Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T21:18:35.810Z Has data issue: false hasContentIssue false
  • English
  • Français

SERIES SOLUTION OF LAPLACE PROBLEMS

Part of: Elliptic equations and systems Two-dimensional theory

Published online by Cambridge University Press:  06 July 2018

LLOYD N. TREFETHEN
LLOYD N. TREFETHEN*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK email trefethen@maths.ox.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

Keywords

Green functionconformal mappingharmonic measureLaplace problemseries expansionleast-squaresCantor set

MSC classification

Primary: 31A99: None of the above, but in this section
Secondary: 35J08: Green's functions
Type
Research Article
Information
The ANZIAM Journal , Volume 60 , Issue 1 , July 2018 , pp. 1 - 26
Copyright
© 2018 Australian Mathematical Society 

References

Amano, K., Okano, D., Ogata, H. and Sugihara, M., “Numerical conformal mappings onto the linear slit domain”, Jpn. J. Ind. Appl. Math. 29 (2012) 165–186; doi:10.1007/s13160-012-0058-0.Google Scholar
Axler, S., “Harmonic functions from a complex analysis viewpoint”, Amer. Math. Monthly 93 (1986) 246–258; doi:10.1080/00029890.1986.11971799.Google Scholar
Austin, A. P., Kravanja, P. and Trefethen, L. N., “Numerical algorithms based on analytic function values at roots of unity”, SIAM J. Numer. Anal. 52 (2014) 1795–1821; doi:10.1137/130931035.Google Scholar
Barnett, A. H. and Betcke, T., “Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains”, J. Comput. Phys. 227 (2008) 7003–7026; doi:10.1016/j.jcp.2008.04.008.Google Scholar
Bergman, S., “Functions satisfying certain partial differential equations of elliptic type and their representation”, Duke Math. J. 14 (1947) 349–366; doi:10.1215/S0012-7094-47-01428-2.Google Scholar
Betcke, T. and Trefethen, L. N., “Reviving the method of particular solutions”, SIAM Rev. 47 (2005) 469–491; doi:10.1137/S0036144503437336.Google Scholar
Betcke, T. and Trefethen, L. N., “Computed eigenmodes of planar regions”, Contemp. Math. 412 (2006) 297–314; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.666.5656&rep=rep1&type=pdf.Google Scholar
Bourot, J. M. and Moreau, F., “Sur l’utilisation de la série cellulaire pour le calcul d’écoulements plans de Stokes en canal indéfini: application au cas d’un cylindre circulaire en translation”, Mech. Res. Commun. 14 (1987) 187–197.Google Scholar
Chapman, S. J., Hewett, D. P. and Trefethen, L. N., “Mathematics of the Faraday cage”, SIAM Rev. 57 (2015) 398–417; doi:10.1137/140984452.Google Scholar
Cheng, H., “On the method of images for systems of closely spaced conducting spheres”, SIAM J. Appl. Math. 61 (2001) 1324–1337; doi:10.1137/S0036139999364992.Google Scholar
Cheng, H. and Greengard, L., “On the numerical evaluation of electrostatic fields in dense random dispersions of cylinders”, J. Comput. Phys. 136 (1997) 629–639; doi:10.1006/jcph.1997.5787.Google Scholar
Cheng, H. and Greengard, L., “A method of images for the evaluation of electrostatic fields in systems of closely spaced conductors”, SIAM J. Appl. Math. 58 (1998) 122–141; doi:10.1137/S0036139996297614.Google Scholar
Crowdy, D. G., “Geometric function theory: a modern view of a classical subject”, Nonlinearity 21 (2008) T205–T219; doi:10.1088/0951-7715/21/10/T04.Google Scholar
Crowdy, D. G., “Conformal slit maps in applied mathematics”, ANZIAM J. 53 (2012) 171–189; doi:10.1017/S1446181112000119.Google Scholar
Crowdy, D. G., Kropf, E. H., Green, C. C. and Nasser, M. M. S., “The Schottky–Klein prime function: a theoretical and computational tool for applications”, IMA J. Appl. Math. 81 (2016) 589–628; doi:10.1093/imamat/hxw028.Google Scholar
