Locally corrected quadratures

Let the surface \(\Gamma\) be defined via a collection of patches \(\Gamma_{j} = 1,2,\ldots N_{\textrm{patches}}\), where each patch \(\Gamma_{j}\) is parametrized by a non-degenerate chart \(\boldsymbol{x}^{j}: B \to \Gamma_{j}\), where \(B\) is standard base element. Given a density \(\sigma\) defined on \(\Gamma\), consider the evaluation of the layer potential \(\mathcal{S}[\sigma](x)\) at a collection of targets \(x=\boldsymbol{t}_{j}\), \(j=1,2\ldots n_{t}\). These targets could be anywhere in the volume including on the surface \(\Gamma\).

\[\mathcal{S}[\sigma](x_{i}) = \int_{\Gamma} G(x_{i},y) \sigma(y) da(y) = \sum_{j=1}^{N_{\textrm{patches}}} \int_{\Gamma_{j}} G_{k}(x_{i},y) \sigma(y) da(y)\]

If a patch \(\Gamma_{j}\) is close to (as compred to the size of the patch) a target \(x_{i}\), then the integrand is nearly singular and the integral becomes difficult to evaluate accurately as compared to when \(x_{i}\) is far from the patch \(\Gamma_{j}\). Locally corrected quadrature methods precompute the quadrature for all near interactions between patches and targets, and use appropriately oversampled quadratures for the rest of the interactions.

Near-far split

Let \(c_{j}\) denote the centroid of a patch given by

\[c_{j} = \int_{\Gamma_{j}} y da(y) \, ,\]

and let \(R_{j}\) denote the smallest radius \(R\) such that a sphere of radius R centered at \(c_{j}\) completely contains \(\Gamma_{j}\), i.e.

\[R_{j} = \min_{R} \{ R : \Gamma_{j} \subset B_{R}(c_{j}) \} \, .\]

Then given \(\eta>0\), the \(\eta\)-scaled near field of the patch \(\Gamma_{j}\) is given by

\[N_{\eta}(\Gamma_{j}) = \{ x : d(c_{j},x) \leq \eta R_{j} \} \, .\]

Given \(N_{\eta}(\Gamma_{j})\), let \(T_{\eta}(x_{i})\) denote the dual list – the collection of patches \(\Gamma_{j}\) for which \(x_{i}\) is in its \(\eta\)-scaled near field,

\[T_{\eta}(x_{i}) = \{ \Gamma_{j} : x_{i} \in N_{\eta}(\Gamma_{j} \} = \{ \Gamma_{j} : d(x_{i},c_{j}) \leq \eta R_{j} \}\]

The integral for \(\mathcal{S}[\sigma](x)\) can be split into two parts

\[\begin{split}\mathcal{S}[\sigma](x) &= \sum_{\Gamma_{j} \in T_{\eta}(x)} \int_{\Gamma_{j}} G(x,y)\sigma(y) da(y) + \sum_{\Gamma_{\ell} \not \in T_{\eta}(x)} \int_{\Gamma_{j}} G(x,y)\sigma(y) da(y) \\ &= \mathcal{S}_{\textrm{near}}[\sigma](x) + \mathcal{S}_{\textrm{far}}[\sigma](x)\end{split}\]