A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

Abstract

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Notes

  1. 1.

    Note that this choice of \(f\) is by no means unique. In [11, Sec. 6.1], \(C^{\infty }(\mathbb R ) \) bump functions are considered (in a different context) which have certain Gevrey regularity. They also could be used for our partition of unity.

References

  1. 1.

    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Applied Mathematics Series, vol. 55. National Bureau of Standards, U.S. Department of Commerce (1972)

  2. 2.

    Babuška, I., Melenk, J.: The partition of unity method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)

    MATH  Article  Google Scholar 

  3. 3.

    Bamberger, A., Ha Duong, T.: Formulation Variationnelle Espace-Temps pur le Calcul par Potientiel Retardé de la Diffraction d’une Onde Acoustique. Math. Methods Appl. Sci. 8, 405–435 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Banjai, L.: Multistep and multistage convolution quadrature for the wave equation: algorithms and experiments. SIAM J. Sci. Comput. 32(5), 2964–2994 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Banjai, L., Melenk, J., Lubich, C.: Runge–Kutta convolution quadrature for operators arising in wave propagation. Numer. Math. 119(1), 1–20 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Banjai, L., Sauter, S.: Rapid solution of the wave equation in unbounded domains. SIAM J. Numer. Anal. 47, 227–249 (2008)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Banjai, L., Schanz, M.: Wave propagation problems treated with convolution quadrature and BEM. Preprint 60/2010, MPI Leipzig.

  8. 8.

    Birgisson, B., Siebrits, E., Peirce, A.: Elastodynamic direct boundary element methods with enhanced numerical stability properties. Int. J. Numer. Methods Eng. 46, 871–888 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Bluck, M., Walker, S.: Analysis of three dimensional transient acoustic wave propagation using the boundary integral equation method. Int. J. Numer. Methods Eng. 39, 1419–1431 (1996)

    MATH  Article  Google Scholar 

  10. 10.

    Chen, Q., Monk, P., Wang, X., Weile, D.: Analysis of convolution quadrature applied to the time-domain electric field integral equation (2012, submitted)

  11. 11.

    Chernov, A., von Petersdorff, T., Schwab, C.: Exponential convergence of hp quadrature for integral operators with Gevrey kernels. ESAIM Math. Model. Numer. Anal. (M2AN) 45(3), 387–422 (2011)

    Google Scholar 

  12. 12.

    Davies, P., Duncan, D.: Averaging techniques for time-marching schemes for retarded potential integral equations. Appl. Numer. Math. 23, 291–310 (May 1997)

    Google Scholar 

  13. 13.

    Davies, P., Duncan, D.: Numerical stability of collocation schemes for time domain boundary integral equations. In: Carstensen, C. (ed.) Computational Electromagnetics, pp. 51–86. Springer, Berlin (2003)

  14. 14.

    Ding, Y., Forestier, A., Ha Duong, T.: A Galerkin scheme for the time domain integral equation of acoustic scattering from a hard surface. J. Acoust. Soc. Am. 86(4), 1566–1572 (1989)

  15. 15.

    Dodson, S., Walker, S., Bluck, M.: Implicitness and stability of time domain integral equation scattering analysis. ACES J. 13, 291–301 (1997)

    Google Scholar 

  16. 16.

    El Gharib, J.: Problèmes de potentiels retardés pour l’acoustique. PhD thesis, École Polytechnique (1999)

  17. 17.

    Gradshteyn, I.: Table of Integrals, Series, and Products. Academic Press, New York (1965)

    Google Scholar 

  18. 18.

    Ha-Duong, T.: On retarded potential boundary integral equations and their discretisation. In: Topics in Computational Wave Propagation: Direct and Inverse Problems. Lecture Notes in Engineering and Computer Science, vol. 31, pp. 301–336. Springer, Berlin (2003)

  19. 19.

    Ha-Duong, T., Ludwig, B., Terrasse, I.: A Galerkin BEM for transient acoustic scattering by an absorbing obstacle. Int. J. Numer. Methods Eng. 57, 1845–1882 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Hackbusch, W., Kress, W., Sauter, S.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation by cutoff and panel-clustering. In: Schanz, M., Steinbach, O. (eds.) Boundary Element Analysis, pp. 113–134. Springer, Berlin (2007)

    Google Scholar 

  21. 21.

