Abstract
In this work we develop an efficient algorithm for the application of the method of fundamental solutions to inhomogeneous polyharmonic problems, that is problems governed by equations of the form Δℓ u=f, ℓ∈ℕ, in circular geometries. Following the ideas of Alves and Chen (Adv. Comput. Math. 23:125–142, 2005), the right hand side of the equation in question is approximated by a linear combination of fundamental solutions of the Helmholtz equation. A particular solution of the inhomogeneous equation is then easily obtained from this approximation and the resulting homogeneous problem in the method of particular solutions is subsequently solved using the method of fundamental solutions. The fact that both the problem of approximating the right hand side and the homogeneous boundary value problem are performed in a circular geometry, makes it possible to develop efficient matrix decomposition algorithms with fast Fourier transforms for their solution. The efficacy of the method is demonstrated on several test problems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33, 1348–1361 (2009)
Alves, C.J.S., Chen, C.S.: A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23, 125–142 (2005)
Alves, C.J.S., Valtchev, S.S.: A Kansa type method using fundamental solutions applied to elliptic PDEs. In: Leitão, V.M.A., Alves, C.J.S., Armando Duarte, C. (eds.) Advances in Meshfree Techniques, Computational Methods in Applied Sciences, pp. 241–256. Springer, Dordrecht (2007)
Alves, C.J.S., Colaço, M.J., Leitão, V.M.A., Martins, N.F.M., Orlande, H.R.B., Roberty, N.C.: Recovering the source term in a linear diffusion problem by the method of fundamental solutions. Inverse Probl. Sci. Eng. 16, 1005–1021 (2005)
Alves, C.J.S., Martins, N.F.M., Roberty, N.C.: Full identification of acoustic sources with multiple frequencies and boundary measurements. Inverse Probl. Imaging 3, 275–294 (2009)
Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer. Algorithms (2010). doi: 10.1007/s11075-010-9384-y
Cheng, A.H.-D.: Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions. Eng. Anal. Bound. Elem. 24, 531–538 (2000)
Cheng, A.H.-D., Lafe, O., Grilli, S.: Dual-reciprocity BEM based on global interpolation functions. Eng. Anal. Bound. Elem. 13, 303–311 (1994)
Cheng, A.H.-D., Antes, H., Ortner, N.: Fundamental solutions of products of Helmholtz and polyharmonic operators. Eng. Anal. Bound. Elem. 14, 187–191 (1994)
Davis, P.J.: Circulant Matrices. Wiley, New York (1979)
Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998)
Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB. World Scientific, Singapore (2007)
Golberg, M.A., Chen, C.S.: Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Southampton (1997)
Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. In: Golberg, M.A. (ed.) Boundary Integral Methods and Mathematical Aspects, pp. 103–176. WIT Press/Computational Mechanics Publications, Boston (1999)
Golberg, M.A., Muleshkov, A.S., Chen, C.S., Cheng, A.H.-D.: Polynomial particular solutions for certain partial differential operators. Numer. Methods Partial Differ. Equ. 19, 112–133 (2003)
Gorzelańczyk, P., Kołodziej, J.A.: Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods. Eng. Anal. Bound. Elem. 37, 64–75 (2008)
Ingber, M.S., Chen, C.S., Tanski, J.A.: A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations. Int. J. Numer. Methods Eng. 60, 2183–2201 (2004)
Jin, B., Zheng, Y.: Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation. Eng. Anal. Bound. Elem. 29, 925–935 (2005)
Kansa, E.J.: Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics-II. Solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19(8/9), 147–161 (1990)
Karageorghis, A.: Efficient MFS algorithms in regular polygonal domains. Numer. Algorithms 50, 215–240 (2009)
Karageorghis, A.: A practical algorithm for determining the optimal pseudoboundary in the MFS. Adv. Appl. Math. Mech. 1, 510–528 (2009)
Karageorghis, A.: Efficient Kansa-type MFS algorithm for elliptic problems. Numer. Algorithms 54, 261–278 (2010)
Karageorghis, A., Smyrlis, Y.-S.: Matrix decomposition algorithms related to the MFS for axisymmetric problems. In: Manolis, G.D., Polyzos, D. (eds.) Recent Advances in Boundary Element Methods, pp. 223–237. Springer, New York (2009)
Karageorghis, A., Chen, C.S., Smyrlis, Y.-S.: A matrix decomposition RBF algorithm: approximation of functions and their derivatives. Appl. Numer. Math. 57, 304–319 (2007)
Karageorghis, A., Chen, C.S., Smyrlis, Y.-S.: Matrix decomposition RBF algorithm for solving 3D elliptic problems. Eng. Anal. Bound. Elem. 33, 1368–1373 (2009)
Kołodziej, J.A., Zieliński, A.P.: Boundary Collocation Techniques and Their Application in Engineering. WIT Press, Southampton (2009)
Lesnic, D.: On the boundary integral equations for a two-dimensional slowly rotating highly viscous fluid flow. Adv. Appl. Math. Mech. 1, 140–150 (2009)
Lonsdale, B., Bloor, M.I.G., Kelmanson, M.A.: An iterative integral-equation method for 6th-order inhomogeneous partial differential equations. In: Ertekin, R.C., Brebbia, C.A., Tanaka, M., Shaw, R.P. (eds.) Boundary Element Technology XI, pp. 369–378. Computational Mechanics Publications, Southampton (1996)
Marin, L.: The method of fundamental solutions for inverse problems associated with the steady-state heat conduction in the presence of sources. Comput. Model. Eng. Sci. 30, 99–122 (2008)
MATLAB, The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA
Smyrlis, Y.-S., Karageorghis, A.: Some aspects of the method of fundamental solutions for certain harmonic problems. J. Sci. Comput. 16, 341–371 (2001)
Smyrlis, Y.-S., Karageorghis, A.: Some aspects of the method of fundamental solutions for certain biharmonic problems. Comput. Model. Eng. Sci. 4, 535–550 (2003)
Tsai, C.C.: Particular solutions of Chebyshev polynomials for polyharmonic and poly-Helmholtz equations. Comput. Model. Eng. Sci. 27, 151–162 (2008)
Tsai, C.C.: The particular solutions for Thin plates resting on Pasternak foundations under arbitrary loadings. Numer. Methods Partial Diff. Equ. 26, 206–220 (2010)
Tsai, C.C., Chen, C.S., Hsu, T.-W.: The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations. Eng. Anal. Bound. Elem. 33, 1396–1402 (2009)
Ushijima, T., Chiba, F.: Error estimates for a fundamental solution method applied to reduced wave problems in a domain exterior to a disc. J. Comput. Appl. Math. 159, 137–148 (2003)
Valle, M.F., Colaço, M.J., Neto, F.S.: Estimation of the heat transfer coefficient by means of the method of fundamental solutions. Inverse Probl. Sci. Eng. 16, 777–795 (2008)
Valtchev, S.S.: Numerical analysis of methods with fundamental solutions for acoustic and elastic wave propagation problems. PhD Thesis, Department of Mathematics, Instituto Superior Téchnico, Universidade Técnica de Lisboa, Lisbon (2008)
Valtchev, S.S., Roberty, N.C.: A time-marching MFS scheme for heat conduction problems. Eng. Anal. Bound. Elem. 32, 480–493 (2008)
Zhang, S., Jin, J.: Computation of Special Functions. Wiley, New York (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karageorghis, A. Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems. J Sci Comput 46, 519–541 (2011). https://doi.org/10.1007/s10915-010-9418-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-010-9418-6