then,
for $-1/2\leq s<-(n-d)/2$, there exists a constant $C>0$, independent of $h$,
$h_{Q}$, and $V$, such that, for all $\psi_{N}\in V_{N}=Y_{h}$,

$\displaystyle|J(\psi_{N})-J^{Q}(\psi_{N})|$ $\displaystyle\leq
Ch_{Q}^{2}h^{s+(n-d)/2}|V|_{2,\mathrm{Hull}(\Gamma)}\|\psi_{N}\|_{H_{\Gamma}^{s}}.$
(103)

  3. (iii)

Suppose that $\Gamma$ is hull-disjoint. Then there exists a constant $C>0$,
independent of $h$ and $h_{Q}$, such that, for $i,j=1,\ldots,N$,

$\displaystyle|A_{ij}-A^{Q}_{ij}|$ $\displaystyle\leq
Ch_{Q}h^{-n}\mu_{{\boldsymbol{m}}(i)}^{1/2}\mu_{{\boldsymbol{m}}(j)}^{1/2},\qquad\text{
and hence }\qquad\|A-A^{Q}\|_{2}\leq Ch_{Q}h^{-n}\mathcal{H}^{d}(\Gamma).$
(104)

If, further, $\Gamma$ is homogeneous, then there exists a constant $C>0$,
independent of $h$ and $h_{Q}$, such that

$\displaystyle|A_{ij}-A^{Q}_{ij}|$ $\displaystyle\leq
Ch^{2}_{Q}h^{-(n+1)}\mu_{{\boldsymbol{m}}(i)}^{1/2}\mu_{{\boldsymbol{m}}(j)}^{1/2},\qquad\text{
and hence }\qquad\|A-A^{Q}\|_{2}\leq
Ch_{Q}^{2}h^{-(n+1)}\mathcal{H}^{d}(\Gamma).$ (105)

###### Proof.

In what follows, when applying results from [25] we are taking $h$, $\Gamma$
and $\Gamma^{\prime}$ in [25] to be respectively $h_{Q}$,
$\Gamma_{{\boldsymbol{m}}(i)}$, and $\Gamma_{{\boldsymbol{m}}(j)}$ from the
current paper. For (i), noting that

$|b_{i}-b^{Q}_{i}|=\mu_{{\boldsymbol{m}}(i)}^{-1/2}\bigg{|}\int_{\Gamma_{{\boldsymbol{m}}(i)}}G(x)\,\mathrm{d}\mathcal{H}^{d}(x)-\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(i)}}\mu_{\boldsymbol{n}}G(x_{\boldsymbol{n}})\bigg{|},$

the first estimate in (102) follows from [25, Thm 3.6(iii)], and the second
then follows from the fact that
$\sum_{i=1}^{N}\mu_{{\boldsymbol{m}}(i)}=\mathcal{H}^{d}(\Gamma)$. For (ii),
given $\psi_{N}\in V_{N}$ let
${\vec{\psi}}=(({\vec{\psi}})_{1},\ldots,({\vec{\psi}})_{N})^{T}\in\mathbb{C}^{N}$
denote the coefficient vector of its expansion with respect to the basis
$(e^{1},\ldots,e^{N})$. By the orthonormality of the basis
$(f^{1},\ldots,f^{N})$ in $\mathbb{L}_{2}(\Gamma)$, the inverse estimate (86)
and the isometry property (16), given $0<t<1$ we can bound

$\displaystyle\|{\vec{\psi}}\|_{2}=\|(\mathrm{tr}_{\Gamma}^{*})^{-1}\psi_{N}\|_{\mathbb{L}_{2}(\Gamma)}\overset{\eqref{eq:InvEst}}{\leq}{c_{t}}h^{-t}\|(\mathrm{tr}_{\Gamma}^{*})^{-1}\psi_{N}\|_{\mathbb{H}^{-t}(\Gamma)}\overset{\eqref{eq:tr*}}{=}{c_{t}}h^{-t}\|\psi_{N}\|_{H^{-t-\frac{n-d}{2}}_{\Gamma}}.$
(106)

Then, to prove (103), using (65), (101), [25, Thm 3.6(iii)], (106), and the
Cauchy-Schwarz inequality,

$\displaystyle|J(\psi_{N})-J^{Q}(\psi_{N})|$ $\displaystyle=$
$\displaystyle\Bigg{|}\sum_{j=1}^{N}({\vec{\psi}})_{j}\mu_{{\boldsymbol{m}}(j)}^{-1/2}\bigg{(}\int_{\Gamma_{{\boldsymbol{m}}(j)}}V(x)\,\mathrm{d}\mathcal{H}^{d}(x)-\sum_{\boldsymbol{n}\in
L_{h_{Q}}^{{\boldsymbol{m}}(j)}}\mu_{\boldsymbol{n}}V(x_{\boldsymbol{n}})\bigg{)}\Bigg{|}$
$\displaystyle\leq$ $\displaystyle
h_{Q}^{2}|V|_{2,\mathrm{Hull}(\Gamma)}(\mathcal{H}^{d}(\Gamma))^{1/2}\|{\vec{\psi}}\|_{2}$
$\displaystyle\leq$ $\displaystyle
c_{t}h_{Q}^{2}h^{-t}|V|_{2,\mathrm{Hull}(\Gamma)}(\mathcal{H}^{d}(\Gamma))^{1/2}\|\psi_{N}\|_{H_{\Gamma}^{-t-\frac{n-d}{2}}}.$

For (iii), the first estimates in (104) and (105) follow from [25, Prop 5.2]
for the off-diagonal terms and [25, Thms 5.7 & 5.11] for the diagonal terms.
The factors of $h^{-(n+1)}$ and $h^{-n}$ come from the fact that, in the
notation of [25, Thms 5.7 & 5.11], $R_{\Gamma,\mathrm{Hull}}$ equals
$\operatorname{diam}(\Gamma)$ times a constant independent of
$\operatorname{diam}(\Gamma)$. We note that, since we are allowing $C$ to be
$k$-dependent, we can use the “$k\operatorname{diam}(\Gamma)\leq c_{\rm osc}$”
estimates in [25, Thms 5.7 & 5.11], since, given $k>0$, the constant $c_{\rm
osc}$ can be chosen as large as is required. To derive the second estimate in
(104) (a similar argument gives the second estimate in (105)), note that, with
$\|\cdot\|_{\mathrm{F}}$ denoting the Frobenius norm,

$\|A-A^{Q}\|_{2}\leq\|A-A^{Q}\|_{\mathrm{F}}\leq
Ch_{Q}h^{-n}\bigg{(}\sum_{i=1}^{N}\sum_{j=1}^{N}\mu_{{\boldsymbol{m}}(i)}\mu_{{\boldsymbol{m}}(j)}\bigg{)}^{1/2}=Ch_{Q}h_{-n}\mathcal{H}^{d}(\Gamma).$

∎

######  Remark 5.15 (Value of $\mathcal{H}^{d}(\Gamma)$).

The proposed quadrature formulas require the values of
$\mu_{{\boldsymbol{m}}}=\mathcal{H}^{d}(\Gamma_{\boldsymbol{m}})$, which are
easily computed in terms of $\mathcal{H}^{d}(\Gamma)$ as
$\mu_{{\boldsymbol{m}}}=\rho_{m_{1}}^{d}\cdots\rho_{m_{\ell}}^{d}\mathcal{H}^{d}(\Gamma)$,
for all ${\boldsymbol{m}}\in I_{\mathbb{N}}$. For $n=1$, the Hausdorff measure
of a class of Cantor sets is shown to be $\mathcal{H}^{d}(\Gamma)=1$ in [23,
Thm. 1.14–1.15]; see, e.g., [47] for more recent related results. However, for
$n>1$ the exact value of the Hausdorff measure of even the simplest IFS
attractors is known only for $d\leq 1$ (see, e.g., [46]), i.e. for the cases
that are not relevant for scattering problems (recall, as discussed above
Lemma 4.4, that $H^{-1/2}_{\Gamma}=\\{0\\}$ and $\phi=0$ for $d\leq n-1$).
Thus, the entries of the Galerkin matrix (60) and right-hand side vector (61)
in our Hausdorff BEM can be computed only up to unknown factors that are
powers of the unknown measure $\mathcal{H}^{d}(\Gamma)$, so that the same is
true for the vector $\vec{c}$ that is the solution of the Galerkin linear
system (57). To be precise, we can’t compute $\vec{c}$, but we can (even if we
don’t know $\mathcal{H}^{d}(\Gamma)$) compute
${\vec{c}}^{\,\star}=(c_{1}^{\star},\ldots,c_{N}^{\star})^{T}$, related to
$\vec{c}$ by ${\vec{c}}^{\,\star}=(\mathcal{H}^{d}(\Gamma))^{1/2}\vec{c}$.

Surprisingly, this inability to compute $\vec{c}$ does not affect in the
slightest the applicability of our Hausdorff BEM. Indeed, it is enough to know
${\vec{c}}^{\,\star}$ (whose value is independent of
$\mathcal{H}^{d}(\Gamma)$) to compute the BEM solution
$\phi_{N}=\sum_{j=1}^{N}c_{j}e^{j}\in H_{\Gamma}^{-1/2}$. For recall that we
know $\phi_{N}$ (as an element of the set of distributions
$H_{\Gamma}^{-1/2}$) if we can compute the duality pairing
$\langle\varphi,\phi_{N}\rangle_{H^{1/2}(\mathbb{R}^{n})\times
H^{-1/2}(\mathbb{R}^{n})}$ for every $\varphi\in H^{1/2}(\mathbb{R}^{n})$.
Further, noting that $\operatorname{supp}(f^{j})=\Gamma_{{\boldsymbol{m}}(j)}$
and that
$\mu_{\boldsymbol{m}}=\mathcal{H}^{d}(\Gamma_{\boldsymbol{m}})=(\operatorname{diam}(\Gamma_{\boldsymbol{m}})/\operatorname{diam}(\Gamma))^{d}\,\mathcal{H}^{d}(\Gamma)$,
we see, using (15), that

$\displaystyle\langle\varphi,\phi_{N}\rangle_{H^{1/2}(\mathbb{R}^{n})\times
H^{-1/2}(\mathbb{R}^{n})}=\sum_{j=1}^{N}\overline{c_{j}}\int_{\Gamma}f^{j}\mathrm{tr}_{\Gamma}\varphi\,\mathrm{d}\mathcal{H}^{d}=\sum_{j=1}^{N}\overline{c^{\star}_{j}}\left(\textstyle{\frac{\operatorname{diam}(\Gamma)}{\operatorname{diam}(\Gamma_{{\boldsymbol{m}}(j)})}}\right)^{d/2}\frac{1}{\mathcal{H}^{d}(\Gamma)}\int_{\Gamma_{{\boldsymbol{m}}(j)}}\mathrm{tr}_{\Gamma}\varphi\,\mathrm{d}\mathcal{H}^{d},$

for every $\varphi\in H^{1/2}(\mathbb{R}^{n})$, and the right hand side of the
above is independent of the value of $\mathcal{H}^{d}(\Gamma)$.