    Hackbusch, W., Kress, W., Sauter, S.: Sparse convolution quadrature for time domain boundary integral formulations of the wave equation. IMA J. Numer. Anal. 29, 158–179 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Nédélec, J.C., Abboud, T., Volakis, J.: Stable solution of the retarded potential equations. In: Applied Computational Electromagnetics Society (ACES) Symposium Digest, 17th Annual Review of Progress, Monterey (2001)

  23. 23.

    Rynne, B., Smith, P.: Stability of time marching algorithms for the electric field integral equation. J. Electromagn. Waves Appl. 4, 1181–1205 (1990)

    Article  Google Scholar 

  24. 24.

    Sauter, S., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics. Springer, Berlin (2010)

    Google Scholar 

  25. 25.

    Sauter, S., Veit, A.: A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions. Part II: implementation and reference solutions. Preprint 03–2011, Universität Zürich

  26. 26.

    Stephan, E., Maischak, M., Ostermann, E.: Transient boundary element method and numerical evaluation of retarded potentials. In: Computational Science—ICCS 2008, vol. 5102, pp. 321–330. Springer, Berlin (2008)

  27. 27.

    Todorov, P.: New explicit formulas for the \(n\)th derivative of composite functions. Pac. J. Math. 92(1), 217–236 (1981)

    MATH  Article  Google Scholar 

  28. 28.

    Trefethen, L.: Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50, 67–87 (February 2008)

  29. 29.

    Veit, A.: A MATLAB code for computing exact solutions of retarded potential equations for a spherical scatterer (2011). https://www.math.uzh.ch/compmath/?exactsolutions

  30. 30.

    Wang, X., Wildman, R., Weile, D., Monk, P.: A finite difference delay modeling approach to the discretization of the time domain integral equations of electromagnetics. IEEE Trans. Antennas Propag. 56(8), 2442–2452 (2008)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Weile, D., Ergin, A., Shanker, B., Michielssen, E.: An accurate discretization scheme for the numerical solution of time domain integral equations. IEEE Antennas Propag. Soc. Int. Symp. 2, 741–744 (2000)

    Google Scholar 

  32. 32.

    Weile, D., Pisharody, G., Chen, N., Shanker, B., Michielssen, E.: A novel scheme for the solution of the time-domain integral equations of electromagnetics. IEEE Trans. Antennas Propag. 52, 283–295 (2004)

    Article  Google Scholar 

  33. 33.

    Weile, D., Shanker, B., Michielssen, E.: An accurate scheme for the numerical solution of the time domain electric field integral equation. IEEE Antennas Propag. Soc. Int. Symp. 4, 516–519 (2001)

    Google Scholar 

  34. 34.

    Wildman, A., Pisharody, G., Weile, D., Balasubramaniam, S., Michielssen, E.: An accurate scheme for the solution of the time-domain integral equations of electromagnetics using higher order vector bases and bandlimited extrapolation. IEEE Trans. Antennas Propag. 52, 2973–2984 (2004)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgments

Thanks are due to Christoph Schwab for fruitful discussions concerning the use of the PUM for the time discretization.

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. Sauter.

Additional information

A. Veit gratefully acknowledges the support given by SNF, No. PDFMP2_127437/1.

Appendix A: Technical estimates

Appendix A: Technical estimates

In this section we want to estimate the \(n\)th derivative of the function \(f\) as defined in (3.5). Therefore let

$$\begin{aligned} h( z) :=\text{ erf}( z) \quad \text{ and} \quad g( x) :=\text{ arctanh}x=\frac{1}{2}\ln \frac{1+x}{1-x} \end{aligned}$$

such that \(f:=h\circ 2g\). Note that [1, (7.1.19)] implies

$$\begin{aligned} h^{( n+1) }( z) =( -1) ^{n}\frac{2}{\sqrt{\pi }}H_{n}( z) \text{ e}^{{-z^{2}}}\quad n=0,1,2, \ldots \end{aligned}$$

