A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation

SIAM Journal on Numerical Analysis

Volumn 33, Issue 3, 1996, Pages 1135-1149

  Farrell, Paul A ^a   Miller, John J H ^b   O'Riordan, Eugene ^c   Shishkin, Grigori I ^d  

^a KENT STATE UNIVERSITY   (United States)

^b TRINITY COLLEGE DUBLIN   (Ireland)

^c DUBLIN CITY UNIVERSITY   (Ireland)

^d INSTITUTE OF MATHEMATICS AND MECHANICS   (Russian Federation)

Author keywords

Finite difference scheme; Piecewise uniform mesh; Semilinear boundary value problem; Singular perturbation

Indexed keywords

EID: 0001372974     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/0733056     Document Type: Article

References (25)

1. A. BRANDT AND I. YAVNEH, Inadequacy of first-order upwind difference schemes for some recirculating flows, J. Comput. Phys., 93 (1991), pp. 128-143.
2. E. P. DOOLAN, J. J. H. MILLER, AND W. H. A. SCHILDERS, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, Ireland, 1980.
3. T. M. EL-MISTIKAWY AND M. J. WERLE, Numerical method for boundary layers with blowing - the exponential box scheme, AIAA J., 16 (1978), pp. 749-751.
4. P. A. FARRELL AND E. C. GARTLAND, JR., On the Scharfetter-Gummel discretization for drift-diffusion continuity equations, in Computational Methods for Boundary and Interior Layers in Several Dimensions, J. J. H. Miller, ed., Boole Press, Dublin, Ireland, 1991, pp. 51-79.
5. P. A. FARRELL AND A. HEGARTY, On the determination of the order of uniform convergence, in Proc. of 13th IMACS World Congress, Dublin, Ireland, 1991, pp. 501-502.
6. P. A. FARRELL, J. J. H. MILLER, E. O'RIORDAN, AND G. I. SHISHKIN, On the nonexistence of ε-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems, to appear.
7. K. W. CHANG AND F. A. HOWES, Nonlinear Singular Perturbation Phenomena: Theory and Application, Appl. Math. Sciences 56, Springer-Verlag, Berlin, New York, 1984.
8. A. M. IL'IN, Difference scheme for a differential equation with a small parameter affecting the highest derivative, Mat. Zametki, 6 (1969), pp. 237-248.
9. O. A. LADYZENSKAYA AND N. N. URALCEVA, Linear and Quasilinear Equations of Elliptic Type, Moscow, Nauka, 1973.
10. V. D. LISEIKIN, On the numerical solution of second order equations with a small parameter affecting the highest derivatives, Chisl. Metody Mechaniki Splosh. Sredy, Novosibirsk, 14 (1983), pp. 98-108.
11. J. A. MACKENIE AND K. W. MORTON, Finite Volume Solutions of Convection-Diffusion Test Problems, Oxford University Computing Lab., Tech. Rep. 90/01, 1990.
12. P. A. MARKOWICH, C. A. RINGHOFER, S. SELBERHERR, AND M. LENTINI, A singular perturbation approach for the analysis of the fundamental semiconductor equations, IEEE Trans. Electron Devices, 30 (1983), pp. 1165-1180.
13. J. J. H. MILLER AND G. I. SHISHKIN, On the construction of uniformly convergent finite difference schemes for singularly perturbed problems for a quasilinear equation, in Computational Methods for Boundary and Interior Layers in Several Dimensions, J. J. H. Miller, ed. Boole Press, Dublin, Ireland, 1991, pp. 103-118.
14. J. J. H. MILLER AND W. SONG, A tetrahedral mixed finite element method for the stationary semiconductor continuity equations, SIAM J. Numer. Anal., 31 (1994), pp. 196-216.
15. J. J. H. MILLER, ED., Applications of Advanced Computational Methods for Boundary and Interior Layers, Boole Press, Dublin, Ireland, 1993.
16. J. D. MURRAY, Lectures on Nonlinear Differential Equation Models in Biology, Clarendon Press, Oxford, 1977.
17. W. V. VAN ROOSBROECK, Theory of flows of electrons and holes in germanium and other semiconductors, Bell Syst Tech. J., 29 (1950), pp. 560-607.
18. D. L. SCHARFETTER AND H. K. GUMMEL, Large-signal analysis of a silicon Read diode oscillator, IEEE Trans. Electron Devices, 16 (1969), pp. 64-77.
19. G. I. SHISHKIN, Approximation of solutions of singularly perturbed boundary value problems with corner boundary layer, Comput. Math. Math. Phys., 27 (1987), pp. 54-63.
20. _, Grid approximation of boundary value problems with regular boundary layer, Part 1, Part 2, preprint, INCA, 1990.
21. _, Difference scheme for solving elliptic equations with small parameters affecting the derivatives, Banach Centre Publ., 3 (1978), pp. 89-92.
22. _, Grid approximation of singularly perturbed boundary value problems with a regular boundary layer, Soviet J. Numer. Anal. Math. Modelling, 4 (1989), pp. 397-417.
23. _, Grid approximation of a singularly perturbed boundary-value problem for a quasi-linear elliptic equation in the completely degenerate case, Comput. Math. Math. Phys., 31 (1991), pp. 33-46.
24. R. VULANOVIĆ, P. A. FARRELL, AND P. LIN, Numerical solution of non-linear singular perturbation problems modeling chemical reactions, in Applications of Advanced Computational Methods for Boundary and Interior Layers, J. J. H. Miller, ed., Boole Press, Dublin, Ireland, 1993, pp. 192-213.
25. V. W. WEEKMAN, JR. AND R. L. GORRING, Influence of volume change on gas-phase reactions in porous catalysts, J. Catalysis, 4 (1965), pp. 260-270.