Global uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems: Higher-order elements

Computer Methods in Applied Mechanics and Engineering

Volumn 171, Issue 1-2, 1999, Pages 1-23

  Li, Jichun ^a   Navon, I M ^a  

^a FLORIDA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; PROBLEM SOLVING; TENSORS; THEOREM PROVING;

GLOBAL UNIFORMLY CONVERGENT FINITE ELEMENT METHOD; SINGULARLY PERTURBED ELLIPTIC BOUNDARY VALUE PROBLEMS;

FINITE ELEMENT METHOD;

EID: 0032675494     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0045-7825(98)00243-6     Document Type: Article

References (36)

1. H.-G. Roos, Layer-adapted grids for singular perturbation problems, Z. Angew. Math. Mech, 78(5) (1998) 291-309.
2. H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer-Verlag, Berlin, 1996).
3. J.J.H. Miller, E. O'Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (World Scientific, Singapore, 1995).
4. K.W. Morton, Numerical Solution of Convection-Diffusion Problems (Chapman and Hall, 1996).
5. A.B. Vasil'eva, V.F. Butuzov and L.V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, Philadelphia, 1995).
6. C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (SIAM, Philadelphia, 1988).
7. M. Van Dyke, Perturbation Methods in Fluid Mechanics (Academic Press, New York, 1964).
8. J.L. Lions, Perturbations Singulieres dans les Problemes aux Limites et en Controle Optimal (Springer-Verlag, Berlin, 1973).
9. D.S. Naidu and A.K. Rao, Singular Perturbation Analysis of Discrete Control Systems (Springer-Verlag, New York, 1985).
10. R. O'Malley, Singular Perturbations Methods for Ordinary Differential Equations (Springer-Verlag, New York, 1991).
11. G.F. Carey and J.T. Oden, Finite Elements: Fluid Mechanics, Vol. VI (Prentice-Hall, Englewood Cliffs, NJ, 1986).
12. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol. 2 (McGraw-Hill Publ., New York, 1991).
13. T.J.R. Hughes, ed., Finite Element Methods for Convection-Dominated Flows, ASME Monograph AMD-34 (ASME, New York, 1979).
14. J.T. Oden, Optimal h-p finite element methods, Comput. Methods Appl. Mech. Engrg. 112 (1994) 309-331.
15. P.W. Hemker, A Numerical Study of Stiff Two-Point Boundary Value Problems (Mathematical Center, Amsterdam, 1977).
16. H.-G. Roos, D. Adam and A. Felgenhauer, A novel nonconforming uniformly convergent finite element method in two dimensions, J. Math. Anal. Appl. 201 (1996) 715-755.
17. C. Schwab and M. Suri, The p and hp versions of the finite element method for problems with boundary layers, Math. Comput. 65 (1996) 1403-1429.
18. C. Schwab, M. Suri and C. Xenophontos, The hp finite element method for problems in mechanics with boundary layers, Technical Report 96-14, Department of Mathematics & Statistics, University of Maryland Baltimore County, December 1996.
19. G.I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Russian J. Numer. Anal. Math. Model. 3 (1988) 393-407.
20. M. Stynes and E. O'Riordan, A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem, J. Math. Anal. Appl. 214 (1997) 36-54.
21. J. Li, Quasi-optimal uniformly convergent finite element methods for the elliptic boundary layer problem, Comput. Math. Applic. (33)10 (1997) 11-22.
22. J. Li and I.M. Navon, Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: reaction-diffusion type, Comput. Math. Applic. (35)3 (1998) 57-70.
23. J. Li and I.M. Navon, Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems II: convection-diffusion type, Comput. Methods Appl. Mech. Engrg. 162 (1998) 49-78.
24. J. Li and I.M. Navon, A global uniformly convergent finite element method for a quasilinear singularly perturbed elliptic problem, SCRI-98T-14, Florida State University, 1998.
25. 2 Δu + ru = f(x, y) in a square, SIAM J. Math. Anal. 21 (1990) 394-408.
26. 2(x, y)u = f(x, y) in a rectangle, Differential Eqs. 9 (1973) 1274-1279.
27. E.B. Becker, G.F. Carey and J.T. Oden, Finite Elements: An Introduction, Vol. I (Prentice-Hall, Englewood Cliffs, NJ, 1981).
28. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York, 1994).
29. I.V. Denisov, Quasilinear singularly perturbed elliptic equations in a rectangle, Comput. Math. Math. Phys. 35 (1995) 1341-1350.
30. I.P. Boglayev, A numerical method for solving quasilinear elliptic equations with a small parameter for the highest derivatives, Comput. Math. Math. Phys. 28 (1988) 118-125.
31. A.M. Schatz and L.B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comput. 40 (1983) 47-89.
32. W.G. Szymczak and I. Babǔska, Adaptivity and error estimation for the finite element method applied to convection diffusion problem, SIAM J. Numer. Anal. 21 (1984) 910-953.
33. R. Verfürth, Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation, Numer. Math. 78 (1998) 479-493.
34. H.-J. Reinhardt, A posteriori error estimates for the finite element solution of a singularly perturbed linear ordinary differential equation, SIAM J. Numer. Anal. 18 (1981) 406-430.
35. J.-Y. Lee, Adaptive numerical methods for singular perturbation problems in one space dimension, Ph.D. Dissertation, Courant Institute of Mathematical Sciences, 1994.
36. L. Demkowicz and J.T. Oden, An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in one space variable, J. Comput. Phys. 67 (1986) 188-213.