where \(H_{n}\) are the Hermite polynomials. Hence,

$$\begin{aligned} f^{( n+1) }( x)&= \left( \frac{d}{dx}\right) ^{n}\left( \frac{4}{\sqrt{\pi }( 1-x^{2}) }\text{ e}^{{-4g^{2}}( x) }\right) \nonumber \\&= \frac{4}{\sqrt{\pi }}\sum _{\ell =0}^{n}\genfrac(){0.0pt}1{n}{\ell }\left( \frac{1}{1-x^{2}}\right) ^{( \ell ) }(\text{ e}^{{-4g^{2}}( x) }) ^{( n-\ell ) }. \end{aligned}$$
(8.1)

IndexTerm 

Lemma 8.1

(Derivatives of \(g\)) It holds

$$\begin{aligned} \left( \frac{1}{1-x^{2}}\right) ^{( \ell ) }=\frac{\ell !p_{\ell }( x) }{( 1-x^{2}) ^{\ell +1}}\quad \forall x\in ( -1,1), \end{aligned}$$

where

$$\begin{aligned} p_{\ell }( x) :=\frac{( x+1) ^{\ell +1}-( x-1) ^{\ell +1}}{2}. \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \left|\, g^{( \ell ) }( x) \right|\le \left\{ \begin{array}{cc} \frac{1}{2}\ln \dfrac{4}{1-x^{2}}&\ell =0\\&\\ \dfrac{( \ell -1) !2^{\ell -1}}{( 1-x^{2}) ^{\ell }}&\ell \in \mathbb N _{\ge 1} \end{array} \right. \quad \forall x\in ( -1,1), \end{aligned}$$
(8.2)

as well as the more generous estimate

$$\begin{aligned} \vert \, g^{( \ell ) }( x) \vert \le q( x) \frac{\ell !2^{\ell -1}}{( 1-x^{2}) ^{\ell } }\quad \forall \ell \in \mathbb N _{0} \end{aligned}$$
(8.3)

with \(q( x) = \ln \frac{4}{1-x^{2}}\).

IndexTerm 

Lemma 8.2

(Derivative of composite functions) For \(n\ge 1\) and \(x\in ( -1,1) \) we have

$$\begin{aligned} ( \text{ e}^{{-4g^{2}}( x) }) ^{( n) }=\text{ e}^{{-4g^{2}}( x) }\sum _{k=1}^{n}A_{n,k}( x) ( -1) ^{k}H_{k}( 2g( x) ), \end{aligned}$$
(8.4a)

where

$$\begin{aligned} A_{n,k}(x) =\frac{2^k}{k!}\sum _{\nu =1}^{k}(-1) ^{k-\nu }\genfrac(){0.0pt}1{k}{\nu }g^{k-\nu }(x) (g^{\nu }) ^{(n) }(x) \end{aligned}$$
(8.4b)

and

$$\begin{aligned} ( g^{\nu }) ^{( n) }&= \sum _{\ell _{\nu -1}=0}^{n} \sum _{\ell _{\nu -2}=0}^{\ell _{\nu -1}}\cdots \sum _{\ell _{1}=0}^{\ell _{2}}\genfrac(){0.0pt}1{n}{\ell _{\nu -1}}\genfrac(){0.0pt}1{\ell _{\nu -1}}{\ell _{n-2}}\cdots \genfrac(){0.0pt}1{\ell _{2}}{\ell _{1}}g^{( n-\ell _{\nu -1}) }g^{( \ell _{\nu -1}-\ell _{\nu -2}) }\nonumber \\&\cdots g^{( \ell _{2}-\ell _{1}) }g^{( \ell _{1}) }. \end{aligned}$$
(8.5)

IndexTerm 

Proof

The representation (8.4a) and (8.4b) follows from [27, formulae (2), (7)], while (8.5) is proved by induction using Leibniz’ product rule for differentiation. \(\square \)

IndexTerm 

Lemma 8.3

(Estimate of derivatives of composite functions) For \(n\ge 1\) and \(x\in ( -1,1) \) we have

$$\begin{aligned} \left|\left( \text{ e}^{{-g^{2}}( x) }\right) ^{\left( n\right) }\right|\le \frac{5}{2}\kappa n!\text{ e}^{{-2g^{2}}\left( x\right)}\left( \frac{C_1 q(x) }{1-x^{2}}\right) ^{n} \end{aligned}$$
(8.6)

with \(\kappa \approx 1.086435\) and \(C_{1}=6\sqrt{2}\text{ e}\).

