An upwind difference approximation is used for a singularly perturbed problem in material science. Based on the discrete Green-s function theory, the error estimate in maximum norm is achieved, which is first-order uniformly convergent with respect to the perturbation parameter. The numerical experimental result is verified the valid of the theoretical analysis.<\/p>\r\n","references":"[1] R. B. Kellogg and A. Tsan., Analysis of some difference approximations\r\nfor a singular perturbation problem without turning points, Math. Comp.,\r\n32(1978) 1025-1039.\r\n[2] Y. Chen, Uniform pointwise convergence for a singularly perturbed\r\nproblem using arc-length equidistribution, J. Comp. Appl. Math., 159\r\n(2003) 25-34.\r\n[3] Y. Chen, Uniform convergence analysis of finite difference approximations\r\nfor singularly perturbed problems on an adapted grid, Advances in Comp.\r\nMath., 24 (2006) 197-212.\r\n[4] N. Kopteva and M. Stynes, A robust adaptive method for a quasi-linear\r\none dimensional convection-diffusion problem, SIAM J. Numer. Anal.,\r\n39(2001) 1446-1467.\r\n[5] T. Linss, Uniforming pointwise convergence of finite difference schemes\r\nusing grid equidistribution, Computing, 66(2001) 27-39.\r\n[6] Y. Qiu and D. M. Sloan, Analysis of difference approximations to a\r\nsingularly perturbed two-point boundary value problem on an adaptively\r\ngenerated grid, J. Comp. Appl. Math., 101(1999) 1-25.\r\n[7] Y. Qiu, D. M. Sloan and T. Tang, Numerical solution of a singularly\r\nperturbed two-point boundary value problem using equidistribution:\r\nanalysis of convergence, J. Comp. Appl. Math., 116(2000) 121-143.\r\n[8] V. B. Andreev and I. A. Savin, On the convergence, uniform with respect\r\nto the small parameter of A. A. Samarskii-s monotone scheme and its\r\nmodifications, Comp. Math. Math. Phys., 35(1995) 581-591.\r\n[9] V. B. Andreev and N. Kopteva, On the convergence, uniform with respect\r\nto a small parameter of monotone three-point difference approximation,\r\nDiffer. Uravn., 34(1998) 921-929.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 52, 2011"}