Abstract
We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected based on the expansion. High-order accuracy can be achieved by utilizing a sufficient number of correction nodes around the singularity to approximate the terms in the series expansion. The derived quadratures are applied to the implicit boundary integral formulation of surface integrals involving the Laplace layer kernels.
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Funding
Open access funding provided by Royal Institute of Technology. Tsai’s research is supported partially by National Science Foundation Grants DMS-1913209 and DMS-2110895.
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Conceptualization: Olof Runborg, Richard Tsai. Methodology: Federico Izzo. Investigation: Federico Izzo. Software: Federico Izzo. Visualization: Federico Izzo. Writing—original draft preparation: Federico Izzo, Olof Runborg, Richard Tsai. Writing—review and editing: Federico Izzo, Olof Runborg, Richard Tsai. Funding acquisition: Richard Tsai. Supervision: Olof Runborg, Richard Tsai.
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Appendices
Appendix 1. Proofs of the lemmas and theorems
In this section, we will prove the Lemmas and Theorems mentioned in Sect. 2 and Sect. 3.
1.1 Proof of Theorem 2.1
Consider a cut-off function \(\psi \in C^\infty _c({\mathbb R}^n)\) such that
Then, we can write f as
The first term is a function compactly supported in \(B_{r_0}\), so by extending it to zero in \({\mathbb {R}}^n\) it satisfies the hypotheses of Theorem 4.1. Hence, the result is valid for the first term.
The second term has regularity \(C^\infty _c({\mathbb {R}}^n)\) and is zero in \(B_{r_0/2}\), so the error for the punctured trapezoidal rule will decrease faster than any polynomial of h.
By combining the results for the two terms, we prove the result.
1.1.1 Results on which Theorem 2.1 depends
Theorem 2.1
Suppose \(v\in C^\infty _c({\mathbb R}^n)\) and \(\ell \in C^{\infty }({\mathbb {R}}\times \mathbb {S}^{n-1})\). Then, for integers \(j\ge 1-n\),
where the constant C is independent of h, but depends on j, \(\ell \) and v.
Proof
Define \(f(\textbf{x}):=\vert \textbf{x}\vert ^j\ell (\vert \textbf{x}\vert ,\textbf{x}/\vert \textbf{x}\vert )v(\textbf{x})\), and consider the cut-off function \(\psi \in C^\infty _c({\mathbb R}^n)\) 61. Then, we can write the punctured trapezoidal rule as
where we cut out the singularity point by multiplying by \(1-\psi \) around \(\textbf{0}\); the scaling by h ensures that, for fixed h, only the node in the singularity point is cut out. This allows us to split the error of the punctured trapezoidal rule as
We will consider the two terms (I), (II) separately, and prove that both can be bounded by \(Ch^{j+n}\).
(I): Given the compact support of \(\psi \), the integral is reduced to an integral over \(\{\vert \textbf{x}\vert \le h\}\):
since \(\vert \textbf{x}\vert ^j\) is integrable as \(j\ge 1-n\). We have proven the estimate for the first term.
(II): For the second term, knowing that the volume of the fundamental parallelepiped of the lattice \(V:=(h\mathbb {Z})^n\) is \(h^n\) and that the dual lattice is \(V^*=(h^{-1}\mathbb {Z})^n\), we use the Poisson summation formula:
Then, the error in (II) is:
where
Using integration by parts separately on each of the variables, we find
For the Laplacian operator applied q times, we therefore have
We use this result to find an expression we can bound using Lemma 4.2; given an integer q, we find
Then, the series of Fourier coefficients is
The series converges if \(2q>n\), and the leading order is \(h^{j+n}\) if \(2q\ge j+n\), so by taking \(q\ge \max (1+n/2,(n+j)/2)\), we find the result sought. Combining the results for (I) and (II), we find the bound
This proves the theorem.
We use the notation \(\textbf{x}=(x_1,x_2,\dots ,x_n)=\sum _{l=1}^n x_l e_l\), and indicate with \(e_l\) the l-th element of the standard \({\mathbb {R}}^n\) basis.
Lemma 2.2
Let \(g,\psi \in C^\infty _c({\mathbb {R}}^n)\), \(\ell \in C^\infty ({\mathbb {R}}\times \mathbb {S}^{n-1})\), where \(\psi \) is such that
Let \(j\ge 1-n\), and \(f(\textbf{x})=\vert \textbf{x}\vert ^j \ell (\vert \textbf{x}\vert ,\textbf{x}/\vert \textbf{x}\vert )g(\textbf{x})\); then, for any multi-index \(\beta \in \mathbb {N}^n_0\) it exists a constant \(C_\beta \) independent of h such that, for \(0<h\le 1\),
Proof
Given \(\beta \in \mathbb {N}^n_0\), we first prove that there exist functions \(f_{\beta }:{\mathbb {R}}\times \mathbb {S}^{n-1}\rightarrow {\mathbb {R}}\) in \(C_c^\infty ({\mathbb {R}}\times \mathbb {S}^{n-1})\) such that
We prove this by induction. The induction base \(\beta =\textbf{0}\) is true because
where \(f_{\textbf{0}}\in C^\infty _c({\mathbb {R}}\times \mathbb {S}^{n-1})\). For the induction step, we assume that (64) is true for \(\beta \) and prove it for \(\beta +e_l\):
By computing the derivative, we find
Because \(f_\beta \in C^\infty _c({\mathbb {R}}\times \mathbb {S}^{n-1})\) the same is also true for \(f_{\beta +e_l}\).