A simple implementation technique which avoids working with an unknown value
for $\mathcal{H}^{d}(\Gamma)$ and produces the correct solution $\phi_{N}$ is
just to set $\mathcal{H}^{d}(\Gamma)=1$ in all calculations. This is
equivalent to introducing a “normalised Hausdorff measure”
$\mathcal{H}^{d}_{\star}(\cdot):=\mathcal{H}^{d}(\cdot)/\mathcal{H}^{d}(\Gamma)$,
so that $\mathcal{H}^{d}_{\star}(\Gamma)=1$, and using it throughout in the
BEM in place of $\mathcal{H}^{d}(\cdot)$.

To study the influence of the quadrature error on the BEM solution we first
adapt the first Strang lemma to our setting.

######  Proposition 5.16.

Let $\Gamma$ be a disjoint IFS attractor with $n-1<d={\rm dim_{H}}(\Gamma)<n$.
Assume that the unique solution $\phi$ of (36) belongs to $H^{s}_{\Gamma}$ for
some $-1/2<s<-(n-d)/2$, so that $0<s+1/2<t_{d}$. Let $V_{N}=Y_{h}$ so that
$N=\dim(V_{N})$. Let $\alpha$ be the coercivity constant of (41) and
$c_{t_{d}}$ the constant in the inverse inequality (86) when $t=t_{d}$. Assume
that $A^{Q}\in\mathbb{C}^{N\times N}$ and ${\vec{b}}^{Q}\in\mathbb{C}^{N}$ are
approximations of the Galerkin matrix $A$ and the right-hand side vector
${\vec{b}}$ in (60)–(61) satisfying

$\|A-A^{Q}\|_{2}\leq
h^{2t_{d}}E_{A}\qquad\text{and}\qquad\|{\vec{b}}-{\vec{b}}^{Q}\|_{2}\leq
h^{t_{d}+s+1/2}E_{b},$ (107)

for some $0\leq E_{A}<\alpha/c_{t_{d}}^{2}$, and $E_{b}\geq 0$. Then the
perturbed linear system $A^{Q}{\vec{c}}^{\,Q}={\vec{b}}^{Q}$ is invertible and
the corresponding solution $\phi_{N}^{Q}:=\sum_{j=1}^{N}c_{j}^{Q}e^{j}\in
V_{N}$ satisfies the error bound

$\displaystyle\|\phi_{N}-\phi_{N}^{Q}\|_{H^{-1/2}_{\Gamma}}$
$\displaystyle\leq\frac{C}{\alpha_{Q}}\Big{(}E_{A}\|\phi\|_{H^{s}_{\Gamma}}+E_{b}\Big{)}h^{s+1/2},$
(108)

where $\phi_{N}=\sum_{j=1}^{N}c_{j}e^{j}\in V_{N}$ is the Galerkin solution
given by (56), $\alpha_{Q}:=\alpha-E_{A}c_{t_{d}}^{2}$, and $C>0$ is a
constant independent of $h$, $\phi$, $E_{a}$, $E_{b}$, and $N$.

###### Proof.

As in the proof of Theorem 5.14, given $\psi_{N}\in V_{N}$ let
${\vec{\psi}}\in\mathbb{C}^{N}$ denote the coefficient vector of its expansion
with respect to the basis $(e^{1},\ldots,e^{N})$. Let $B(\psi_{N}):=-\langle
g,\psi_{N}\rangle_{H^{1/2}(\Gamma_{\infty})\times
H^{-1/2}(\Gamma_{\infty})}={\vec{\psi}}^{H}\vec{b}$ and denote the perturbed
sesquilinear form and antilinear functional by
$a^{Q}(\xi_{N},\psi_{N}):={\vec{\psi}}^{H}A^{Q}{\vec{\xi}}$ and
$B^{Q}(\psi_{N}):={\vec{\psi}}^{H}\vec{b}^{Q}$, for $\xi_{N},\psi_{N}\in
V_{N}$. The first bound in (107) gives

$\displaystyle|a(\xi_{N},\psi_{N})-a^{Q}(\xi_{N},\psi_{N})|={|{\vec{\psi}}^{H}(A-A^{Q}){\vec{\xi}}|}$
$\displaystyle\leq\|A-A^{Q}\|_{2}\|{\vec{\xi}}\|_{2}\|{\vec{\psi}}\|_{2}$
$\displaystyle\leq
h^{t_{d}}E_{A}{c_{t_{d}}}\|{\vec{\xi}}\|_{2}\|\psi_{N}\|_{H^{-1/2}_{\Gamma}}$
(109) $\displaystyle\leq
E_{A}{c_{t_{d}}^{2}}\|\xi_{N}\|_{H^{-1/2}_{\Gamma}}\|\psi_{N}\|_{H^{-1/2}_{\Gamma}},\qquad\forall\xi_{N},\psi_{N}\in
V_{N}.$

From this and the coercivity of $a(\cdot,\cdot)$ in (41) follows the
coercivity of the perturbed form, and hence the invertibility of $A^{Q}$:

$|a^{Q}(\psi_{N},\psi_{N})|\geq|a(\psi_{N},\psi_{N})|-|a(\psi_{N},\psi_{N})-a^{Q}(\psi_{N},\psi_{N})|\geq\alpha_{Q}\|\psi_{N}\|_{H^{-1/2}_{\Gamma}}^{2},\quad\forall\psi_{h}\in
V_{N}.$

The second bound in (107), combined with (106) for $t=t_{d}$, gives

$\displaystyle|B(\psi_{N})-B^{Q}(\psi_{N})|=|{\vec{\psi}}^{H}({\vec{b}}-{\vec{b}}^{Q})|\leq\|{\vec{b}}-{\vec{b}}^{Q}\|_{2}\|{\vec{\psi}}\|_{2}\leq
E_{b}c_{t_{d}}h^{s+1/2}\|\psi_{N}\|_{H^{-1/2}_{\Gamma}}^{2},\quad\forall\psi_{h}\in
V_{N}.$ (110)

Now $\phi^{Q}_{N}\in V_{N}$ is the solution of
$a^{Q}(\phi^{Q}_{N},\psi_{N})=B^{Q}(\psi_{N})$, $\forall\psi_{N}\in V_{N}$.
Strang’s first lemma (e.g. [5, III.1.1]), applied in the special case that
$V=S_{h}$ (in the notation of [5, III.1.1]), combined with (109) and (110),
gives

$\displaystyle\|\phi_{N}-\phi^{Q}_{N}\|_{H^{-1/2}_{\Gamma}}$
$\displaystyle\leq\frac{1}{\alpha_{Q}}\bigg{(}\sup_{\psi_{N}\in
V_{N}}\frac{|a(\phi_{N},\psi_{N})-a^{Q}(\phi_{N},\psi_{N})|}{\|\psi_{N}\|_{H^{-1/2}_{\Gamma}}}+\sup_{\psi_{N}\in
V_{N}}\frac{|B(\psi_{N})-B^{Q}(\psi_{N})|}{\|\psi_{N}\|_{H^{-1/2}_{\Gamma}}}\bigg{)}$
$\displaystyle\leq\frac{c_{t_{d}}}{\alpha_{Q}}\bigg{(}E_{A}h^{t_{d}}\|{\vec{c}}\|_{2}+E_{b}h^{s+1/2}\bigg{)}.$

Applying (106) with $t=-s-(n-d)/2$, we obtain that

$\displaystyle\|\phi_{N}-\phi^{Q}_{N}\|_{H^{-1/2}_{\Gamma}}\leq\frac{c_{t_{d}}}{\alpha_{Q}}\bigg{(}c_{-s-\frac{n-d}{2}}E_{A}\|\phi_{N}\|_{H^{s}_{\Gamma}}+E_{b}\bigg{)}h^{s+1/2},$

and the bound (108) follows on applying Corollary 5.12 with $s_{2}=s$. ∎

By combining Theorem 5.6, Proposition 5.16 and the quadrature error bounds in
Theorem 5.14, we can complete the convergence analysis of our fully discrete
Hausdorff BEM for a hull-disjoint IFS attractor.

######  Corollary 5.17.

Let $\Gamma$ be a hull-disjoint IFS attractor with $n-1<d={\rm
dim_{H}}(\Gamma)<n$. Assume that the unique solution $\phi$ of (36) belongs to
$H^{s}_{\Gamma}$ for some $-1/2<s<-(n-d)/2$. Let $V_{N}=Y_{h}$. Let the
entries of the Galerkin matrix $A$ (60) and right-hand side ${\vec{b}}$ (61)
be approximated with the quadrature formulas outlined in (93)-(99). Let $g$
satisfy the assumptions of Theorem 5.14(i), and suppose that

$\displaystyle h_{Q}\leq C_{Q}h^{d+1},$ (111)

for some sufficiently small $C_{Q}>0$, independent of $h_{Q}$ and $h$. Then
the approximated linear system is invertible and the fully discrete solution
$\phi_{N}^{Q}$ satisfies the error bound

$\displaystyle\|\phi-\phi_{N}^{Q}\|_{H^{-1/2}_{\Gamma}}$ $\displaystyle\leq
C\big{(}\|\phi\|_{H^{s}_{\Gamma}}+|G|_{2,\mathrm{Hull}(\Gamma)}\big{)}h^{s+1/2},$
(112)

for some $C>0$ independent of $h$, $\phi$, and $G$.

If, further, $\Gamma$ is homogeneous the above result holds with (111)
replaced by the weaker condition

$\displaystyle h_{Q}\leq C_{Q}h^{d/2+1}.$ (113)

###### Proof.

The two conditions (111) and (113) can be written as

$\displaystyle h_{Q}\leq C_{Q}h^{\frac{d}{1+p}+1},$ (114)

where $p=0$ in the general case, and $p=1$ when $\Gamma$ is also homogeneous.
By Theorem 5.14(i) and (iii) our quadrature formulas achieve (107) with

$E_{A}=C_{*}h_{Q}^{1+p}h^{-d-1-p},\qquad
E_{b}=C_{*}h_{Q}^{2}h^{-s-1+\frac{n-d}{2}}|G|_{2,\mathrm{Hull}(\Gamma)},$
(115)

for some constant $C_{*}$ independent of both $h$ and $h_{Q}$. Hence (114)
implies that $E_{A}$ is bounded independently of $h$, and, furthermore, by
choosing the constant $C_{Q}$ in (114) to be sufficiently small, one can
ensure that $E_{A}\leq\alpha/(2c_{t_{d}}^{2})$, so that the discrete system is
invertible by Proposition 5.16, with $\alpha_{Q}:=\alpha-
E_{A}c_{t_{d}}^{2}\geq\alpha/2$. The assumption of (114) also ensures that
$E_{b}$ is bounded independently of $h$, with

$E_{b}\leq
C_{*}C_{Q}^{2}h^{\frac{2d}{1+p}-s+\frac{n-d}{2}+1}|G|_{2,\mathrm{Hull}(\Gamma)}\leq
C_{*}C_{Q}^{2}(\operatorname{diam}(\Gamma))^{\frac{2d}{1+p}-s+\frac{n-d}{2}+1}|G|_{2,\mathrm{Hull}(\Gamma)},$

since $h\leq\operatorname{diam}(\Gamma)$ and
$\frac{2d}{1+p}-s+\frac{n-d}{2}+1>0$ for both $p=0$ and $p=1$. Then, by (108),

$\displaystyle\|\phi_{N}-\phi_{N}^{Q}\|_{H^{-1/2}_{\Gamma}}$
$\displaystyle\leq
C\big{(}\|\phi\|_{H^{s}_{\Gamma}}+|G|_{2,\mathrm{Hull}(\Gamma)}\big{)}h^{s+1/2},$
(116)

for some $C>0$ independent of $h$, $\phi$, and $G$, and by combining this with
(80) we deduce (112). ∎

Under a stronger condition on $h_{Q}$ we can also prove a superconvergence
result for the fully discrete approximation $J^{Q}(\phi_{N}^{Q})$, given by
(101), to the linear functional $J(\phi)$ given by (62) (cf. [35, Thm.
4.2.18]). The significance of this result for the computation of $u(x)$ and
$u^{\infty}(\hat{x})$ is as spelled out in and above Remark 5.7.

