Previous |  Up |  Next

Article

Keywords:
numerical analysis; convection-diffusion problems; boundary layers; uniform convergence
Summary:
Singularly perturbed problems of convection-diffusion type cannot be solved numerically in a completely satisfactory manner by standard numerical methods. This indicates the need for robust or $\epsilon $-uniform methods. In this paper we derive new conditions for such schemes with special emphasize to parabolic layers.
References:
[AS64] Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards, 1964.
[Ec73] Eckhaus, W.: Matched asymptotic expansions and singular perturbations. North-Holland, Amsterdam, 1973. MR 0670800 | Zbl 0255.34002
[Em73] Emel’janov, K.V.: A difference scheme for a three-dimensional elliptic equation with a small parameter multiplying the highest derivative. Boundary value problems for equations of mathematical physics, USSR Academy of Sciences, Ural Scientific Centre, 1973, pp. 30–42. (Russian)
[Gu93] Guo, W.: Uniformly convergent finite element methods for singularly perturbed parabolic problems. Ph.D. Dissertation, National University of Ireland, 1993.
[HK90] Han, H., Kellogg, R.B.: Differentiability properties of solutions of the equation $-\epsilon ^2\Delta u+ru=f(x,y)$ in a square. SIAM J. Math. Anal., 21 (1990), 394–408. DOI 10.1137/0521022 | MR 1038899
[La61] Lax, P.D.: On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients. Comm. Pure Appl. Math. 14 (1961), 497–520. DOI 10.1002/cpa.3160140324 | MR 0145686 | Zbl 0102.11701
[Le76] Lelikova, E.F.: On the asymptotic solution of an elliptic equation of the second order with a small parameter effecting the highest derivative. Differential Equations 12 (1976), 1852–1865. (Russian) MR 0445100
[Ro85] Roos, H.-G.: Necessary convergence conditions for upwind schemes in the two-dimensional case. Int. J. Numer. Meth. Eng. 21 (1985), 1459–1469. DOI 10.1002/nme.1620210808 | MR 0799066 | Zbl 0578.65098
[SK87] Shih, S.D., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18 (1987), 1467–1511. DOI 10.1137/0518107 | MR 0902346
[Sh89] Shishkin, G.I.: Approximation of the solutions of singularly perturbed boundary-value problems with a parabolic boundary layer. U.S.S.R. Comput. Maths. Math. Physics 29 (1989), 1–10. DOI 10.1016/0041-5553(89)90109-2 | MR 1011021 | Zbl 0709.65073
[Si90] Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems with convective terms. Sov. J. Numer. Anal. Math. Modelling 5 (1990), 173–187. DOI 10.1515/rnam.1990.5.2.173 | MR 1122367 | Zbl 0816.65051
[Si92] Shishkin, G.I.: Methods of constructing grid approximations for singularly perturbed boundary value problems. Sov. J. Numer. Anal. Math. Modelling 7 (1992), 537–562. MR 1202653 | Zbl 0816.65072
[ST92] Stynes, M., Tobiska, L.: Necessary $L_2$-uniform conditions for difference schemes for two-dimensional convection-diffusion problems. Computers Math. Applic. 29 (1995), 45–53. DOI 10.1016/0898-1221(94)00237-F | MR 1321058
[Ys83] Yserentant, H.: Die maximale Konsistenzordnung von Differenzapproximationen nichtnegativer Art. Numer. Math. 42 (1983), 119–123. DOI 10.1007/BF01400922 | MR 0716478
Partner of
EuDML logo