Crowdy, D. and Marshall, J., “Conformal mappings between canonical multiply connected domains”, Comput. Methods Funct. Theory 6 (2006) 59–76; doi:10.1007/BF03321118.Google Scholar
Crowdy, D. G. and Marshall, J. S., “Computing the Schottky–Klein prime function on the Schottky double of planar domains”, Comput. Methods Funct. Theory 7 (2007) 293–308; doi:10.1007/BF03321646.Google Scholar
Crowdy, D. G., Tanveer, S. and DeLillo, T., “Hybrid basis scheme for computing electrostatic fields exterior to close-to-touching disks”, IMA J. Numer. Anal. 36 (2016) 743–769; doi:10.1093/imanum/drv030.Google Scholar
Curtiss, J. H., “Interpolation by harmonic polynomials”, J. SIAM 10 (1962) 709–736; doi:10.1137/0110055.Google Scholar
DeLillo, T. K., Driscoll, T. A., Elcrat, A. R. and Pfaltzgraff, J. A., “Radial and circular slit maps of unbounded multiply connected circle domains”, Proc. R. Soc. Lond. A 464 (2008) 1719–1737; doi:10.1098/rspa.2008.0006.Google Scholar
DeLillo, T. K., Elcrat, A. R., Kropf, E. H. and Pfaltzgraff, J. A., “Efficient calculation of Schwarz–Christoffel transformations for multiply connected domains using Laurent series”, Comput. Methods Funct. Theory 13 (2013) 307–336; doi:10.1007/s40315-013-0023-1.Google Scholar
Driscoll, T. A. and Trefethen, L. N., Schwarz–Christoffel mapping (Cambridge University Press, Cambridge, 2002).Google Scholar
Finn, M. D., Cox, S. M. and Byrne, H. M., “Topological chaos in inviscid and viscous mixers”, J. Fluid Mech. 493 (2003) 345–361; doi:10.1017/S0022112003005858.Google Scholar
Fisher, S. D., Function theory on planar domains: a second course in complex analysis (Dover Publ., New York, 2007).Google Scholar
Fox, L., Henrici, P. and Moler, C., “Approximations and bounds for eigenvalues of elliptic operators”, SIAM J. Numer. Anal. 4 (1967) 89–102; doi:10.1137/0704008.Google Scholar
Gaier, D., Lectures on complex approximation (Birkhäuser, Basel, 1987).Google Scholar
Gander, W. and Hřebíček, J., Solving problems in scientific computing using Maple and MATLAB, 4th edn (Springer, Berlin, Heidelberg, 2004).Google Scholar
Garnett, J. B. and Marshall, D. E., Harmonic measure (Cambridge University Press, Cambridge, 2005).Google Scholar
Green, C. C., Snipes, M. A. and Ward, L. A., “Using the Schottky–Klein prime function to compute harmonic measure distribution functions of a class of multiply connected planar domains”, ANZIAM Conference 2018, 8 February 2018; http://www.maths.utas.edu.au/anziam2018/ documents/ANZIAM2018Handbook.pdf.Google Scholar
Green, C. C., Snipes, M. A. and Ward, L. A., “Harmonic measure distribution functions for a class of multiply connected symmetric slit domains”, March 2018, In preparation.Google Scholar
Greengard, L. and Rokhlin, V., “A new version of the fast multipole method for the Laplace equation in three dimensions”, Acta Numer. 6 (1997) 229–269; doi:10.1017/S0962492900002725.Google Scholar
Hackbusch, W., Hierarchical matrices: algorithms and analysis (Springer, Berlin–Heidelberg, 2015).Google Scholar
Helsing, J. and Ojala, R., “On the evaluation of layer potentials close to their sources”, J. Comput. Phys. 227 (2008) 2899–2921; doi:10.1016/j.jcp.2007.11.024.Google Scholar
Henrici, P., “A survey of I. N. Vekua’s theory of elliptic partial differential equations with analytic coefficients”, Z. Angew. Math. Phys. 8 (1957) 169–203; doi:10.1007/BF01600500.Google Scholar
Hockney, R. W., “A solution of Laplace’s equation for a round hole in a square peg”, J. SIAM 12 (1964) 1–14; doi:10.1137/0112001.Google Scholar
Kantorovich, L. V. and Krylov, V. I., Approximate methods of higher analysis (Interscience, New York, 1960).Google Scholar
Karageorghis, A. and Fairweather, G., “The method of fundamental solutions for the numerical solution of the biharmonic equation”, J. Comput. Phys. 69 (1987) 434–459; doi:10.1016/0021-9991(87)90176-8.Google Scholar