######  Corollary 5.18.

Suppose that $\Gamma$, $\phi$, $V_{N}=Y_{h}$, and $g$ are defined, and satisfy
the same assumptions, as in Corollary 5.17. Let $J$ and $J^{Q}$ be defined by
(62) and (101) and suppose that $\varphi$ satisfies the assumptions of Theorem
5.14(ii). Suppose also that $\zeta\in H_{\Gamma}^{s}$, where $\zeta$ is the
solution of (56) with $-g$ replaced by $\varphi$, and suppose that

$h_{Q}\leq C^{\prime}_{Q}h^{d+s+3/2},$ (117)

for some sufficiently small $C^{\prime}_{Q}>0$, independent of $h$ and
$h_{Q}$. Then the approximated linear system is invertible and

$\displaystyle|J(\phi)-J^{Q}(\phi_{N}^{Q})|\leq$ $\displaystyle\hskip
12.91663ptC\Big{(}\|\phi\|_{H^{s}_{\Gamma}}(\|\zeta\|_{H^{s}_{\Gamma}}+\|\varphi\|_{H^{1/2}(\Gamma_{\infty})}+|V|_{2,\mathrm{Hull}(\Gamma)})+(\|\varphi\|_{H^{1/2}(\Gamma_{\infty})}+|V|_{2,\mathrm{Hull}(\Gamma)})|G|_{2,\mathrm{Hull}(\Gamma)}\Big{)}h^{2s+1},$
(118)

for some constant $C>0$ independent of $h$, $\phi$, $\zeta$, $\varphi$, $G$,
and $V$.

If, further, $\Gamma$ is homogeneous the above result holds with (117)
replaced by the weaker condition

$\displaystyle h_{Q}\leq C^{\prime}_{Q}h^{d/2+s/2+5/4}.$ (119)

###### Proof.

The two conditions (117) and (119) can be written as

$\displaystyle h_{Q}\leq C^{\prime}_{Q}h^{\frac{d+s+1/2}{1+p}+1},$ (120)

where $p$ is as in the proof of Corollary 5.17. We first note that, for every
$C_{Q}>0$, since $s>-1/2$, (120) implies (114) provided $C^{\prime}_{Q}$ is
sufficiently small. Thus, if (120) holds with $C^{\prime}_{Q}$ sufficiently
small, then the approximated linear system is invertible by Corollary 5.17.
Next we estimate

$\displaystyle|J(\phi)-J^{Q}(\phi_{N}^{Q})|$
$\displaystyle\leq|J(\phi)-J(\phi_{N})|+|J(\phi_{N})-J(\phi_{N}^{Q})|+|J(\phi_{N}^{Q})-J^{Q}(\phi_{N}^{Q})|$
$\displaystyle\leq
ch^{2s+1}\|\phi\|_{H^{s}_{\Gamma}}\|\zeta\|_{H^{s}_{\Gamma}}+\|\varphi\|_{H^{1/2}(\Gamma_{\infty})}\|\phi_{N}-\phi_{N}^{Q}\|_{H^{-1/2}_{\Gamma}}+Ch_{Q}^{2}h^{s+(n-d)/2}|V|_{2,\mathrm{Hull}(\Gamma)}\|\phi^{Q}_{N}\|_{H_{\Gamma}^{s}},$

by (81) and Theorem 5.14(ii). We note moreover that
$\|\phi^{Q}_{N}\|_{H_{\Gamma}^{s}}\leq
C_{1}(\|\phi\|_{H_{\Gamma}^{s}}+|G|_{2,\mathrm{Hull}(\Gamma)})$, for some
constant $C_{1}>0$ independent of $\phi$, $G$, and $h$, as a consequence of
Corollary 5.17 and Proposition 5.11, so that (120) implies that the last term
in the above equation is

$\displaystyle\leq$ $\displaystyle
CC_{1}(C^{\prime}_{Q})^{2}h^{\frac{2}{1+p}(d+s+1/2)+2+s+(n-d)/2}|V|_{2,\mathrm{Hull}(\Gamma)}(\|\phi\|_{H_{\Gamma}^{s}}+|G|_{2,\mathrm{Hull}(\Gamma)})$
$\displaystyle\leq$ $\displaystyle
CC_{1}(C^{\prime}_{Q})^{2}(\operatorname{diam}(\Gamma))^{s(\frac{2}{1+p}-1)+1+\frac{2}{1+p}(d+1/2)+(n-d)/2}|V|_{2,\mathrm{Hull}(\Gamma)}(\|\phi\|_{H_{\Gamma}^{s}}+|G|_{2,\mathrm{Hull}(\Gamma)})h^{2s+1},$

recalling that $s>-1/2$ so that
$s(\frac{2}{1+p}-1)+1+\frac{2}{1+p}(d+1/2)+(n-d)/2>0$ for both $p=0$ and
$p=1$. Further, arguing as in the proof of Corollary 5.17, the conditions
(107) are satisfied with $E_{A}$ and $E_{b}$ given by (115), for some constant
$C_{*}$ independent of $h$ and $h_{Q}$. If also (120) holds then

$E_{A}\leq C_{*}(C^{\prime}_{Q})^{1+p}h^{s+1/2}$

and

$\displaystyle E_{b}$ $\displaystyle\leq
C_{*}(C^{\prime}_{Q})^{2}h^{\frac{2}{1+p}(d+s+1/2)+1-s+(n-d)/2}|G|_{2,\mathrm{Hull}(\Gamma)}$
(121) $\displaystyle\leq
C^{*}(C^{\prime}_{Q})^{2}(\operatorname{diam}(\Gamma))^{(2-\frac{2}{1+p})(-s)+\frac{2}{1+p}(d+1/2)+1/2+(n-d)/2}|G|_{2,\mathrm{Hull}(\Gamma)}h^{s+1/2},$

recalling that $s<-(n-d)/2$, so that
$(2-\frac{2}{1+p})(-s)+\frac{2}{1+p}(d+1/2)+1/2+(n-d)/2>0$ for $p=0,1$. Thus,
if $C^{\prime}_{Q}$ is chosen sufficiently small so that
$E_{A}\leq\alpha/(2c_{t_{d}}^{2})$ for $0<h\leq\operatorname{diam}(\Gamma)$,
it follows from Proposition 5.16 that

$\|\phi_{N}-\phi_{N}^{Q}\|_{H^{-1/2}_{\Gamma}}\leq
C_{2}(\|\phi\|_{H_{\Gamma}^{s}}+|G|_{2,\mathrm{Hull}(\Gamma)})h^{2s+1},$

for some constant $C_{2}>0$ independent of $h$, $\phi$ and $G$. The bound
(118) follows by combining the above inequalities. ∎

######  Remark 5.19 (Reduced quadrature).

The quadrature rule defined by (94) uses the same maximum mesh width $h_{Q}$
for all off-diagonal elements $A_{ij}$, $i\neq j$. Since the magnitude of the
integrand $\Phi(x,y)$ and its derivatives blows up as one approaches the
diagonal $x=y$, and decays away from it, it is possible to save computational
effort, while maintaining the error bounds (104) and (105), and hence the
error estimates (112) and (118). To achieve this we increase the quadrature
mesh width (and hence potentially decrease the number of quadrature points)
for the computation of $A_{ij}$ when the elements
$\Gamma_{{\boldsymbol{m}}(i)}$ and $\Gamma_{{\boldsymbol{m}}(j)}$ are
sufficiently well-separated spatially. In more detail, from [25, Prop 5.2] we
have that, for $i\neq j$, the error in the quadrature approximation (94), with
$h_{Q}$ replaced by a local quadrature mesh width $h_{Q,i,j}$, satisfies

$|A_{ij}-A_{ij}^{Q}|\leq
C\mu_{{\boldsymbol{m}}(i)}\mu_{{\boldsymbol{m}}(j)}h_{Q,i,j}^{2}\Upsilon(R_{ij}),$

for some constant $C$ independent of $i$ and $j$, where

$\Upsilon(R):=\frac{1+(kR)^{n/2+1}}{R^{n+1}}\qquad\text{and}\qquad
R_{ij}:=\operatorname{dist}(\mathrm{Hull}(\Gamma_{{\boldsymbol{m}}(i)}),\mathrm{Hull}(\Gamma_{{\boldsymbol{m}}(j)})).$

Given $h_{Q}$, one can therefore maintain the bounds (104) and (105) by
replacing $h_{Q}$ in (94) by

$h_{Q,i,j}=h_{Q}\left(\frac{\max_{p\neq
q}\Upsilon(R_{pq})}{\Upsilon(R_{ij})}\right)^{1/2}\geq h_{Q}.$

If the quantities $R_{ij}$ are not known exactly, but satisfy

$\displaystyle R^{-}_{ij}\leq R_{ij}\leq R^{+}_{ij},\qquad i\neq j,$ (122)

for some known quantities $R^{-}_{ij}\geq 0$ and $R^{+}_{ij}<\infty$, then one
can maintain (104) and (105) by replacing $h_{Q}$ in (94) by

$\displaystyle h_{Q,i,j}=h_{Q}\max\left(\left(\frac{\max_{p\neq
q}\Upsilon(R^{+}_{pq})}{\Upsilon(R^{-}_{ij})}\right)^{1/2},1\right)\geq
h_{Q}.$ (123)

Here we are using the fact that $\Upsilon(R)$ is a monotonically decreasing
function of $R$, and the $\max(\cdot,1)$ is needed to ensure that
$h_{Q,i,j}\geq h_{Q}$ (so that we are not increasing the computational effort
unnecessarily) because the quantity inside the square root may be smaller than
1. (This holds in particular if $R^{-}_{ij}=0$, in which case we are
interpreting $1/\Upsilon(R^{-}_{ij})$ as $1/\Upsilon(0)=1/\infty=0$.)

In our numerical results in §6 we will compute $A_{ij}^{Q}$ as described
above, choosing $h_{Q,i,j}$ according to (123) with

$\displaystyle
R^{-}_{ij}:=\max(\operatorname{dist}(x_{{\boldsymbol{m}}(i)},x_{{\boldsymbol{m}}(j)})-\operatorname{diam}(\Gamma_{{\boldsymbol{m}}(i)})-\operatorname{diam}(\Gamma_{{\boldsymbol{m}}(j)}),0)$
(124)

and

$\displaystyle
R^{+}_{ij}:=\operatorname{dist}(x_{{\boldsymbol{m}}(i)},x_{{\boldsymbol{m}}(j)}),$
(125)

which satisfy (122).