IndexTerm 

Proof

From (8.3) and (8.5) we conclude for all \(n\ge 1\), \(\nu \ge 1\), and \(x\in (-1,1) \)

$$\begin{aligned} \left|( g^{\nu }) ^{( n) }( x) \right|&\le n!2^{n-\nu }\frac{q^{\nu }( x) }{( 1-x^{2}) ^{n}}\sum _{\ell _{\nu -1}=0}^{n}\sum _{\ell _{\nu -2}=0}^{\ell _{\nu -1}}\cdots \sum _{\ell _{1}=0}^{\ell _{2}}1\nonumber \\&= n!2^{n-\nu }\frac{q^{\nu }(x) }{(1-x^{2}) ^{n} }\genfrac(){0.0pt}{}{n+\nu -1}{\nu -1}. \end{aligned}$$
(8.7)

Thus, from (8.4b) we get that

$$\begin{aligned} \vert A_{n,k}(x) \vert&\le \frac{2^k n!}{k!} \frac{q^{k}(x) }{(1-x^{2}) ^{n}}\sum _{\nu =1} ^{k}\genfrac(){0.0pt}1{k}{\nu }2^{n-\nu }\genfrac(){0.0pt}{}{n+\nu -1}{\nu -1}\nonumber \\&\le \frac{2^{n}n!}{k!}\frac{q^{k}(x) }{(1-x^{2}) ^{n}}\left( \frac{n+k}{k}\right) ^{k}\sum _{\nu =1}^{k} \genfrac(){0.0pt}1{k}{\nu }2^{k-\nu }\nonumber \\&\le \frac{2^{n}n!}{k!}\frac{1}{(1-x^{2}) ^{n}}\left( \frac{3(n+k) q(x) }{k}\right) ^{k}. \end{aligned}$$
(8.8)

From [1, (22.14.17)] we obtain

$$\begin{aligned} H_{k}(2 g(x) ) \le \text{ e}^{{2g^{2}} ( x)}\kappa 2^{k/2}\sqrt{k!}. \end{aligned}$$

The combination of (8.4a) and (8.4b), (8.5), (8.7) and (8.8) results in the estimate for the \(n\)th derivative of \(\text{ e}^{-4g^{2}(x) }\):

$$\begin{aligned} \left|\left( \text{ e}^{{-4g^{2}}(x) }\right) ^{(n) }\right|&\le \kappa 2^{n}n!\frac{\text{ e}^{{-2g^{2}}(x)}}{( 1-x^{2}) ^{n}}\sum _{k=1}^{n}\frac{1}{\sqrt{k!}}\left( \frac{3\sqrt{2} (n+k) q(x) }{k}\right) ^{k}\\&\le \kappa n!\text{ e}^{{-2g^{2}}(x)}\left( \frac{6\sqrt{2}q(x) }{1-x^{2}}\right) ^{n}\sum _{k=1}^{n} \frac{1}{\sqrt{k!}}\left( \frac{n+k}{k}\right) ^{k} \\ \nonumber&\le \kappa n!\text{ e}^{{-2g^{2}}(x)}\left( \frac{6\sqrt{2}\text{ e} q(x) }{1-x^{2}}\right) ^{n}\sum _{k=1}^{n} \frac{1}{\sqrt{k!}} \\&\le \frac{5}{2}\kappa n!\text{ e}^{{-2g^{2}}(x)}\left( \frac{6\sqrt{2}\text{ e} q(x) }{1-x^{2}}\right) ^{n}. \end{aligned}$$

\(\square \)

IndexTerm 

Theorem 8.4

(Estimate of \(n\)th derivative of \(f\)) We have

$$\begin{aligned} |f^{(n+1) }(x)| \le C_{2} C_1^n n! \frac{q(x)^n}{(1-x^2)^{n+1}}\text{ e}^{-2g^2(x)} \end{aligned}$$

with \(C_{2}=\frac{10\kappa }{\sqrt{\pi }} \frac{C_1\ln (4)}{C_1\ln (4)-2}\).

