Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem
ABSTRACT
A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.
Cite this paper
S. Chen, W. Hou and X. Jiang, "Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 643-646. doi: 10.4236/apm.2013.37085.
S. Chen, W. Hou and X. Jiang, "Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 643-646. doi: 10.4236/apm.2013.37085.
References
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http://dx.doi.org/10.1007/978-1-4757-4338-8 [3] H. G. Roos, M. Stynes and L. Tobiska, “Numerical Methods for Singularly Perturbed Differential Equations,” Springer, Berlin, 1996.
http://dx.doi.org/10.1007/978-3-662-03206-0 [4] R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem without Turning Points,” Mathematics of Computation, Vol. 32, No. 144, 1978, pp. 1025-1039.
http://dx.doi.org/10.1090/S0025-5718-1978-0483484-9 [5] F. Brezzi, L. D. Marini and A. Russo, “On the Choice of Stabilizing Subgrid for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 2-5, 2005, pp. 127-148.
http://dx.doi.org/10.1016/j.cma.2004.02.022 [6] T. Linβ and N. Madden, “A Finite Element Analysis of Coupled System of Singularly Perturbed Reaction-Diffusion Equations,” Applied Mathematics and Computation, Vol. 148, No. 3, 2004, pp. 869-880.
http://dx.doi.org/10.1016/S0096-3003(02)00955-4 [7] M. K. Kadalbajoo, A. S. Yadaw and D. Kumar, “Comparative Study of Singularly Perturbed Two-Point BVPs via: Fitted-Mesh Finite Difference Method, B-Spline Collocation Method and Finite Element Method,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 713-725. http://dx.doi.org/10.1016/j.amc.2008.07.014 [8] L. P. Franca and E. G. Dutra do Carmo, “The Galerkin Gradient Least Squares Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 74, No. 1, 1989, pp. 41-54.
http://dx.doi.org/10.1016/0045-7825(89)90085-6 [9] F. Ilinca and J.-F. Hétu, “Galerkin Gradient Least-Squares Formulations for Transient Conduction Heat Transfer,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 27-28, 2002, pp. 3073-3097.
http://dx.doi.org/10.1016/S0045-7825(02)00242-6 [10] T. Linβ, “Layer-Adapted Meshes for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 9-10, 2003, pp. 1061-1105. http://dx.doi.org/10.1016/S0045-7825(02)00630-8 [11] M. Stynes, “Steady-State Convection-Diffusion Problems,” Acta Numerica, Vol. 14, 2005, pp. 445-508.
http://dx.doi.org/10.1017/S0962492904000261 [12] M. Stynes and L. Tobiska, “The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy,” SIAM Journal on Numerical Analysis, Vol. 41, No. 5, 2003, pp. 1620-1642. [13] Z. Zhang, “Finite Element Superconvergence Approximation of One Dimensional Singularly Perturbed Problems,” Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, 2002, pp. 374-395.
http://dx.doi.org/10.1002/num.10001 [14] Z. Zhang, “Finite Element Super-Convergence on Shishkin Mesh for 2-d Convection-Diffusion Problems,” Mathematical and Computer Modelling, Vol. 72, No. 243, 2003, pp. 1147-1177.
http://dx.doi.org/10.1090/S0025-5718-03-01486-8
[1] K. W. Chang and F. A. Howes, “Nonlinear Singular Perturbation Phenomena: Theory and Applications,” Spring-Verlag, New York, 1984. [2] S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Methods,” Springer, Berlin, 1994.
http://dx.doi.org/10.1007/978-1-4757-4338-8 [3] H. G. Roos, M. Stynes and L. Tobiska, “Numerical Methods for Singularly Perturbed Differential Equations,” Springer, Berlin, 1996.
http://dx.doi.org/10.1007/978-3-662-03206-0 [4] R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem without Turning Points,” Mathematics of Computation, Vol. 32, No. 144, 1978, pp. 1025-1039.
http://dx.doi.org/10.1090/S0025-5718-1978-0483484-9 [5] F. Brezzi, L. D. Marini and A. Russo, “On the Choice of Stabilizing Subgrid for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 2-5, 2005, pp. 127-148.
http://dx.doi.org/10.1016/j.cma.2004.02.022 [6] T. Linβ and N. Madden, “A Finite Element Analysis of Coupled System of Singularly Perturbed Reaction-Diffusion Equations,” Applied Mathematics and Computation, Vol. 148, No. 3, 2004, pp. 869-880.
http://dx.doi.org/10.1016/S0096-3003(02)00955-4 [7] M. K. Kadalbajoo, A. S. Yadaw and D. Kumar, “Comparative Study of Singularly Perturbed Two-Point BVPs via: Fitted-Mesh Finite Difference Method, B-Spline Collocation Method and Finite Element Method,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 713-725. http://dx.doi.org/10.1016/j.amc.2008.07.014 [8] L. P. Franca and E. G. Dutra do Carmo, “The Galerkin Gradient Least Squares Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 74, No. 1, 1989, pp. 41-54.
http://dx.doi.org/10.1016/0045-7825(89)90085-6 [9] F. Ilinca and J.-F. Hétu, “Galerkin Gradient Least-Squares Formulations for Transient Conduction Heat Transfer,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 27-28, 2002, pp. 3073-3097.
http://dx.doi.org/10.1016/S0045-7825(02)00242-6 [10] T. Linβ, “Layer-Adapted Meshes for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 9-10, 2003, pp. 1061-1105. http://dx.doi.org/10.1016/S0045-7825(02)00630-8 [11] M. Stynes, “Steady-State Convection-Diffusion Problems,” Acta Numerica, Vol. 14, 2005, pp. 445-508.
http://dx.doi.org/10.1017/S0962492904000261 [12] M. Stynes and L. Tobiska, “The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy,” SIAM Journal on Numerical Analysis, Vol. 41, No. 5, 2003, pp. 1620-1642. [13] Z. Zhang, “Finite Element Superconvergence Approximation of One Dimensional Singularly Perturbed Problems,” Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, 2002, pp. 374-395.
http://dx.doi.org/10.1002/num.10001 [14] Z. Zhang, “Finite Element Super-Convergence on Shishkin Mesh for 2-d Convection-Diffusion Problems,” Mathematical and Computer Modelling, Vol. 72, No. 243, 2003, pp. 1147-1177.
http://dx.doi.org/10.1090/S0025-5718-03-01486-8