The next step is to expand the derivative in (63) and use (64), and then bound it:
We use the properties of \(\psi \), and the compact support of \(f_\nu \). Let \(L>0\) be such that \(\forall \nu \le \beta \), supp\(\,f_\nu \) is contained in the ball \(B_L(\textbf{0})\). Note furthermore that the derivatives of \(\psi \) are compactly supported in the annulus \(\{\textbf{x}\in {\mathbb {R}}^n:\,\frac{1}{2}\le \vert \textbf{x}\vert \le 1\}\). From this, we can say that
We use these bounds in the evaluation of the integral, and after passing to polar coordinates, we arrive at (63) via
The lemma is proven.
1.2 Proof of Lemma 2.2
For any \(\textbf{u}\in \mathbb {S}^1\), we expand \(\ell \) around \(r=0\) and write the remainder in integral form:
Then,
where \(\sigma \in C^\infty ((-r_0,r_0)\times \mathbb {S}^1)\) because \(\ell \in C^\infty ((-r_0,r_0)\times \mathbb {S}^1)\). The lemma is thus proven.
1.3 Proof of Lemma 3.2
The first two identities in (53) follows since \(P_\Gamma (\bar{\textbf{x}}^*+(\textbf{0},z')_B) = \bar{\textbf{x}}^*\) for all \(z'\), as was already pointed out in Sect. 3.2.1. For the second part, we note that the surface normal at the point \(\bar{\textbf{x}}^*+ \big ( {\textbf{y}}_{\text {p}},f({\textbf{y}}_{\text {p}})\big )_B\) is parallell to \((-\nabla f({\textbf{y}}_{\text {p}}),1)_B\). Therefore, there is a \(t\in {\mathbb {R}}\) such that
which implies that
Using the fact that \({\textbf{y}}_{\text {p}}=\textbf{h}({\textbf{y}}',z')\) and differentiating both sides with respect to \({\textbf{y}}'\) gives us,
and the result follows upon evaluating at \({\textbf{y}}'={\textbf{y}}_{\text {p}}=\textbf{0}\) and using (50). Since \(\textbf{h}\) is smooth on \({\mathcal M}_L\) the matrix D must thus be well-defined.
For the second order term in the Taylor expansion, we write \(\textbf{y}_{\text {p}}=(y_1,y_2)\), \(\textbf{h}=(h_1,h_2)^T\) and \(\textbf{F}=(F_1,F_2)^T\). We then get for \(j=1,2\),
From the expressions above, we have that \(\frac{\partial \textbf{F}(\textbf{0})}{\partial \textbf{y}_{\text {p}}}=D^{-1}(z)\). Therefore, evaluating at \({\textbf{y}}'={\textbf{y}}_{\text {p}}=\textbf{0}\), yields
Since
we finally get
This gives (54) and the lemma is proven.
1.4 Proof of Lemma 3.3
For the first function, using the hypothesis \(\bar{\textbf{g}}(\textbf{0})=\textbf{0}\) and the notation \(\textbf{x}=\vert \textbf{x}\vert \textbf{u}\) with \(\textbf{x}/\vert \textbf{x}\vert =:\textbf{u}\in \mathbb {S}^{m-1}\), we write the expansion around \(\textbf{x}=\textbf{0}\) as
where \(E_{\bar{\textbf{g}},\nu }(\textbf{x}):=\frac{2}{\nu !}\int _0^1(1-t)\partial ^\nu \bar{\textbf{g}}(t\textbf{x})\text {d}t\) is given by the integral form of the remainder term. Using the full rank of \(D\bar{\textbf{g}}(\textbf{0})\), there exists \(0<r_1\le r_0\) be such that \(f(\vert \textbf{x}\vert ,\textbf{u})\ne 0\) in \((-r_1,r_1)\times \mathbb {S}^{m-1}\). Then,
and from the hypotheses on \(D\bar{\textbf{g}}(\textbf{0})\) and on the smoothness of \(\bar{\textbf{g}}\), \(\ell _1\) is \(C^\infty ((-r_1,r_1)\times \mathbb {S}^{m-1})\).