##  6 Numerical results

In this section we present numerical results that illustrate our methods and
assess the sharpness of our theoretical predictions. The Hausdorff BEM (57)
has been implemented in the Julia language [4] and the code is available at
https://github.com/AndrewGibbs/IFSintegrals. In all examples described in this
section, we consider the scattering of plane waves, i.e. the datum $g$ is as
in (31) with $u^{i}(x)=e^{{\mathrm{i}}k\vartheta\cdot x}$ and $|\vartheta|=1$.
We validate our implementation against a different method in §6.4, but in the
rest of our experiments we use as a reference solution a more accurate
Hausdorff-BEM solution with a large number of degrees of freedom
$N_{\mathrm{ref}}$. Most of our experiments are for homogeneous attractors, in
which case $N_{\mathrm{ref}}=M^{\ell_{\mathrm{ref}}}$ for some
$\ell_{\mathrm{ref}}\in\mathbb{N}$. For Cantor sets $(M=2)$ we choose
$\ell_{\mathrm{ref}}=15$, so that $N_{\mathrm{ref}}=32768$. For Cantor Dusts
($M=4$) we choose $\ell_{\mathrm{ref}}=8$, so that $N_{\mathrm{ref}}=65536$.

Our implementation uses the quadrature rules described in §5.4. Precisely, we
approximate the right-hand side in the linear system and linear functionals of
the solution (the scattered field and far field) using the quadrature rules
(99) and (100), respectively. We approximate the diagonal matrix elements
$A_{ii}$ by (95)-(98). To approximate $A_{ij}$ with $i\neq j$ we use (94) with
$h_{Q}$ replaced by $h_{Q,i,j}\geq h_{Q}$ given (in terms of $h_{Q}$) by
(123)-(125). All these quadrature rules depend on the parameter $h_{Q}$. We
choose $h_{Q}=C_{Q}h$, where the constant $C_{Q}$ is chosen to depend on the
wavenumber as detailed below. While the requirement $h_{Q}=C_{Q}h$ is weaker
(i.e. it requires fewer quadrature points) than the conditions $h_{Q}\leq
C_{Q}h^{1+d/2}$ (see (113)) and $h_{Q}\leq C_{Q}h^{1+d}$ (see (111)) required
by our theory, our numerical experiments suggest that, in practice,
$h_{Q}=C_{Q}h$ is sufficient to achieve our theoretical convergence rates,
even when using the reduced quadrature of Remark 5.19.

We measure the accuracy of our BEM solutions in the $H^{-1/2}_{\Gamma}$ norm.
These $H^{-1/2}_{\Gamma}$ norms are computed by expressing them in terms of a
single-layer BIO with wavenumber $k={\mathrm{i}}$, which we denote
$S^{\mathrm{i}}$, as in [16, Table 1]. Practically, we achieve this by
assembling an approximation $A^{{\mathrm{i}},Q}$ to the Galerkin matrix
$A^{\mathrm{i}}$ for the operator $S^{\mathrm{i}}$, analogous to (93), with
$k={\mathrm{i}}$ and $N_{\mathrm{ref}}$ degrees of freedom, using quadrature
approximations analogous to (95)-(98) and (94), choosing $h_{Q}$ the same as
for the reference solution, with the reduced quadrature formulae (123)-(125)
applied as if $k=1$. (While the quadrature convergence analysis in [25, §5]
was presented only for $k>0$, one can check that the relevant results also
extend, mutatis mutandis, to the case $k={\mathrm{i}}$.) Next, we view
$\phi_{N}^{Q}$ as an element of the larger space $V_{N_{\mathrm{ref}}}$ and
define ${\vec{v}}$ as the coefficient vector of
$\phi_{N}^{Q}-\phi_{N_{\mathrm{ref}}}^{Q}$ in $V_{N_{\mathrm{ref}}}$. Then,
arguing as in [16, Table 1] (and see Footnote 16), it follows that

$\|\phi_{N}^{Q}-\phi_{N_{\mathrm{ref}}}^{Q}\|_{H^{-1/2}_{\Gamma}}^{2}=2\left\langle
S^{\mathrm{i}}(\phi_{N}^{Q}-\phi_{N_{\mathrm{ref}}}^{Q}),\phi_{N}^{Q}-\phi_{N_{\mathrm{ref}}}^{Q}\right\rangle_{H^{1/2}(\Gamma_{\infty})\times
H^{-1/2}(\Gamma_{\infty})}=2{\vec{v}}^{H}A^{\mathrm{i}}{\vec{v}}\approx
2{\vec{v}}^{H}A^{{\mathrm{i}},Q}{\vec{v}}.$

###  6.1 $H^{-1/2}_{\Gamma}$-norm convergence, Cantor set ($n=1$)

Figure 7: Scattering in $\mathbb{R}^{2}$ by a middle-third Cantor set screen
(a subset of the line segment $[0,1]\times\\{0\\}$) with wavenumber $k=50$ and
incident plane wave direction $\vartheta=(1,-1)/\sqrt{2}$, showing the total
(left panel) and scattered (right panel) fields computed using our Hausdorff
BEM with $\ell=10$ and quadrature parameters as in §6.1.

We consider first the case where $n=1$ (corresponding to scattering in
$\mathbb{R}^{2}$) and $\Gamma\subset\Gamma_{\infty}\cong\mathbb{R}$ is a
“middle-$(1-2\rho)$ Cantor set” screen, for some $0<\rho<1/2$. Precisely,
$\Gamma$ is the attractor of the disjoint homogeneous IFS with $M=2$,

$s_{1}(x)=\rho x,\qquad s_{2}(x)=1-\rho+\rho x,\qquad x\in\mathbb{R},$ (126)

a $d$-set with $d={\rm dim_{H}}(\Gamma)=\log{2}/\log{(1/\rho)}$. We denote by
$\phi_{\ell}$ the Hausdorff-BEM solution “at level $\ell$”, for
$\ell\in\mathbb{N}_{0}$, i.e. with $h=\rho^{\ell}$ and $N=2^{\ell}$.

In Figure 7 we show an example of the total and scattered fields for the
middle-third case ($\rho=1/3$). In Figure 8 we present plots of the relative
errors in the Hausdorff-BEM solution for $\ell=0,1,\ldots,14$, measured in
$H^{-1/2}_{\Gamma}$ norm, using $\phi_{15}$ as reference solution, for various
$\rho$ and $k$ values. In all cases the incident wave is the plane wave
$u^{i}(x)={\mathrm{e}}^{{\mathrm{i}}k\vartheta\cdot x}$ with
$\vartheta=(1,-1)/\sqrt{2}$. The experiments in Figures 7 and 8 are carried
out using the quadrature rules described in §5.4, with the reduced quadrature
of Remark 5.19, and with $h_{Q}=\rho^{2}h$ for $k=0.1$ and $h_{Q}=\rho^{4}h$
for $k=50$. This means that the off-diagonal entries of the BEM matrix require
up to $16$ and $256$ evaluations of the Helmholtz fundamental solution for
$k=0.1$ and $k=50$, respectively.

In Remark 5.8 we noted that our theoretical analysis suggests we should expect
convergence approximately like $2^{-\ell/2}$ for the BEM solution, which is
what we observe in the numerical results (for each error convergence plot we
also plot $C2^{-\ell/2}$ – the black dashed lines – choosing the constant $C$
so that the two curves coincide for the largest value of $\ell$). Higher
wavenumbers give different pre-asymptotic behaviour (in particular, the
approximation is not accurate for small values of $\ell$) but do not affect
the asymptotic rates for large $\ell$, see Figure 8.

Figure 8: $H^{-1/2}_{\Gamma}$ relative errors for Hausdorff-BEM solutions on
Cantor sets.

For $\rho=0.5$ our Hausdorff BEM coincides with classical piecewise-constant
BEM on a uniform mesh applied to the Lipschitz screen that is the unit
interval $[0,1]$; our implementation uses the special choice (95)-(98) of
quadrature rules to evaluate the matrix entries (which is simply the composite
midpoint rule for the off-diagonal entries). Running our Hausdorff-BEM code
with $\rho=0.5$ we observe error curves (not reported here) that are almost
identical to those for $\rho=0.49$. Hence, at least in this case, the
piecewise-constant Hausdorff-BEM approximation of the integral equation on a
fractal (the Cantor set with $\rho=0.49$) is no less accurate, for the same
number of degrees of freedom, than a classical piecewise-constant BEM
approximation on an adjacent, more regular set (the interval $[0,1]$).

###  6.2 $H^{-1/2}_{\Gamma}$-norm convergence, Cantor dust ($n=2$)

We consider next the case where $n=2$ (corresponding to scattering in
$\mathbb{R}^{3}$) and $\Gamma\subset\Gamma_{\infty}\cong\mathbb{R}^{2}$ is a
“middle-$(1-2\rho)$ Cantor dust” for some $0<\rho<1/2$, defined by the
disjoint homogeneous IFS with $M=4$,

$\displaystyle\begin{split}s_{1}(x,y)=\rho(x,y),\qquad&s_{2}(x,y)=(1-\rho,0)+\rho(x,y),\\\
s_{3}(x,y)=(1-\rho,1-\rho)+\rho(x,y),\qquad&s_{4}(x,y)=(0,1-\rho)+\rho(x,y),\qquad(x,y)\in\mathbb{R}^{2},\end{split}$
(127)

a $d$-set with $d={\rm dim_{H}}(\Gamma)=\log{4}/\log{(1/\rho)}$, see Figure 1.
Such a screen generates a non-zero scattered field if and only if $\rho>1/4$
(see [12, Example 8.2] and the discussion before Lemma 4.4). In Figure 9 we
present results similar to those in Figure 8, albeit for a more restricted
range of $\ell$ and a lower value of $k$ in the right-hand panel, due to the
increased computational cost associated with the change from $M=2$ (Cantor
set) to $M=4$ (Cantor dust). The Hausdorff-BEM solution $\phi_{\ell}$
corresponds to a mesh of $N=4^{\ell}$ elements of diameter
$h=\sqrt{2}\rho^{\ell}$. The incident wave is the plane wave
$u^{i}(x)={\mathrm{e}}^{{\mathrm{i}}k\vartheta\cdot x}$ with
$\vartheta=(0,1,-1)/\sqrt{2}$. We use the quadrature rules described in §5.4,
with the reduced quadrature of Remark 5.19, and with $h_{Q}=\rho^{2}h$ for
both $k=0.1$ and $k=5$. As a result, the off-diagonal entries of the BEM
matrix require up to $256$ evaluations of $\Phi$.

Figure 9: $H^{-1/2}_{\Gamma}$ relative errors for Hausdorff-BEM solutions on
Cantor dusts. The legend inset in the right-hand panel applies to both panels.

In Remark 5.8 we noted that our theoretical analysis suggests we should expect
convergence approximately like $(4\rho)^{-\ell/2}$, whereas the convergence we
observe in Figure 9 appears to be faster than this, especially for lower
values of $\rho$. In fact, these numerical results are consistent with the
theoretical bounds being sharp, as we now explain. The issue is that the
convergence rate $(4\rho)^{-\ell/2}$ is slow, so the reference solution (with
$\ell=8$) used in Figure 9 is still quite far from the exact solution $\phi$
of the integral equation, which affects the shape of the error plot. Let us
assume that the Galerkin error bound (84) holds with $\epsilon=0$ and in fact
is exact, i.e.

