# zbMATH — the first resource for mathematics

Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers. (English) Zbl 0870.65091
The authors derive necessary conditions for uniform convergence of the finite difference method for solving elliptic boundary value problems with a small diffusion term of the unit square. It is known that these problems have hyperbolic character, which is the main source of numerical difficulties. Exponential and parabolic layers are investigated. The authors prove the non-existence of a five-point scheme, which would be uniformly convergent with respect to the perturbation parameter $$\epsilon$$ and discretization parameter $$h.$$ They also present necessary conditions for coefficients of a nine-point scheme to ensure uniform convergence.

##### MSC:
 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards, 1964. · Zbl 0171.38503 [2] Eckhaus, W.: Matched asymptotic expansions and singular perturbations. North-Holland, Amsterdam, 1973. · Zbl 0255.34002 [3] 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. [4] Guo, W.: Uniformly convergent finite element methods for singularly perturbed parabolic problems. Ph.D. Dissertation, National University of Ireland, 1993. [5] 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. · Zbl 0732.35020 · doi:10.1137/0521022 [6] 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. · Zbl 0102.11701 · doi:10.1002/cpa.3160140324 [7] 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. · Zbl 0338.35006 [8] Roos, H.-G.: Necessary convergence conditions for upwind schemes in the two-dimensional case. Int. J. Numer. Meth. Eng. 21 (1985), 1459-1469. · Zbl 0578.65098 · doi:10.1002/nme.1620210808 [9] Shih, S.D., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18 (1987), 1467-1511. · Zbl 0642.35006 · doi:10.1137/0518107 [10] 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. · Zbl 0709.65073 · doi:10.1016/0041-5553(89)90109-2 [11] Shishkin, G.I.: Grid approximation of singularly perturbed boundary value problems with convective terms. Sov. J. Numer. Anal. Math. Modelling 5 (1990), 173-187. · Zbl 0816.65051 · doi:10.1515/rnam.1990.5.2.173 [12] Shishkin, G.I.: Methods of constructing grid approximations for singularly perturbed boundary value problems. Sov. J. Numer. Anal. Math. Modelling 7 (1992), 537-562. · Zbl 0816.65072 · doi:10.1515/rnam.1992.7.6.537 [13] 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. · Zbl 0822.65077 · doi:10.1016/0898-1221(94)00237-F [14] Yserentant, H.: Die maximale Konsistenzordnung von Differenzapproximationen nichtnegativer Art. Numer. Math. 42 (1983), 119-123. · Zbl 0536.65074 · doi:10.1007/BF01400922 · eudml:132865
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.