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A numerical scheme based on
mean value solutions for
the helmholtz equation
on triangular grids


Authors: M. G. Andrade and J. B. R. do Val
Journal: Math. Comp. 66 (1997), 477-493
MSC (1991): Primary 35A40, 65N06; Secondary 35J25, 65N15, 65N22
MathSciNet review: 1401937
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Abstract | References | Similar Articles | Additional Information

Abstract: A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding $O(h^{4})$ accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.


References [Enhancements On Off] (What's this?)

  • 1. Garrett Birkhoff and Surender Gulati, Optimal few-point discretizations of linear source problems, SIAM J. Numer. Anal. 11 (1974), 700–728. MR 0362933
  • 2. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962. MR 0140802
  • 3. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers] New York-London, 1975. Computer Science and Applied Mathematics. MR 0448814
  • 4. E. C. Gartland Jr., Discrete weighted mean approximation of a model convection-diffusion equation, SIAM J. Sci. Statist. Comput. 3 (1982), no. 4, 460–472. MR 677099, 10.1137/0903030
  • 5. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • 6. G. H. Golub and C. F. V. Loan, Matrix computations, The John Hopkins Univ. Press, Baltimore, 1984.
  • 7. Jan Górowski, On some properties of the solution of the Dirichlet problem for the Helmholtz equation in the interior and exterior of a circle, Demonstratio Math. 19 (1986), no. 2, 303–315. MR 895003
  • 8. Murli M. Gupta, Ram P. Manohar, and John W. Stephenson, A single cell high order scheme for the convection-diffusion equation with variable coefficients, Internat. J. Numer. Methods Fluids 4 (1984), no. 7, 641–651. MR 754894, 10.1002/fld.1650040704
  • 9. Murli M. Gupta, Ram P. Manohar, and John W. Stephenson, High-order difference schemes for two-dimensional elliptic equations, Numer. Methods Partial Differential Equations 1 (1985), no. 1, 71–80. MR 868052, 10.1002/num.1690010108
  • 10. L. V. Kantorovich and V. I. Krylov, Approximate methods of higher analysis, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York; P. Noordhoff Ltd., Groningen, 1958. MR 0106537
  • 11. R. P. Manohar and J. W. Stephenson, Single cell high order difference methods for the Helmholtz equation, J. Comput. Phys. 51 (1983), 444-453.
  • 12. Tarek P. Mathew, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems. I. Algorithms and numerical results, Numer. Math. 65 (1993), no. 4, 445–468. MR 1231895, 10.1007/BF01385762
    Tarek P. Mathew, Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems. II. Convergence theory, Numer. Math. 65 (1993), no. 4, 469–492. MR 1231896, 10.1007/BF01385763
  • 13. G. D. Stubley, G. D. Raithby and A. B. Strong, Proposal for a new discrete method based on an assessment of discretization errors, Num. Heat Transfer 3 (1980), 411-428.
  • 14. J. B. R. do Val and M. G. Andrade Fo., On the numerical solution of the Dirichlet problem for Helmholtz equation, Applied Math. Letters 9 (1996), 85-89.

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Additional Information

M. G. Andrade
Affiliation: Depto. de Ciencias de Computacao e Estatistica, Instituto de Ciencias Matematica de Sao Carlos, Universidade de Sao Paulo, C.P. 668 - Sao Carlos - SP, 13.560-970 - Brasil
Email: Marinho@icmsc.usp.br

J. B. R. do Val
Affiliation: Depto. de Telemática, Fac. de Eng. Elétrica, Universidade Estadual de Campinas - UNICAMP, C.P. 6101, 13081-970 - Campinas - SP, Brasil
Email: jbosco@dt.fee.unicamp.br

DOI: https://doi.org/10.1090/S0025-5718-97-00825-9
Keywords: Numerical solutions for partial differential equations, elliptic differential equations, Helmholtz equations, non-standard difference approximation, convection-diffusion equations
Received by editor(s): July 31, 1995
Additional Notes: This work was partially supported by CNPq, Conselho Nacional de Desenvolvimento Científico e Tecnológico, grant number 300573/95-2(NV) and 300721/86-2(NV)
Article copyright: © Copyright 1997 American Mathematical Society