$\|\phi-\phi_{\ell}\|_{H^{-1/2}_{\Gamma}}=C\alpha^{\ell},\quad\ell\in\mathbb{N},$
(128)

for $\alpha:=(4\rho)^{-1/2}\in(1/\sqrt{2},1)$ and some $C>0$. Then, by the
triangle inequality,

$C\alpha^{\ell}(1-\alpha^{\ell^{\prime}-\ell})\leq\|\phi_{\ell}-\phi_{\ell^{\prime}}\|_{H^{-1/2}_{\Gamma}}\leq
C\alpha^{\ell}(1+\alpha^{\ell^{\prime}-\ell}),\qquad\ell,\ell^{\prime}\in\mathbb{N}_{0}\;\text{with}\;\ell<\ell^{\prime},$
(129)

and, for any given $\ell^{\prime}\in\mathbb{N}$, (128) imposes no constraint
on $\|\phi_{\ell}-\phi_{\ell^{\prime}}\|_{H^{-1/2}_{\Gamma}}$, for
$\ell=0,\ldots,\ell^{\prime}-1$, beyond (129); in particular, it may be that
either the lower bound or the upper bound in (129) is attained. In Figure 10
we plot $\|\phi_{\ell}-\phi_{8}\|_{H^{-1/2}_{\Gamma}}$, for $\ell=1,\ldots,7$,
together with the upper and lower bounds of (129) for a suitable $C$ and
$\ell^{\prime}=8$. This plot makes clear that the values we compute for
$\|\phi_{\ell}-\phi_{8}\|_{H^{-1/2}_{\Gamma}}$ are consistent with (129) and
so with (128). (This reasoning, i.e. the bounds (129), also explains the
little “dip” at the right endpoints of the error curves for the Cantor set
experiments in Figure 8.)

In Figure 11 we plot, as a better test of the sharpness of our theory, the
norm $\|\phi_{\ell}-\phi_{\ell+1}\|_{H^{-1/2}_{\Gamma}}$, for
$\ell=0,\ldots,7$, of the difference between Galerkin solutions at consecutive
levels $\ell$. We observe, for each choice of $\rho$ and $k$, that, for large
enough $\ell$ (how large depending on $\rho$ and $k$),
$\|\phi_{\ell}-\phi_{\ell+1}\|_{H^{-1/2}_{\Gamma}}=c\alpha^{\ell}=c(4\rho)^{-\ell/2}$,
for some constant $c>0$, also depending on $\rho$ and $k$, in agreement with
(129) with $\ell^{\prime}=\ell+1$. Moreover, since
$\|\phi-\phi_{\ell}\|_{H^{-1/2}_{\Gamma}}\to 0$ as $\ell\to\infty$ by Theorem
5.6, this observed convergence implies that

$\|\phi-\phi_{\ell}\|_{H^{-1/2}_{\Gamma}}\leq\sum_{m=\ell}^{\infty}\|\phi_{m}-\phi_{m+1}\|_{H^{-1/2}_{\Gamma}}=\frac{c}{1-\alpha}\,\alpha^{\ell},$

confirming, experimentally, the convergence rates for the Cantor dust case
predicted in Remark 5.8, and confirming the predicted dependence of these
convergence rates on $\rho$.

Figure 10: Relative $H^{-1/2}_{\Gamma}$ errors for Hausdorff-BEM solutions on
Cantor dusts using $\phi_{8}$ as the reference solution, and the bounds from
(129) with $\ell^{\prime}=8$, for $k=0.1$ and different contraction factors
$\rho$.

Figure 11: $H^{-1/2}_{\Gamma}$ errors for Hausdorff-BEM solutions on Cantor
dusts: the norm of the difference $\phi_{\ell}-\phi_{\ell+1}$ between
consecutive levels decays like $(4\rho)^{-\ell/2}$.

For $\rho=0.5$ our Hausdorff BEM coincides with a piecewise-constant BEM on a
uniform mesh applied to the screen $[0,1]^{2}$; our implementation uses the
quadrature rules (95)-(98) (which reduce to a composite product midpoint rule
for the off-diagonal entries). As similarly reported for the Cantor set in
§6.1, running our Hausdorff-BEM code with $\rho=0.5$ we observe error curves
(not reported here) that are almost identical to those for $\rho=0.49$.

For this Cantor dust example we also compare numerical results with the
theoretical predictions of Theorem 5.13. In Figure 12 we plot the norms of the
Hausdorff-BEM Galerkin matrix $A$ and its inverse against $\ell$ (the matrix
dimension is $N=4^{\ell}$) for $\rho=1/3$ and different values of $k$. We
observe that $\|A\|_{2}$ is approximately independent of $\ell$ and
$h=\sqrt{2}\rho^{\ell}$, and that, for $\ell\geq 2$,
$\|A^{-1}\|_{2}\approx(\rho
M)^{\ell}=(4\rho)^{\ell}=\rho^{(1-d)\ell}=c^{*}h^{1-d}=c^{*}h^{-2t_{d}}$,
where $c^{*}=2^{(d-1)/2}$ and $t_{d}$ is defined by (44), in agreement with
Theorem 5.13. (We note that $-2t_{d}=1-d=1-\log 4/\log 3\approx-0.262$.)

Figure 12: The norm of the Hausdorff-BEM Galerkin matrix and its inverse. Here
$-2t_{d}\approx-0.262$.

###  6.3 Convergence of the scattered field

Figure 13: Convergence of the scattered near- and far-field for Cantor sets
and dusts, measured as
$\|u_{\ell}-u_{\ell_{\mathrm{ref}}}\|_{L_{\infty}}/\|u_{\ell_{\mathrm{ref}}}\|_{L_{\infty}}$
and
$\|u^{\infty}_{\ell}-u^{\infty}_{\ell_{\mathrm{ref}}}\|_{L_{\infty}}/\|u^{\infty}_{\ell_{\mathrm{ref}}}\|_{L_{\infty}}$,
respectively, with $\ell_{\mathrm{ref}}=15$ for Cantor sets and
$\ell_{\mathrm{ref}}=8$ for Cantor dusts.

Figure 14: Incremental convergence of the scattered near- and far-field for
Cantor dusts, measured as $\|u_{\ell}-u_{\ell+1}\|_{L_{\infty}}$ and
$\|u^{\infty}_{\ell}-u^{\infty}_{\ell+1}\|_{L_{\infty}}$, respectively.

We now study the convergence of the Hausdorff-BEM approximation of the field
scattered by $\Gamma$, for the same scatterers (Cantor sets and dusts),
wavenumbers, incident waves and quadrature parameters considered in §6.1–6.2.
The near field at $x\in\mathbb{R}^{n+1}$ and far-field pattern at
$\hat{x}\in\mathbb{S}^{n}$ are computed using (100), choosing
$\varphi=\Phi(x,\cdot)$ and $\varphi=\Phi^{\infty}(\hat{x},\cdot)$
respectively, using the same $h_{Q}$ values used to construct the associated
BEM system.

In Figure 13 we show $L_{\infty}$ errors in the near- and far-field for
scattering by Cantor sets and Cantor dusts, obtained by computing the maximum
error over a suitable set of sample points. In more detail, define the
parameter $\mathcal{N}(k):=10\max(k,2)$, which is always an even integer for
the values of $k$ in our experiments. For the near-field, when $n=1$ we sample
at $4\mathcal{N}$ points on the boundary of the square
$(-1,2)\times(-1.5,1.5)$, and when $n=2$ we sample at $\mathcal{N}^{2}$ points
on a uniform grid on the square $(-1,2)\times(-1,2)\times\\{-1\\}$ (recall
that $\Gamma\subset[0,1]\times[0,1]\times\\{0\\}$). For the far-field, when
$n=1$ we sample at $\mathcal{N}$ points on the circle $\mathbb{S}^{1}$, and
when $n=2$ we sample at $\frac{\mathcal{N}}{2}\times\mathcal{N}$ points on the
sphere $\mathbb{S}^{2}$, chosen such that the points form a uniform grid in
spherical coordinate space $[0,\pi]\times[0,2\pi]$.

For Cantor sets, we observe that near- and far-field errors converge to zero
with rates $2^{-\ell}$, precisely as predicted by the theory in (84). For
Cantor dusts, we observe convergence that is apparently faster than the
predicted theory, similar to the observation made in relation to Figure 9,
which was explained in §6.2. Applying the same reasoning as in §6.2 to achieve
a robust comparison with the theoretical error bounds, in Figure 14 we plot
$\|u_{\ell}-u_{\ell+1}\|_{L_{\infty}}$ and
$\|u^{\infty}_{\ell}-u^{\infty}_{\ell+1}\|_{L_{\infty}}$ against $\ell$. Both
agree with the predicted convergence rate of $(\rho M)^{-\ell}$.

###  6.4 Comparison against “prefractal BEM”

Figure 15: Absolute $L_{\infty}$ errors of Hausdorff-BEM
(${u}_{\ell}^{\infty}$, blue curves) and prefractal-BEM
($\tilde{u}_{\ell}^{\infty}$, red curves) approximations of the far-field
pattern $u^{\infty}$ for scattering by three Cantor sets with $k=0.1$ (left
panels) and $k=50$ (right panels) and $\vartheta=(1/2,-\sqrt{3}/2)$. In all
cases $\ell_{\mathrm{ref}}=13$.

Figure 16:  Ratio of errors of Hausdorff-BEM (${u}_{10}^{\infty}$) and
prefractal-BEM ($\tilde{u}_{10}^{\infty}$) approximations of the far-field
pattern $u^{\infty}$ with $\ell=10$ for scattering by Cantor sets with
$k=0.1$, $k=30$ and $k=50$ and $\vartheta=(1/2,-\sqrt{3}/2)$, for a range of
values of $\rho\in(0,1/2)$, measured in the $L_{\infty}$ (left panel) and
$L_{2}$ (right panel) norms on $\mathbb{S}^{1}$. A value below $1$ indicates
that the Hausdorff BEM is more accurate, while a value above $1$ indicates
that the prefractal BEM is more accurate. In both cases
$\ell_{\mathrm{ref}}=13$.

In this section we compare the approximations produced by our Hausdorff BEM
with those produced by the method of [16]. This provides a stronger validation
of our Hausdorff BEM than the experiments so far presented, which compare our
Hausdorff-BEM approximations against higher-accuracy approximations produced
using the same Hausdorff-BEM code. We shall refer to the method of [16] as
“prefractal BEM”, since it involves applying a standard BEM approximation on a
prefractal approximation of $\Gamma$, which is the closure of a finite union
of disjoint Lipschitz open sets. For brevity we focus only on the case of
scattering by Cantor sets. In this case, for the prefractal BEM we take as
prefractals the sequence of sets $\Gamma^{(0)}=[0,1]$,
$\Gamma^{(\ell)}:=s(\Gamma^{(\ell-1)})$, $\ell\in\mathbb{N}$, where $s$ is as
in (4) for the IFS (126).

