Abstract
A boundary integral equation in general form will be considered, which can be used to solve Dirichlet problems for the Helmholtz equation. The goal of this paper is to develop a fast Fourier-Galerkin method solving these boundary integral equations. To this aim, a scheme for splitting integral operators is presented, which splits the corresponding integral operator into a convolution operator and a compact operator. A truncation strategy is presented, which can compress the dense coefficient matrix to a sparse one having only \(\mathcal {O}(n)\) nonzero entries, where n is the order of the Fourier basis functions used in the method. The proposed fast method preserves the stability and optimal convergence order. Moreover, exponential convergence can be obtained under suitable assumptions. Numerical examples are presented to confirm the theoretical results for the approximation accuracy and computational complexity of the proposed method.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)
Borm, S., Sauter, S.: BEM with linear complexity for the classical boundary integral operators. Math Comp. 74, 1139–1177 (2005)
Cai, H., Xu, Y.: A fast Fourier-Galerkin method for solving singualr boundary integral equations. SIAM J. Numer. Anal. 46, 1965–1984 (2008)
Galkowski, J., Spence, E.: Wavenumber-Explicit Regularity estimates on the acoustic single- and Double-Layer operators. Integr. Equat. Oper. Th., 91 (2019)
Han, X., Tacy, M.: Sharp norm estimates of layer potentials and operators at high frequency. J. Funct. Anal. 269(9), 2890–2926 (2015)
Helsing, J., Karlsson, A.: An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces. J. Comput. Phys. 272, 686–703 (2014)
Jiang, Y., Wang, B., Xu, Y.: A fast Fourier-Galerkin method solving a boundary integral equation for the biharmonic equation. SIAM J. Numer. Anal. 52(5), 2530–2554 (2014)
Jiang, Y., Wang, B., Xu, Y.: A fully discrete fast fourier-Galerkin method solving a boundary integral equation for the biharmonic equation. J. Sci. Comput. 76(3), 1594–1632 (2018)
Jiang, Y., Xu, Y.: Fast fourier-Galerkin methods for solving singular boundary integral equations: numerical integration and precondition. J. Comput. Appl. Math. 234(9), 2792–2807 (2010)
Krantz, S.G., Parks, H.: A Primer of Real Analytic Functions. Birkhauser, Boston (2002)
Kress, R.: Linear Integral Equations. Springer, New York (1999)
Kress, R., Sloan, I.H.: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numer. Math. 66 (2), 199–214 (1993)
Mclean, W.: A spectral Galerkin method for a boundary integral equation. Math. Comp. 47, 597–607 (1986)
Nosich, A.I.: The method of analytical regularization in wave-scattering and eigenvalue problems: Foundations and review of solutions. IEEE Antennas Propagat. Mag. 41(3), 34–49 (1999)
Saranen, J., Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer, Berlin (2002)
Wang, B., Wang, R., Xu, Y.: Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs. Sci. China Math. 53 (1), 1–22 (2010)
Xu, Y., Zhao, Y.: An extrapolation method for a class of boundary integral equations. Math Comput. 65(214), 587–610 (1996)
Young, P., Hao, S., Martinsson, P.G.: A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces. J. Comput. Phys. 231, 4142–4159 (2012)
Funding
Supported partially by National Natural Science Foundation of China under grants 11501087 and 11571383, the Special Project on High-performance Computing under the National Key R&D Program (No. 2016YFB0200602), the Science and Technology Program of Guangzhou, China (No. 201804020053) and the Science Strength Promotion Programme of UESTC.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors of this paper are ordered alphabetically.
Appendix: Technical lemmas used in the proofs
Appendix: Technical lemmas used in the proofs
For given p ≥ 0, and q > 0, when \(\mathcal {A}+{\mathscr{B}}\) is injective from Hp(I) to Hp+m+ 1(I) with a bounded inversion, and \({\mathscr{B}}\in B(H^{p}(I),H^{p+m+q+1}(I))\), we define
Lemma 8
Let p ≥ 0 and q > 0. If \(\mathcal {A}+{\mathscr{B}}\) is injective from Hp(I) to Hp+m+ 1(I) with a bounded inversion, and \({\mathscr{B}}\in B(H^{p}(I),H^{p+m+q+1}(I))\), then there holds that for all n ≥ Cm,p,q, and all w ∈ Xn,
Proof
Since \({\mathscr{B}}\in B(H^{p}(I),H^{p+m+q+1}(I))\), for all w ∈ Xn,
Thus, when n ≥ Cm,p,q, there holds that for all w ∈ Xn,
□
For all \(x\in \mathbb {R}\), \(r_{1},r_{2}\in \mathbb {R}\) with r1 − 2 > r2 > 0, define
It is easy to see that for all \(x\in \mathbb {R}\), \(G_{r_{1},r_{2}}(x)<+\infty \). In the next lemma, we discuss the decay rate of function \(G_{r_{1},r_{2}}\) as variant x going to infinite.