Kupradze, V., Potential methods in the theory of elasticity (Israel Program for Scientific Translations, Jerusalem, 1965).Google Scholar
Liesen, J., Sète, O. and Nasser, M., “Fast and accurate computation of the logarithmic capacity of compact sets”, Comput. Methods Funct. Theory 17 (2017) 689–713; doi:10.1007/s40315-017-0207-1.Google Scholar
Malik, N. H., “A review of the charge simulation method and its applications”, IEEE Trans. Elect. Insul. 24.1 (1989) 3–20; doi:10.1109/14.19861.Google Scholar
Moler, C. B., “Accurate bounds for the eigenvalues of the Laplacian and applications to rhombical domains”, Technical Report CS 121, Department of Computer Science, Stanford University, 1969,http://i.stanford.edu/pub/cstr/reports/cs/tr/69/121/CS-TR-69-121.pdf.Google Scholar
Nasser, M. M. S., “Fast solution of boundary integral equations with the generalized Neumann kernel”, Electron. Trans. Numer. Anal. 44 (2015) 189–229; https://www.researchgate.net/publication/256097368_Fast_solution_of_boundary_integral_equations_with_the_generalized_Neumann_kernel.Google Scholar
Nasser, M. M. S. and Green, C. C., “A fast numerical method for ideal fluid flow in domains with multiple stirrers”, Nonlinearity 31 (2018); 815–837; doi:10.1088/1361-6544/aa99a5.Google Scholar
Nasser, M. M. S., Murid, A. H. M., Ismail, M. and Alejaily, E. M. A., “Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions”, Appl. Math. Comput. 217 (2011) 4710–4727; doi:10.1016/j.amc.2010.11.027.Google Scholar
Ogata, H. and Katsurada, M., “Convergence of the invariant scheme of the method of fundamental solutions for two-dimensional potential problems in a Jordan region”, Jpn. J. Ind. Appl. Math. 31 (2014) 231–262; doi:10.1007/s13160-013-0131-3.Google Scholar
Price, T. J., Mullin, T. and Kobine, J. J., “Numerical and experimental characterization of a family of two-roll-mill flows”, Proc. R. Soc. Lond. A 459 (2003) 117–135; doi:10.1098/rspa.2002.1022.Google Scholar
Prosnak, W. J., Computation of fluid motions in multiply connected domains (G. Braun, Karlsruhe, 1987).Google Scholar
Ransford, T., Potential theory in the complex plane (Cambridge University Press, Cambridge, 1995).Google Scholar
Ransford, T. and Rostand, J., “Computation of capacity”, Math. Comp. 76 (2007) 1499–1520; doi:10.1090/S0025-5718-07-01941-2.Google Scholar
Lord Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium”, Phil. Mag. 34 (1892) 481–502; doi:10.1080/14786449208620364.Google Scholar
Rostand, J., “Computing logarithmic capacity with linear programming”, Exp. Math. 6 (1997) 221–238; doi:10.1080/10586458.1997.10504611.Google Scholar
Runge, C., “Zur Theorie der eindeutigen analytischen Functionen”, Acta Math. 6 (1885) 229–244; doi:10.1007/BF02400416.Google Scholar
Runge, C., “Uber empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten”, Z. Math. Phys. 46 (1901) 224–243.Google Scholar
Singer, H., Steinbigler, H. and Weiss, P., “A charge simulation method for the calculation of high voltage fields”, IEEE Trans. Pow. App. Syst. 5 (1974) 1660–1668; doi:10.1109/TPAS.1974.293898.Google Scholar
Trefethen, L. N., “Ten digit algorithms”, Numer. Anal. Rep. 05/13, Oxford U. Computing Lab., 2005, https://protect-au.mimecast.com/s/MwT8CoVzE9HKDkzgsViQE4?domain=people.maths.ox.ac.uk.Google Scholar
Trefethen, L. N., Approximation theory and approximation practice (SIAM, Philadelphia, PA, 2013).Google Scholar
Walsh, J. L., “The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions”, Bull. Amer. Math. Soc. (N.S.) 35 (1929) 499–544; doi:10.1090/S0002-9904-1929-04753-0.Google Scholar
Záviška, F., “Über die Beugung elektromagnetischer Wellen an parallelen, unendlich langen Kreiszylindern”, Ann. Phys. 345 (1913) 1023–1056; doi:10.1002/andp.19133450511.Google Scholar