For each $\ell\in\mathbb{N}_{0}$ we denote by $\tilde{u}^{\infty}_{\ell}$ the
prefractal-BEM approximation to the far-field pattern $u^{\infty}$, computed
on the prefractal $\Gamma^{(\ell)}$ using a standard piecewise-constant
Galerkin BEM with one degree of freedom for each connected component of
$\Gamma^{(\ell)}$. Quadrature was carried out using high-order Gauss and
product Gauss rules, and quadrature parameters were chosen so that increasing
quadrature accuracy did not noticeably change the results presented below.
This leads to a total number of degrees of freedom $N(\ell)=2^{\ell}$, which
is the same number of degrees of freedom used in our Hausdorff BEM at level
$\ell$. As before, we write $u^{\infty}_{\ell}$ to denote the approximation to
the far-field pattern for the Hausdorff BEM, which is computed as described in
§6.3, except that now we use smaller values of the quadrature parameter
$h_{Q}$ (as detailed below), to ensure that the results presented are not
polluted by effects of insufficiently accurate quadrature, so our focus is on
the Galerkin error per se.

In Figure 15 we show $L_{\infty}$ errors in the far-field pattern between the
two methods for $\rho=0.1$, $1/3$ and $0.49$, with incident direction
$\vartheta=(1/2,-\sqrt{3}/2)$ and wavenumbers $k=0.1$ and $k=50$ (with
$h_{Q}=\rho^{6}h$ for $k=0.1$ and $h_{Q}=\rho^{8}h$ for $k=50$ for the
Hausdorff BEM), obtained by computing the maximum error over 300 equally-
spaced observation angles in $\mathbb{S}^{1}$. For each method we calculate
the errors in two different ways, using, firstly, a Hausdorff-BEM reference
solution, and, secondly, a prefractal-BEM reference solution; in both cases
the reference solution is computed with $\ell_{\mathrm{ref}}=13$. In all
cases, it is clear that both methods are converging to the same solution,
validating both methods. Indeed, it appears that both methods converge at the
same $2^{-\ell}$ rate. However, the results suggest that the relative accuracy
of the two methods is dependent on the value of $\rho$, with the two methods
having essentially the same accuracy for $\rho=0.49$, the prefractal BEM being
more accurate for $\rho=1/3$, and the Hausdorff BEM being more accurate for
$\rho=0.1$. To investigate this further we carried out similar experiments for
more values of $\rho$, plotting the ratio of the errors obtained (measured in
the $L_{\infty}$ and $L_{2}$ norms) with $\ell=10$ in Figure 16. We observe
that:

  * •

for $\rho$ between $0.4$ and $0.5$ the two methods are very similar in
accuracy;

  * •

for $\rho$ between $0.3$ and $0.4$ the prefractal BEM appears to be more
accurate, significantly so, by a factor $>100$, for $\rho\approx 1/3$ (this
appears to be due entirely to some unexpected enhanced accuracy of the
prefractal BEM for $\rho\approx 1/3$);

  * •

for $\rho$ between $0$ and $0.3$ the Hausdorff BEM appears to be more
accurate, significantly so, by a factor $>1000$, for the lowest value of
$\rho$;

  * •

the ratio of the errors for the two methods appears to be essentially
independent of $k$ for the range of $k$ tested. To illustrate this we have
also included in Figure 16 results for $k=30$ (computed with
$h_{Q}=\rho^{8}h$), alongside the results for $k=0.1$ and $k=50$; we observe
that the results for all three $k$ values are almost identical.

These observations about the accuracy of the two methods (particularly the
“spike” in Figure 16 near $\rho=1/3$) merit further investigation, but we
leave this to future work.

###  6.5 Non-homogeneous or non-disjoint IFS attractors

Figure 17: The scattered field induced by a plane wave, with wavenumber $k=50$
and direction vector $\vartheta=(0,1,-1)/\sqrt{2}$, incident on the “non-
homogenous dust” of [25, Fig. 8b, eq. (70)], plotted on three faces of a cube,
computed with mesh parameter $h=h_{\mathrm{ref}}=\sqrt{2}/16384$. The
scatterer, which is a subset of the plane $\mathbb{R}^{2}\times\\{0\\}$, is
shown in black. Further details are given in §6.5.

Figure 18: $H^{-1/2}_{\Gamma}$ norms of the errors for the two IFS attractors
described in §6.5. Top left:
$\|\phi_{h}-\phi_{h_{\mathrm{ref}}}\|_{H^{-1/2}_{\Gamma}}/\|\phi_{h_{\mathrm{ref}}}\|_{H^{-1/2}_{\Gamma}}$,
bottom left:
$\|\phi_{\ell}-\phi_{\ell_{\mathrm{ref}}}\|_{H^{-1/2}_{\Gamma}}/\|\phi_{\ell_{\mathrm{ref}}}\|_{H^{-1/2}_{\Gamma}}$,
top right: $\|\phi_{h}-\phi_{h/4}\|_{H^{-1/2}_{\Gamma}}$ (see Footnote 18),
bottom right: $\|\phi_{\ell}-\phi_{\ell+1}\|_{H^{-1/2}_{\Gamma}}$.

We now consider two IFS attractors that are not Cantor sets/dusts, both for
$n=2$:

  * (i)

the “non-homogeneous dust”, shown in Figure 17, and defined in [25, Fig. 8b,
eq. (70)] (with $M=4$, $\rho_{1}=\rho_{2}=\rho_{3}=\frac{1}{4}$,
$\rho_{4}=\frac{1}{2}$ and Hausdorff dimension $d\approx 1.20$);

  * (ii)

the Sierpinski triangle [16, §6.3] (with $M=3$, $s_{1}(x)=\frac{1}{2}x$,
$s_{2}(x)=\frac{1}{2}x+(\frac{1}{2},0)$,
$s_{3}(x)=\frac{1}{2}x+(\frac{1}{4},\frac{\sqrt{3}}{4})$ and Hausdorff
dimension $d=\log 3/\,\log 2\approx 1.58$);

Both satisfy the OSC (5) and are hence $d$-sets for the stated values of $d$.
But only (ii) is homogeneous, and only (i) is disjoint and satisfies the
assumptions of the Hausdorff-BEM convergence theory of §5.2. Figure 18 shows
the $H^{-1/2}_{\Gamma}$ norms of the Hausdorff-BEM errors computed against a
fixed reference solution (left column) and the norms of the differences
between Hausdorff-BEM solutions on consecutive meshes as in Figure 11 (right
column), for a range of wavenumbers.181818 In §6.2 we motivated the study of
the errors $\|\phi_{\ell}-\phi_{\ell+1}\|_{H^{-1/2}_{\Gamma}}$ between
consecutive levels with the bound (129). In the case of non-homogeneous IFSs,
denoting by $\phi_{h}$ the Hausdorff-BEM solution corresponding to a mesh of
size $h$, assuming that the theoretical error bound (80) is sharp, i.e.
$\|\phi-\phi_{h}\|_{H^{-1/2}_{\Gamma}}=Ch^{\alpha}$ for some $C>0$ and
$\alpha=(1-n+d)/2$, it follows that
$Ch^{\alpha}\big{(}1-(h^{\prime}/h)^{\alpha}\big{)}\leq\|\phi_{h}-\phi_{h^{\prime}}\|_{H^{-1/2}_{\Gamma}}\leq
Ch^{\alpha}\big{(}1+(h^{\prime}/h)^{\alpha}\big{)}$ for $0<h^{\prime}<h$. In
the example (i), two consecutive meshes have $h^{\prime}/h=1/4$; thus we
expect theoretically that $\|\phi_{h}-\phi_{h^{\prime}}\|_{H^{-1/2}_{\Gamma}}$
is approximately proportional to $h^{\alpha}$.  For the Sierpinski triangle
(ii) we use meshes of $N=3^{\ell}$ elements of diameter $2^{-\ell}$ and we
plot the errors against the level $\ell$ for $\ell=0,\ldots,9$, using a
reference solution with $\ell_{\mathrm{ref}}=10$, i.e.
$N_{\mathrm{ref}}=3^{10}=59049$. For (i), since the IFS is non-homogeneous the
BEM mesh is not parametrised by the level $\ell$ but rather by the mesh size
$h=\frac{\sqrt{2}}{4^{j}}$, $j=0,\ldots,5$, giving the number of degrees of
freedom as $N=1,7,40,217,1159,6160,$ respectively. The reference solution has
$h=h_{\mathrm{ref}}=\frac{\sqrt{2}}{4^{6}}=\frac{\sqrt{2}}{4096}$ and
$N_{\mathrm{ref}}=75316$. For quadrature, for (i) we use $h_{Q}=\frac{h}{16}$
and for (ii) $h_{Q}=\frac{h}{4}$.

For (i), as we found for the homogeneous Cantor dust in §6.2, at first sight
the convergence in the top-left panel of Figure 18 appears slightly faster
than the theoretical rate $h^{(1-n+d)/2}$ predicted by Theorem 5.1. As before,
this apparent mismatch is due to the limited accuracy of the reference
solution; when we plot incremental errors (in the top right panel) we see much
clearer agreement with the theory. Example (ii) is not covered by our
convergence theory, but if our theory were to extend to this case our
predicted convergence rate of $h^{(1-n+d)/2}$ would evaluate to
$(2/3)^{\ell}$. Similarly to what we observed for (i), the relative errors in
the bottom left panel of Figure 18 converge slightly faster than
$(2/3)^{\ell}$, while the increments in the bottom right panel converge
approximately in agreement with $(2/3)^{\ell}$. However, we leave theoretical
justification of this empirical observation for future work.

##  Appendix A Besov spaces on $d$-sets

In this appendix we show that the spaces denoted by
$B_{\alpha}^{p,q}(\Gamma)$, and defined in terms of atoms in [30, section 6],
coincide with the spaces denoted by $\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)$
defined in [8, Def. 6.3], under the assumptions $\alpha>0$, $1\leq
p,q<\infty$, and $\Gamma$ being a $d$-set, $0<d<n$, preserving Markov’s
inequality in the sense of [30, §4] (or see [31, p. 34], and note Remark A.6
below). As a consequence (see Corollary A.4) we have that our space
$\mathbb{H}^{\alpha}(\Gamma)$, defined in §2.4, coincides with
$B_{\alpha}^{2,2}(\Gamma)$ under the same assumptions on $\alpha$ and
$\Gamma$.191919We recall (see the discussion in §2.4 and [8, Rem. 6.4]) that,
for $0<\alpha<1$, $\mathbb{H}^{\alpha}(\Gamma)$ also coincides with the Besov
space $B^{\alpha}_{2,2}(\Gamma)$ of [31].

We start by rephrasing the above mentioned definition of
$B_{\alpha}^{p,q}(\Gamma)$ in terms closer to the notation used in [7], which
we want to use to make the announced connection. In order to do that, we need
first to recall the notion of atom as used in [30]. For each
$\nu\in{\mathbb{N}_{0}}$, consider the family of cubes

$Q^{\nu m}:=\prod_{i=1}^{n}[2^{-\nu}m_{i},2^{-\nu}(m_{i}+1)],\quad
m\in\mathbb{Z}^{n}.$

Then, given $\alpha$, $p$, $d$, and $K$ as in Definition A.1 below, an
$(\alpha,p)$-atom $a^{\nu m}$ associated with $Q^{\nu m}$ is any $a^{\nu m}\in
C^{K}(\mathbb{R}^{n})$ such that

$\operatorname{supp}a^{\nu m}\subset 3Q^{\nu m},$ $|D^{\beta}a^{\nu m}(x)|\leq
2^{-\nu(\alpha-|\beta|-d/p)},\quad x\in\mathbb{R}^{n},\;|\beta|\leq K,$

where for a cube $Q\subset\mathbb{R}^{n}$ and a scalar $r>0$ the cube $rQ$ is
defined to have the same centre $x_{Q}$ as $Q$ and side length $r$ times that
of $Q$, i.e., $rQ:=\\{x_{Q}+r(x-x_{Q}):x\in Q\\}$.