Lemma 9
If \(m_{1},m_{2}\in \mathbb {R}\) with m1 − 1 > m2 > 1, then for all \(x\in \mathbb {R}\),
Proof
Since \(G_{m_{1},m_{2}}(x)=G_{m_{1},m_{2}}(-x)\), it is sufficient to prove this lemma by showing that (50) holds when x ≥ 0. Note that for x ≥ 0,
and
Meanwhile, since for 0 ≤ k < x,
there holds
Note that m1 − 1 > m2 > 1. Combining (51), (52), and (53), from the definition of \(G_{m_{1},m_{2}}\) we obtain the desired estimate (50). □
Lemma 10
Let 0 < r < 1 and α > 0. Then, there is a positive constant c such that for all k ≥ 0,
where c depends on r.
Proof
Define
It is easy to see that \( {\sum }_{l=0}^{k} r^{l} (k+1-l)^{-\alpha }=(k+1)^{-\alpha }S_{k}. \) Thus, it is sufficient to prove this lemma by showing that the sequence \(\{S_{k}:k\in \mathbb {N}\}\) is bounded. Note that
i.e., \(S_{k+1}=1+\left (\frac {k+2}{k+1}\right )^{\alpha }rS_{k}\). Since \(\lim _{k\rightarrow +\infty } \left (\frac {k+2}{k+1}\right )^{\alpha }=1\) and r < 1, for any r < R < 1 there exists an integer k∗ such that for all k > k∗, \(\left (\frac {k+2}{k+1}\right )^{\alpha }r<R\). Thus, for all k > k∗, there is Sk+ 1 = 1 + RSk. Then, by noting that 0 < R < 1, we know that the sequence \(\{S_{k}:k\in \mathbb {N}\}\) is bounded. □
Lemma 11
If \(\gamma _{1}, \gamma _{2}\in \tilde {C}_{2\pi }(I)\), then function h, defined by (4), is in \(\tilde {C}_{2\pi }(I^{2})\).
Proof
We know that, for any d > c > 0, the function \(\log \) is real analytic in [c,d]. Note that function h is in \({C}^{\infty }_{2\pi }(I^{2})\), which implies that \(\inf \left \{\frac {|\gamma (t)-\gamma (s)|^{2}}{4e^{-2}\sin \limits ^{2}\frac {s-t}{2}}:t,s\in I\right \}>0\). We rewrite h as,
where,
Therefore, to prove this lemma, it suffices to show \(h_{1}, h_{2} \in \tilde {C}_{2\pi }(I^{2})\).
It is clear that the extension of h1 is real analytic on \(\mathbb {R}^{2}\) on \(\{(t,s)\in \mathbb {R}^{2}: t\neq s+2k\pi , k\in \mathbb {Z}\}\). Thus, by the 2π-biperiodicity of h1, to prove \(h_{1}\in \tilde {C}_{2\pi }(I^{2})\), we only need to show that for all \(t_{0}\in \mathbb {R}\), the extension of function h1 can be represented by a convergent power series in some neighborhood of point (t0,t0). Define
for all \(t,s\in \mathbb {R}\). Note that for all \(t,s\in \mathbb {R}\) with |t − s| < π/2, \(h_{1}(t,s)=\tilde {h}(t,s)\bar {h}(t,s)\), and \(\bar {h}\) is real analytic on \(\{(t,s)\in \mathbb {R}^{2}:|t-s|<\pi /2\}\). Thus, we only need to show that for all \(t_{0}\in \mathbb {R}\), function \(\tilde {h}\) can be represented by a convergent power series in some neighborhood of point (t0,t0). Since \(\gamma _{1}\in \tilde {C}_{2\pi }(I)\), there is an neighborhood Bε(t0) := {t ∈ R : |t − t0| < ε} such that for all t ∈Bε(t0),
Thus, there holds that for all t,s ∈Bε(t0) with t≠s,
For t,s ∈Bε(t0) with t = s, it is easy to see that
and
Thus, (54) holds for all t,s ∈Bε(t0). This ensures \(h_{1} \in \tilde {C}_{2\pi }(I^{2})\).