For the second function form, let \(r(\textbf{x}):=\bar{\textbf{p}}(\textbf{x})^T\bar{\textbf{g}}(\textbf{x})\); then, \(\nabla r(\textbf{x})=\bar{\textbf{g}}(\textbf{x})^TD\bar{\textbf{p}}(\textbf{x})+\bar{\textbf{p}}(\textbf{x})^TD\bar{\textbf{g}}(\textbf{x})\). Using the hypothesis \(\bar{\textbf{p}}(\textbf{0})^TD\bar{\textbf{g}}(\textbf{0})=\textbf{0}\), we write the expansion of r around \(\textbf{x}=\textbf{0}\) using the integral form of the remainder:
where \(E_{r,\nu }(\textbf{x}):=\frac{2}{\nu !}\int _0^1(1-t)\partial ^\nu r(t\textbf{x})\text {d}t\), so that we find
From the hypotheses on the smoothness of \(\bar{\textbf{g}}\) and \(\bar{\textbf{p}}\), \(\ell _2\) is \(C^\infty ((-r_1,r_1)\times \mathbb {S}^{m-1})\) and the result is proven.
Appendix 2. Computation of the derivatives of the local surface function
In this section, we will show how to find numerically the derivatives of f in the Implicit Boundary Integral Methods setting of Sect. 3. The derivatives are needed to evaluate the functions B and C of (52), which are used in the approximated kernels (41).
The first derivatives and the mixed second derivatives are zero by construction, so we will show how to find the pure second derivatives and all the third derivatives.
Let \(\bar{\textbf{z}}\) be an arbitrary point in \(T_\varepsilon \), and \(\eta =d_\Gamma (\bar{\textbf{z}})\). Let \(\Gamma _\eta :=\{\bar{\textbf{z}}\in T_\varepsilon :\,d_\Gamma (\bar{\textbf{z}})=\eta \}\) be the surface parallel to \(\Gamma \) at signed distance \(\eta \).
The pure second derivatives of f at \(P_\Gamma (\bar{\textbf{z}})\), \(f_{xx},f_{yy}\), are the principal directions \(\kappa _1,\kappa _2\) of \(\Gamma \) at \(P_\Gamma (\bar{\textbf{z}})\). We find the principal curvatures \(g_1,g_2\) of \(\Gamma _\eta \) in \(\bar{\textbf{z}}\) via the Hessian of \(d_\Gamma \) at \(\bar{\textbf{z}}\):
where \(\bar{\varvec{\tau }}_1\), \(\bar{\varvec{\tau }}_2\) are the principal directions and \(\bar{\textbf{n}}\) is the normal to \(\Gamma \) in \(P_\Gamma (\textbf{z})\). In practice, the values of either \(P_\Gamma \) or \(d_\Gamma \) are given on the grid nodes. The principal directions and curvatures are computed from eigendecomposition of third order numerical approximations of the Hessian, \(H_{d_\Gamma }\). Alternatively, one can obtain this information from the derivative matrix of \(P_\Gamma \), see [17]. Then, the following relation lets us find the principal curvatures \(\kappa _1,\kappa _2\) from \(g_1,g_2\) and \(\eta \):
The third derivatives of f can be found by computing the second derivatives with respect to \({\textbf{y}}'\) of \(\textbf{h}({\textbf{y}}',z')\) from Sect. 3.2.1. By differentiating twice (65) with respect to \({\textbf{y}}'=(x,y)\) with \(\textbf{h}({\textbf{y}}',z')={\textbf{y}}_{\text {p}}=(h_1,h_2)\) and evaluating in \({\textbf{y}}'=\textbf{0}\), we find the following two linear systems:
We find the first and second derivatives of \(\textbf{h}({\textbf{y}}',z')\) by computing the derivatives of \(P_\Gamma \) in \(\bar{\textbf{z}}\) and applying a change of basis transformation.
By construction \(\bar{\textbf{z}}=P_\Gamma (\bar{\textbf{z}})+\eta \bar{\textbf{n}}\). Then, we use the closest point projections of the grid nodes around \(\bar{\textbf{z}}\),
In the B basis, these points are expressed as \(\bar{\textbf{v}}_{ijk}=\bar{\textbf{x}}^*+(\bar{\textbf{w}}_{ijk})_B\), where \(\bar{\textbf{w}}_{ijk}=Q^{-1}(\bar{\textbf{v}}_{ijk}-\bar{\textbf{x}}^*)\). We apply finite differences (central differences of 4th order in this case) to the component of the nodes \(\bar{\textbf{w}}_{ijk}=(X_{ijk},Y_{ijk},Z_{ijk})\) to compute
We can then use these approximations to find the derivatives of \(h_i\), \(i=1,2\) by applying the following transformations:
Finally, we solve the two systems 66 with these values and \(z'=\eta \).
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Izzo, F., Runborg, O. & Tsai, R. High-order corrected trapezoidal rules for a class of singular integrals. Adv Comput Math 49, 60 (2023). https://doi.org/10.1007/s10444-023-10060-0
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DOI: https://doi.org/10.1007/s10444-023-10060-0
Keywords
- Singular integrals
- Trapezoidal rules
- Level set methods
- Closest point projection
- Boundary integral formulations