The following definition of $B_{\alpha}^{p,q}(\Gamma)$ is the second,
equivalent, definition of [30, §6].

######  Definition A.1.

Let $\alpha>0$, $1\leq p,q<\infty$, and $\Gamma$ be a $d$-set, $0<d<n$,
preserving Markov’s inequality. Assume that $\mathbb{N}\ni
K>\lfloor\alpha\rfloor$ Then $f\in B_{\alpha}^{p,q}(\Gamma)$ iff

$f=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda^{\nu m}a^{\nu
m},\quad\mbox{ convergence in }\mathbb{L}_{p}(\Gamma),$ (130)

for some family of $(\alpha,p)$-atoms $a^{\nu m}$ and numbers
$\lambda=(\lambda^{\nu m})_{\nu\in\mathbb{N}_{0},\,m\in\mathbb{Z}^{n}}$
satisfying

$\|\lambda\|_{b_{p,q}}:=\left(\sum_{\nu=0}^{\infty}\left(\sum_{m\in\mathbb{Z}^{n}}|\lambda^{\nu
m}|^{p}\right)^{q/p}\right)^{1/q}<\infty,$ (131)

the norm of $f$ in $B_{\alpha}^{p,q}(\Gamma)$ being defined as the infimum of
the left-hand side of (131) taken over all possible representations (130) of
$f$.

In [7], atoms are also considered, but the definition is different: for each
$\nu\in{\mathbb{N}_{0}}$, consider the family of cubes

$Q_{\nu m}:=\prod_{i=1}^{n}[2^{-\nu}(m_{i}-1),2^{-\nu}(m_{i}+1)],\quad
m\in\mathbb{Z}^{n}.$

Then, given $s\in\mathbb{R}$, $0<p<\infty$, $K\in{\mathbb{N}_{0}}$, and $c\geq
1$, an $(s,p)_{K,0,c}$-atom $a_{\nu m}$ associated with $Q_{\nu m}$ is any
continuous function with (classical) partial derivatives up to the order $K$
such that

$\operatorname{supp}a_{\nu m}\subset cQ_{\nu m},$ $|D^{\beta}a_{\nu m}(x)|\leq
2^{-\nu(s-|\beta|-n/p)},\quad x\in\mathbb{R}^{n},\;|\beta|\leq K.$

######  Remark A.2.

It is easily seen that, for each $\nu\in{\mathbb{N}_{0}}$ and
$m\in\mathbb{Z}^{n}$, $Q_{\nu m}\subset 3Q^{\nu m}\subset 2Q_{\nu m}$. And
from this it is straightforward to show that:

  1. 1.

Let $\alpha>0$, $1\leq p<\infty$, $0<d<n$, and $\mathbb{N}\ni
K>\lfloor\alpha\rfloor$. Any $(\alpha,p)$-atom associated with $Q^{\nu m}$ is
an $(\alpha+\frac{n-d}{p},p)_{K,0,2}$-atom $a_{\nu m}$ associated with $Q_{\nu
m}$.

  2. 2.

Let $0<d<n$, $s>\frac{n-d}{p}$, $1\leq p<\infty$, $\mathbb{N}\ni K>\lfloor
s-\frac{n-d}{p}\rfloor$. Any $(s,p)_{K+1,0,1}$-atom associated with $Q_{\nu
m}$ is an $(s-\frac{n-d}{p},p)$-atom associated with $Q^{\nu m}$.

######  Proposition A.3.

Let $\alpha>0$, $1\leq p,q<\infty$, and $\Gamma$ be a $d$-set, $0<d<n$,
preserving Markov’s inequality. Then the space
$\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)$ defined in [8, Def. 6.3] coincides with
the space $B_{\alpha}^{p,q}(\Gamma)$ of Definition A.1 above, with equivalence
of norms.

###### Proof.

Consider $\mathbb{N}\ni K>\alpha+\frac{n-d}{p}$. We have, in particular, that
$K>\lfloor\alpha\rfloor$. Given $f\in B_{\alpha}^{p,q}(\Gamma)$, we have (130)
with (131) for some family of $(\alpha,p)$-atoms $a^{\nu m}$ and numbers
$\lambda^{\nu m}$ such that

$\|\lambda\|_{b_{p,q}}\leq 2\|f\|_{B_{\alpha}^{p,q}(\Gamma)}.$

On the other hand, since $K>\alpha+\frac{n-d}{p}$ and
$0>-\alpha-\frac{n-d}{p}$, it follows from [7, Thm. 2.3, Rem. 2.4 and Thm.
2.5] and part 1 of the above Remark A.2 that

$g:=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda^{\nu m}a^{\nu m}\in
B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}$

with convergence in the sense of
$B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}$ and

$\|g\|_{B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}}\leq
c\|\lambda\|_{b_{p,q}},$ (132)

for some constant $c>0$ independent of $f$. From elementary embeddings and the
linearity and continuity of the trace operator $\mathrm{tr}_{\Gamma}$ from [8,
Prop. 6.2],

$\mathrm{tr}_{\Gamma}g=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda^{\nu
m}\mathrm{tr}_{\Gamma}a^{\nu
m}=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda^{\nu m}a^{\nu
m}|_{\Gamma},\quad\mbox{ convergence in }\mathbb{L}_{p}(\Gamma),$

the latter equality being justified by the arguments in [8, Rem. 6.4] and the
fact that each $a^{\nu m}$ is continuous. But then $\mathrm{tr}_{\Gamma}g$
must be the same as $f$ (recall the representation (130) and the fact that
$a^{\nu m}|_{\Gamma}$ and $a^{\nu m}$ define the same class in
$\mathbb{L}_{p}(\Gamma)$. So, we have proved that $f$ also belongs to
$\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)$. Additionally, using (132),

$\|f\|_{\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)}\leq\|g\|_{B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}}\leq
c\|\lambda\|_{b_{p,q}}\leq 2c\|f\|_{B_{\alpha}^{p,q}(\Gamma)}.$

To see the opposite inclusion, given
$f\in\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)$, choose $g\in
B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}$ such that
$f=\mathrm{tr}_{\Gamma}g$ and

$\displaystyle\|g\|_{B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}}\leq
2\|f\|_{\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)}.$ (133)

Since $K+1>\alpha+\frac{n-d}{p}$ and $0>-\alpha-\frac{n-d}{p}$, it follows
from [7, Thm. 2.3 and Thm. 2.5] that

$g=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda_{\nu m}a_{\nu
m},\quad\mbox{ convergence in
}B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})},$

for some family of $(\alpha+\frac{n-d}{p},p)_{K+1,0,1}$-atoms $a_{\nu m}$ and
numbers $\lambda_{\nu m}$ which we can choose to satisfy the estimate

$\left(\sum_{\nu=0}^{\infty}\left(\sum_{m\in\mathbb{Z}^{n}}|\lambda_{\nu
m}|^{p}\right)^{q/p}\right)^{1/q}\leq
c^{\prime}\|g\|_{B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}},$ (134)

for some constant $c^{\prime}>0$ independent of $f$. Again using elementary
embeddings and the linearity and continuity of the trace operator
$\mathrm{tr}_{\Gamma}$ from [8, Prop. 6.2],

$f=\mathrm{tr}_{\Gamma}g=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda_{\nu
m}\mathrm{tr}_{\Gamma}a_{\nu
m}=\sum_{\nu=0}^{\infty}\sum_{m\in\mathbb{Z}^{n}}\lambda_{\nu m}a_{\nu
m}|_{\Gamma},\quad\mbox{ convergence in }\mathbb{L}_{p}(\Gamma).$

But then we see from part 2 of the above Remark A.2 that we have what is
needed to conclude that $f$ also belongs to $B_{\alpha}^{p,q}(\Gamma)$.
Moreover, from (133) and (134) it also follows that

$\|f\|_{B_{\alpha}^{p,q}(\Gamma)}\leq
c^{\prime}\|g\|_{B_{p,q}^{\alpha+\frac{n-d}{p}}{(\mathbb{R}^{n})}}\leq
2c^{\prime}\|f\|_{\mathbb{B}_{p,q,0}^{\alpha}(\Gamma)}.$

∎

######  Corollary A.4.

Let $\alpha>0$ and $\Gamma$ be a $d$-set, $0<d<n$, preserving Markov’s
inequality. Then $\mathbb{H}^{\alpha}(\Gamma)=B_{\alpha}^{2,2}(\Gamma)$ with
equivalence of norms.

###### Proof.

Our space $\mathbb{H}^{\alpha}(\Gamma)$ coincides with the spaces
$\mathbb{H}^{\alpha}_{2,0}(\Gamma)=\mathbb{F}^{\alpha}_{2,2,0}(\Gamma)$
defined in [8, Def. 6.3]. Moreover, since (for all $s\in\mathbb{R}$) the
Triebel-Lizorkin space $F_{2,2}^{s}{(\mathbb{R}^{n})}$ coincides with the
Besov space $B_{2,2}^{s}{(\mathbb{R}^{n})}$, with equivalent norms, it
follows, inspecting the definitions of $\mathbb{F}^{\alpha}_{2,2,0}(\Gamma)$
and $\mathbb{B}_{2,2,0}^{\alpha}(\Gamma)$ in [8, Def. 6.3], that
$\mathbb{F}^{\alpha}_{2,2,0}(\Gamma)=\mathbb{B}_{2,2,0}^{\alpha}(\Gamma)$,
also with equivalence of norms, and the result follows immediately from the
previous proposition. ∎

######  Remark A.5.

Characterizations of trace spaces in terms of atomic representations, like the
one given in Proposition A.3 above, are not new. With a somewhat different
approach and in the larger setting of the so-called $h$-sets, they can already
be seen in [6, Prop. 3.5.4].

######  Remark A.6.

If $\Gamma\subset\mathbb{R}^{n}$ is a $d$-set and $d>n-1$, then the hypothesis
that $\Gamma$ preserves Markov’s inequality is automatically satisfied [31,
Thm. 3 on p. 39].

##  Appendix B Inverse estimates

In this appendix we prove that the inverse estimate (86), established for
$0<t<1$ in Theorem 5.10, holds in fact for the extended range $0<t\leq J$, for
any $J\in\mathbb{N}$, moreover with a constant in the inverse estimate that is
independent of $t$. Our proof is modelled on standard inverse estimate
arguments for negative exponent Sobolev spaces, e.g. [20, Theorem 4.6], with
the important difference that it is, of course, impossible to support the
smooth “bubble functions” (in the terminology of [20, §4.3]), that are a key
tool in the arguments of [20] and in the proof of earlier inverse estimate
results, inside an element $\Gamma_{{\boldsymbol{m}}}\subset\Gamma$, as
$\Gamma$ has empty interior. In the case that $\Gamma$ is a disjoint IFS
attractor, it turns out that, to carry through an analogous argument, it is
enough to replace the bubble function by a smooth function supported in a
carefully chosen neighbourhood of $\Gamma_{{\boldsymbol{m}}}$ (the function
$\sigma_{{\boldsymbol{m}}}\in C_{0}^{\infty}(\mathbb{R}^{n})$ in the proof
below). The assumption that $\Gamma$ is a disjoint IFS attractor implies that
$\Gamma$ is a $d$-set with $0<d<n$ (see Lemma 2.6), but, in contrast to
Theorem 5.10, we make no additional constraint on $d$. We continue to assume
in this appendix that $h$ lies in the range (78), i.e. that

$0<h\leq h_{0}:=\operatorname{diam}(\Gamma),$

and to use the notations of the main part of the paper, notably $L_{h}$,
$\chi_{\boldsymbol{m}}$, and $\mathbb{Y}_{h}$ defined by (76), (18), and (75),
respectively.