IndexTerm 

Proof

From (8.1), (8.2) and (8.6) we get

$$\begin{aligned} |f^{(n+1) }(x)|&\le \frac{10\kappa }{\sqrt{\pi }} \sum _{l=0}^{n}\genfrac(){0.0pt}{}{n}{l}\frac{l! 2^l}{(1-x^2)^{l+1}} (n-l)! \left( \frac{C_1 q(x)}{1-x^2} \right)^{n-l}\text{ e}^{-2g^2(x)}\\&\le \frac{10\kappa }{\sqrt{\pi }} C_1^n n! \frac{q(x)^n}{(1-x^2)^{n+1}}\text{ e}^{-2g^2(x)} \sum _{l=0}^{n} \left(\frac{ 2}{C_1 q(x)} \right)^{l}\\&\le \frac{10\kappa }{\sqrt{\pi }} \frac{C_1\ln (4)}{C_1\ln (4)-2} C_1^n n! \frac{q(x)^n}{(1-x^2)^{n+1}}\text{ e}^{-2g^2(x)}, \end{aligned}$$

which leads to the desired result. \(\square \)

IndexTerm 

Lemma 8.5

For \(x\in (-1,1)\) and \(\alpha \ge 2\), we have

$$\begin{aligned} \left\Vert \frac{\text{ e}^{{-2g^{2}}(x)}}{(1-x^{2}) ^{\alpha }} \right\Vert_\infty \le \text{ e}^{\sigma _\alpha } \end{aligned}$$

with

$$\begin{aligned} \sigma _\alpha :=\frac{1}{4}\alpha ^2+\frac{1}{2} -\ln \left(\frac{1}{2}\alpha +\frac{1}{2}\sqrt{\alpha ^2-4}\right) . \end{aligned}$$

IndexTerm 

Proof

We set

$$\begin{aligned} \frac{\text{ e}^{{-2g^{2}}(x)}}{(1-x^{2}) ^{\alpha }} = \text{ e}^{s_n(x)}, \end{aligned}$$

where

$$\begin{aligned} s_n(x) := -2\text{ arctanh}(x)^2 - \alpha \text{ ln}(1-x^2). \end{aligned}$$

With the definition of \(\text{ arctanh}(x)\) we get

$$\begin{aligned} s_n(x)&= -2\left[ \frac{1}{2}\ln (1+x)-\frac{1}{2}\ln (1-x) \right]^2 - \alpha \ln (1-x) - \alpha \ln (1+x) \\&= -\frac{1}{2}[\ln (1+x)]^2 + \ln (1+x)\ln (1-x)-\frac{1}{2}[\ln (1-x)]^2 \\&-\, \alpha \ln (1-x) - \alpha \ln (1+x). \end{aligned}$$

Since \(s_n(x)\) is symmetric we assume \(0\le x < 1\) and get

$$\begin{aligned} s_n(x)&\le -\frac{1}{2}[\ln (1-x)]^2 - \alpha \text{ ln}(1-x) + \text{ ln}(1+x)\text{ ln}(1-x) =:\tilde{s}_n(x) . \end{aligned}$$