Replacing γ1 and h1 in above discussion by γ2 and h2 respectively, we obtain that \(h_{2} \in \tilde {C}_{2\pi }(I^{2})\). Then, we can claim that \(h=\frac {1}{2}\log ({h^{2}_{1}}+{h^{2}_{2}})\in \tilde {C}_{2\pi }(I^{2})\). □
Lemma 12
For 0 < r < 1 and m ≥ 0, there exists a positive constant c such that for all \(k,l\in \mathbb {Z}\) with kl ≤ 0,
and for all \(k,l\in \mathbb {Z}\) with kl > 0,
Proof
By noting that
it is sufficient for us to consider the case with k ≥−l. In this case, there holds that
When kl ≤ 0, we continue the estimate (57) from the following two sub-cases.
-
(i)
In the first sub-case, let − l ≤ k ≤ 0. There holds that
$$ \sum\limits_{k^{\prime} \leq-l} r^{-2\left( k^{\prime}+l\right)}\left( 1+\left|k^{\prime}\right|\right)^{-m} \leq(1+|l|)^{-m} \sum\limits_{k^{\prime} \leq-l} r^{-2\left( k^{\prime}+l\right)}=\frac{(1+|l|)^{-m}}{1-r^{2}} \leq \frac{(1+|k|)^{-m}}{1-r^{2}}, $$$$ \sum\limits_{-l< k^{\prime}< k} (1+|k^{\prime}|)^{-m}\leq (k+l) (1+|k|)^{-m}, $$and by Lemma 10
$$ \begin{array}{@{}rcl@{}} \sum\limits_{k^{\prime}\geq k} r^{2(k^{\prime}-k)}(1+|k^{\prime}|)^{-m}&\leq& \sum\limits_{k\leq k^{\prime}\leq 0} r^{2(k^{\prime}-k)}(1-k^{\prime})^{-m}+{}\sum\limits_{k^{\prime}> 0} r^{2(k^{\prime}-k)}(1+k^{\prime})^{-m} \\ &\leq& \sum\limits_{u=0}^{-k} r^{2 u}(1-k-u)^{-m}+\sum\limits_{k^{\prime}>0} r^{2\left( k^{\prime}-k\right)}\\ &\leq& c_{1} (1+|k|)^{-m}, \end{array} $$where c1 depends only on r and m. Substituting the above inequalities into (57) shows that there is a positive constant c2 such that for all \(k, l \in \mathbb {Z}\) with − l ≤ k ≤ 0,
$$ \sum\limits_{k^{\prime}\in\mathbb{Z}} r^{|k-k^{\prime}|+|l+k^{\prime}|}(1+|k^{\prime}|)^{-{m}} \leq c_{2} r^{k+l}(k+l+1) (1+|k|)^{-m}. $$(58) -
(ii)
In the second sub-case, let 0 ≤−l ≤ k. By an argument similar to the proof for (58), we can obtain that there is a positive constant c3 such that for all \(k, l\in \mathbb {Z}\) with 0 ≤−l ≤ k,
$$ \sum\limits_{k^{\prime}\in\mathbb{Z}} r^{|k-k^{\prime}|+|l+k^{\prime}|}(1+|k^{\prime}|)^{-{m}} \leq c_{3} r^{k+l}(k+l+1) (1+|l|)^{-m}. $$(59)Combining (58) and (59), we know that when kl ≤ 0 and − l ≤ k, inequality (55) holds.
When kl > 0, there holds that
$$ \sum\limits_{k^{\prime}\geq l} r^{2(k^{\prime}-l)}(1+|k^{\prime}|)^{-m}\leq \frac{(1+|l|)^{-m}}{1-r^{2}}, $$$$ \sum\limits_{-l< k^{\prime}< k} (1+|k^{\prime}|)^{-m}\leq k+l, $$and
$$ \sum\limits_{k^{\prime}\geq k} r^{2(k^{\prime}-k)}(1+|k^{\prime}|)^{-m}\leq \frac{(1+|k|)^{-m}}{1-r^{2}}. $$Thus, combining the above inequalities with (57) implies that (56) holds when kl > 0 and − l ≤ k.
□
Rights and permissions
About this article
Cite this article
Jiang, Y., Wang, B. & Yu, D. A fast Fourier-Galerkin method solving boundary integral equations for the Helmholtz equation with exponential convergence. Numer Algor 88, 1457–1491 (2021). https://doi.org/10.1007/s11075-021-01082-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-021-01082-0