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Convergence and Error of Some Numerical Methods for Solving a Convection-Diffusion Problem

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DOI: 10.4236/am.2013.45A009    3,265 Downloads   4,983 Views  


We use the local Fourier analysis to determine the properties of the multigrid method when used in modeling the skin penetration of a drug. The analyses of these properties can be very in designing an efficient structure of the multigrid method and in comparing the element and finite difference discretization techniques. After the theoretical results obtained, we also present some numerical results for a problem for which the solution is known.

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The authors declare no conflicts of interest.

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G. Nut, I. Chiorean and P. Blaga, "Convergence and Error of Some Numerical Methods for Solving a Convection-Diffusion Problem," Applied Mathematics, Vol. 4 No. 5A, 2013, pp. 72-79. doi: 10.4236/am.2013.45A009.


[1] K. W. Morton and D. F. Mayers, “Numerical Solution of Partial Differential Equations, An Introduction,” Cam bridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511812248
[2] E. Becker, G. Carey and J. Oden, “Finite Elements. An Introduction,” Prentice-Hall, Englewood Cliffs, 1981.
[3] H. C. Elman, D. J. Silvester and A. J. Wathen, “Finite Elements and Fast Iterative Solvers,” Oxford University Press, Oxford, 2005.
[4] R. P. Fedorenko, “A Relaxation Method for Solving Elliptic Difference Equations,” USSR Computational Mathematics and Mathematical Physics, Vol. 1, 1962, pp. 1092-1096.
[5] R. P. Fedorenko, “The Speed of Convergence of One Iterative Process,” USSR Computational Mathematics and Mathematical Physics, Vol. 4, 1964, pp. 227-235.
[6] A. Brandt, “Multilevel Adaptive Solutions to Boundary Value Problems,” Mathematics of Computation, Vol. 31, 1977, pp. 333-390. doi:10.1090/S0025-5718-1977-0431719-X
[7] A. Brandt, “Multigrid Techniques: 1984 Guide with Ap plications to Fluid Dynamics,” GMD-Studien Nr. 85, Gesellschaft für Matematik und Datenverarbeitung, St. Augustin, Bonn, 1984.
[8] W. Hackbush, “Elliptic Differential Equations,” Springer Verlag, New York, 1992. doi:10.1007/978-3-642-11490-8
[9] U. Trottenberg, C. Oosterlee and A. Schuller, “Multigrid,” Elsevier Academic Press, London, 2001.
[10] P. Weseling, “An Introduction to Multigrid Method,” John Wiley & Sons, New York, 1991.
[11] M. A. Olshanski and A. Reusken, “On a Robust Multigrid Method for Connection-Diffusion Finite Element Problems.” molshan/ftp/pub/proceed_cd.pdf
[12] A. Reusken, “Convergence Analysis of a Multigrid Method for Convection-Diffusion Equations,” Numerische Mathematik, Vol. 91, No. 2, 2002, pp. 323-349. doi:10.1007/s002110100312
[13] R. Wienands and W. Joppich, “Practical Fourier Analysis for Multigrid Methods,” Chapman& Hall/CRC Press, Bo ca Raton, 2005.
[14] R. W. Lewis, P. Nithiarasu and K. N. Seetharamu, “Fundamentals of the Finite Element Method for Heat and Fluid Flow,” John Wiley & Sons Ltd, The Atrium, 2004. doi:10.1002/0470014164
[15] W. L. Briggs, V. E. Henson and S. McCormick, “A Multigrid Tutorial,”2nd Edition, Siam, Philadelphia, 2000. doi:10.1137/1.9780898719505
[16] D. Neumann, “Modeling Transdermal Absorption,” Biotechnology: Pharmaceutical Aspects, Springer, New York, 2008.
[17] B. Al-Qallaf, D. Bhusan Das, D. Mori and Z. Cui, “Modelling Transdermal Delivery of High Molecular Weight Drugs from Microneedle Systems,” Philosophical Trans actions of the Royal Society A, Vol. 365, 2007, pp. 2951-2967. doi:10.1098/rsta.2007.0003

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