######  Theorem B.1.

Suppose that $\Gamma$ is the attractor of an IFS, satisfying (4), that
$\Gamma_{1}$, …, $\Gamma_{M}$ are disjoint (i.e., $\Gamma$ is disjoint), and
that $J\in\mathbb{N}$. Then there exists $c_{I}>0$ such that, for every
$\psi_{h}\in\mathbb{Y}_{h}$,

$\|\psi_{h}\|_{\mathbb{L}_{2}(\Gamma)}\leq
c_{I}h^{-t}\|\psi_{h}\|_{\mathbb{H}^{-t}(\Gamma)},\quad 0<t\leq J.$ (135)

###### Proof.

Let

$\rho_{\min}:=\min_{j=1,...,M}\rho_{j}\quad\mbox{ and }\quad
c_{0}:=\mathcal{H}^{d}(\Gamma).$

Since $\Gamma_{1}$, …, $\Gamma_{M}$ are disjoint, $\Gamma$ satisfies the OSC
for some open set $O\supset\Gamma$, by Lemma 2.5. By a standard mollification
we can construct a $\sigma\in C_{0}^{\infty}(\mathbb{R}^{n})$ supported in $O$
such that $\sigma=1$ in a neighbourhood of $\Gamma$. For each
${\boldsymbol{m}}=(m_{1},...,m_{\ell})\in L_{h}$, let
$\rho_{{\boldsymbol{m}}}:=\prod_{j=1}^{\ell}\rho_{m_{j}}$,

$h_{{\boldsymbol{m}}}:=\rho_{{\boldsymbol{m}}}h_{0}=\operatorname{diam}(\Gamma_{{\boldsymbol{m}}})\in(\rho_{\min}h,h],$
(136)

and

$\sigma_{{\boldsymbol{m}}}:=\sigma\circ s_{m_{\ell}}^{-1}\circ
s_{{m_{\ell-1}}}^{-1}\circ\cdots\circ s_{m_{1}}^{-1},$

and note that, as a consequence of the OSC, the supports of
$\sigma_{{\boldsymbol{m}}}$ and $\sigma_{{\boldsymbol{m}}^{\prime}}$ are
disjoint, for ${\boldsymbol{m}}\neq{\boldsymbol{m}}^{\prime}$.

Now suppose that $0<t\leq J$, and define $s$ in terms of $t$ by (10). For
${\boldsymbol{m}}\in L_{h}$, noting (15) and (18), we have that

$\langle\sigma_{{\boldsymbol{m}}},\mathrm{tr}_{\Gamma}^{*}\chi_{{\boldsymbol{m}}}\rangle_{H^{s}(\mathbb{R}^{n})\times
H^{-s}(\mathbb{R}^{n})}=(\mathrm{tr}_{\Gamma}\sigma_{{\boldsymbol{m}}},\chi_{{\boldsymbol{m}}})_{\mathbb{L}_{2}(\Gamma)}=(\mathcal{H}^{d}(\Gamma_{{\boldsymbol{m}}}))^{1/2}=c_{0}^{1/2}\,\rho_{{\boldsymbol{m}}}^{d/2}.$
(137)

Clearly (135) holds for $\psi_{h}=0$. Now suppose that
$\psi_{h}\in\mathbb{Y}_{h}\setminus\\{0\\}$, so that, for some coefficients
$a=(a_{{\boldsymbol{m}}})_{{\boldsymbol{m}}\in L_{h}}$, we have

$\psi_{h}=\sum_{{\boldsymbol{m}}\in
L_{h}}a_{{\boldsymbol{m}}}\chi_{{\boldsymbol{m}}},$

and

$\|\psi_{h}\|_{\mathbb{L}_{2}(\Gamma)}=\|a\|_{2}:=\left(\sum_{{\boldsymbol{m}}\in
L_{h}}|a_{{\boldsymbol{m}}}|^{2}\right)^{1/2}.$

Define $u_{h}\in Y_{h}\subset H^{-s}(\mathbb{R}^{n})$ and $v_{h}\in
H^{s}(\mathbb{R}^{n})$ by

$u_{h}:=\mathrm{tr}_{\Gamma}^{*}\psi_{h}=\sum_{{\boldsymbol{m}}\in
L_{h}}a_{{\boldsymbol{m}}}\mathrm{tr}_{\Gamma}^{*}\chi_{{\boldsymbol{m}}}\quad\mbox{and}\quad
v_{h}:=\sum_{{\boldsymbol{m}}\in
L_{h}}a_{{\boldsymbol{m}}}\sigma_{{\boldsymbol{m}}}.$

Then, since the supports of the $\sigma_{{\boldsymbol{m}}}$ are disjoint and
again noting (15), we have, using (137) and (136), that

$\langle v_{h},u_{h}\rangle_{H^{s}(\mathbb{R}^{n})\times
H^{-s}(\mathbb{R}^{n})}=c_{0}^{1/2}\,\sum_{{\boldsymbol{m}}\in
L_{h}}|a_{{\boldsymbol{m}}}|^{2}\rho_{{\boldsymbol{m}}}^{d/2}\geq
c_{0}^{1/2}\,(\rho_{\min}h/h_{0})^{d/2}\,\|a\|_{2}^{2}.$ (138)

For $j\in\mathbb{N}_{0}$ and ${\boldsymbol{m}}\in L_{h}$, using standard
properties of Fourier transforms (e.g., [26, Prop. 2.2.11]),

$\displaystyle\|\sigma_{{\boldsymbol{m}}}\|^{2}_{H^{j}(\mathbb{R}^{n})}$
$\displaystyle=$
$\displaystyle\rho_{\boldsymbol{m}}^{2n}\int_{\mathbb{R}^{n}}\left(1+|\xi|^{2}\right)^{j}\left|\widehat{\sigma}(\rho_{\boldsymbol{m}}\xi)\right|^{2}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\rho_{\boldsymbol{m}}^{n}\int_{\mathbb{R}^{n}}\left(1+|\xi/\rho_{\boldsymbol{m}}|^{2}\right)^{j}\left|\widehat{\sigma}(\xi)\right|^{2}\,\mathrm{d}\xi$
$\displaystyle\leq$
$\displaystyle\rho_{{\boldsymbol{m}}}^{n-2j}\|\sigma\|^{2}_{H^{j}(\mathbb{R}^{n})},$

since $\rho_{\boldsymbol{m}}\leq 1$, so, using that
$\rho_{{\boldsymbol{m}}}\in(\rho_{\min}h/h_{0},h/h_{0}]$ by (136), and again
that the supports of the $\sigma_{{\boldsymbol{m}}}$ are disjoint,

$\|v_{h}\|^{2}_{H^{j}(\mathbb{R}^{n})}=\sum_{{\boldsymbol{m}}\in
L_{h}}|a_{{\boldsymbol{m}}}|^{2}\|\sigma_{{\boldsymbol{m}}}\|^{2}_{H^{j}(\mathbb{R}^{n})}\leq\left\\{\begin{array}[]{ll}\|\sigma\|^{2}_{H^{j}(\mathbb{R}^{n})}(\rho_{\min}h/h_{0})^{n-2j}\,\|a\|_{2}^{2},&\mbox{if
}j\geq n/2,\\\
\|\sigma\|^{2}_{H^{j}(\mathbb{R}^{n})}(h/h_{0})^{n-2j}\,\|a\|_{2}^{2},&\mbox{if
}j<n/2.\end{array}\right.$

In particular this applies for $j=0,L$, where
$L:=\lceil\max(J+(n-d)/2,n/2)\rceil$, so that $L\geq\max(s,n/2)$. Hence, and
applying Hölder’s inequality to the definition of the $H^{s}(\mathbb{R}^{n})$
norm and noting that
$\|\sigma\|_{H^{0}(\mathbb{R}^{n})}\leq\|\sigma\|_{H^{L}(\mathbb{R}^{n})}$,

$\|v_{h}\|_{H^{s}(\mathbb{R}^{n})}\leq\|v_{h}\|^{1-s/L}_{H^{0}(\mathbb{R}^{n})}\|v_{h}\|^{s/L}_{H^{L}(\mathbb{R}^{n})}\leq\|\sigma\|_{H^{L}(\mathbb{R}^{n})}\,\rho_{\min}^{(n/(2L)-1)s}\,(h/h_{0})^{n/2-s}\,\|a\|_{2}.$
(139)

Now, recalling from §2.4 that
$\mathrm{tr}_{\Gamma}^{*}:\mathbb{H}^{-t}(\Gamma)\to H^{-s}(\mathbb{R}^{n})$
has unit norm,

$\|\psi_{h}\|_{\mathbb{H}^{-t}(\Gamma)}\geq\|u_{h}\|_{H^{-s}(\mathbb{R}^{n})}=\sup_{v\in
H^{s}(\mathbb{R}^{n})\setminus\\{0\\}}\frac{|\langle
v,u_{h}\rangle_{H^{s}(\mathbb{R}^{n})\times
H^{-s}(\mathbb{R}^{n})}|}{\|v\|_{H^{s}(\mathbb{R}^{n})}}\geq\frac{|\langle
v_{h},u_{h}\rangle_{H^{s}(\mathbb{R}^{n})\times
H^{-s}(\mathbb{R}^{n})}|}{\|v_{h}\|_{H^{s}(\mathbb{R}^{n})}}.$

Combining this inequality with (138) and (139) and recalling (10) we obtain
that

$\|\psi_{h}\|_{\mathbb{H}^{-t}(\Gamma)}\geq\frac{c_{0}^{1/2}}{\|\sigma\|_{H^{L}(\mathbb{R}^{n})}}\,\rho^{t+n/2}_{\min}\,(h/h_{0})^{t}\,\|a\|_{2},$

and the result follows on recalling that
$\|a\|_{2}=\|\psi_{h}\|_{\mathbb{L}_{2}(\Gamma)}$. ∎

Acknowledgements. We are grateful to Martin Costabel for instructive
discussions related to this project. SC-W was supported by Engineering and
Physical Sciences Research Council (EPSRC) Grant EP/V007866/1, and DH and AG
by EPSRC grants EP/S01375X/1 and EP/V053868/1. AM was supported by the PRIN
project “NA-FROM-PDEs” and by the MIUR through the “Dipartimenti di
Excellenza” Program (2018–2022)–Dept. of Mathematics, University of Pavia. AC
was supported by CIDMA (Center for Research and Development in Mathematics and
Applications) and FCT (Foundation for Science and Technology) within project
UIDB/04106/2020.

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