\(\tilde{s}_n(x)\) is strictly increasing in the interval \([0,0.5]\) for arbitrary \(\alpha \in \mathbb R _{\ge 2}\). Therefore we may restrict to find an upper bound for \(\tilde{s}_n(x)\) in the interval \([0.5,1[\). With the inequality \(\ln (1+x)\ln (1-x)\le -\ln (-\ln (1-x))\) we get

$$\begin{aligned} \tilde{s}_n(x)&\le -\frac{1}{2}[\ln (1-x)]^2 - \alpha \text{ ln}(1-x) -\ln (-\ln (1-x)) =:\hat{s}_n(x) \end{aligned}$$

in \([0.5,1[\). The derivative of \(\hat{s}_n(x) \) is given by

$$\begin{aligned} \hat{s}_n^\prime (x)=\frac{[\ln (1-x)]^2+\alpha \ln (1-x)+1}{(1-x)\ln (1-x)} \end{aligned}$$

which has the root

$$\begin{aligned} x_0 = 1-\text{ e}^{-\theta _\alpha }, \end{aligned}$$

where \(\theta _\alpha := \frac{1}{2}\alpha +\frac{1}{2}\sqrt{\alpha ^2-4}\). Inserting this above shows that

$$\begin{aligned} s_n(x)\le \alpha \theta _\alpha - \frac{1}{2} \theta ^2_\alpha -\ln {\theta _\alpha } \end{aligned}$$

which leads to the desired result after some straightforward manipulations. \(\square \)

IndexTerm 

Lemma 8.6

It holds

$$\begin{aligned} \int _{-1}^1 \left(\ln \frac{4}{1-t^2}\right)^n dt \le 16 n! \end{aligned}$$

for \(n\in \mathbb N \).

IndexTerm 

Proof

We first note that

$$\begin{aligned}&\int _{-1}^1|\ln (1-t)|^i |\ln (1+t)|^{k-i} dt \\&\quad = \int _{-1}^0 |\ln (1-x)|^i |\ln (1+t)|^{k-i} dt + \int _{0}^1 |\ln (1-x)|^i |\ln (1+t)|^{k-i} dt\\&\quad \le (\ln 2)^i\int _{-1}^0 |\ln (1+t)|^{k-i} dt + (\ln 2)^{k-i}\int _{0}^1 |\ln (1-t)|^i dt\\&\quad = (\ln 2)^i\int _{0}^1 |\ln (t)|^{k-i} dt + (\ln 2)^{k-i}\int _{0}^1 |\ln (t)|^i dt\\&\quad = (\ln 2)^i (k-i)!+ (\ln 2)^{k-i}i!, \end{aligned}$$

where we used [17, (2.711)] in the last step. With these computations we get

$$\begin{aligned} \int _{-1}^1 \left|\left( \ln \frac{4}{1-t^2}\right)^n \right|dt&\le \sum _{k=0}^n \genfrac(){0.0pt}{}{n}{k}\int _{-1}^1 |\ln (1-t^2)|^k(\ln 4)^{n-k} dt\\&\le \sum _{k=0}^n \sum _{i=0}^k \genfrac(){0.0pt}{}{n}{k} \genfrac(){0.0pt}{}{k}{i}(\ln 4)^{n-k} \int _{-1}^1 |\ln (1-t)|^i |\ln (1+t)|^{k-i} dt\\&\le \sum _{k=0}^n \sum _{i=0}^k \genfrac(){0.0pt}{}{n}{k} \genfrac(){0.0pt}{}{k}{i}(\ln 4)^{n-k}( (\ln 2)^i (k-i)!+ (\ln 2)^{k-i}i!)\\&\le \sum _{k=0}^n \genfrac(){0.0pt}{}{n}{k}(\ln 4)^{n-k}\left(k!\sum _{i=0}^k \frac{(\ln 2)^i}{i!} +k!\sum _{i=0}^k \frac{(\ln 2)^{k-i}}{(k-i)!}\right)\\&\le 4\sum _{k=0}^n \genfrac(){0.0pt}{}{n}{k}(\ln 4)^{n-k}k!\\&\le 4 n! \sum _{k=0}^n \frac{(\ln 4)^{n-k}}{(n-k)!} \le 16 n!. \end{aligned}$$

\(\square \)

IndexTerm 

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sauter, S., Veit, A. A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions. Numer. Math. 123, 145–176 (2013). https://doi.org/10.1007/s00211-012-0483-7

Download citation

Mathematics Subject Classification

  • 35L05
  • 65N38
  • 65R20