Estimating the error of numerical solutions of systems of reaction-diffusion equations

Memoirs of the American Mathematical Society

Volumn , Issue 696, 2000, Pages 1-109

  Estep, Donald J ^a   Larson, Mats G ^a   Williams, Roy D ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034216158     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: 10.1090/memo/0696     Document Type: Article

References (73)

1. [AF88] S. Adjerid and J. Flaherty, A local refinement finite element method for two-dimensional parabolic systems, SIAM J. Sci. Statist. Comput. 9 (1988), 792-810.
2. [AF93] S. Adjerid and J. Flaherty, A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems, Numer. Math. 65 (1993), 1-21.
3. [AW75] D. Aronson and H. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, Proceedings of the Tulane Program in P.D.E.'s (New York), Springer Lecture Notes, 446, New York, 1975, pp. 5-49.
4. [BB82] M. Bieterman and I. Babuška, The finite element method for parabolic equations, II. A posteriori error estimations and adaptive approach, Numer. Math. 40 (1982), 373-406.
5. [BB86] M. Bieterman and I. Babuška, An adaptive method of lines with error control for parabolic equations of the reaction-diffusion type, J. Comput. Phys. 63 (1986), 33-66.
6. [BBRW97] M. Braack, R. Becker, R. Rannacher, and J. Warnatz, An adaptive finite element method for combustion problems, Preprint 98-10, SFB 359, University of Heidelberg, to appear in parts in the Proceedings of the Summer Conference on Numerical Modelling in Continuum Mechanics, Prague, Sept. 8-11, 1997 and the Proceedings of ENUMATH-97, Heidelberg Sept. 29 - Oct. 3, 1997, World Scientific Publishing.
7. [BDG81] K. Brown, P. Donne, and R. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Diff. Eqns. 40 (1981), 232-252.
8. [BK90] L. Bronsard and R. Kohn, On the slowness of phase-boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990), 983-997.
9. [BK91] L. Bronsard and R. Kohn, Motion by mean-curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Eqns. 90 (1991), 211-237.
10. [Bra98] M. Braack, An adaptive finite element method for reactive flow problems, Ph.D. thesis, University of Heidelberg, Heidelberg, Germany, 1998.
11. [CCS77] K. Chueh, C. Conley, and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Ind. Uni. Math. J. 26 (1977), 373-392.
12. [CD66] J. Cooley and F. Dodge, Digital computer solutions for excitation and propagation of the nerve impulse, Biophys. J. 6 (1966), 583-599.
13. [Cha81] N. Chafee, The electric ballast resistor: homogeneous and nonhomogeneous equilibria, Nonlinear Differential Equations: Invariance Stability and Bifurcation (New York) (P. De Motoni and L. Salvadori, eds.), Academic Press, New York, 1981.
14. [Cia78] P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, New York, 1978.
15. [Coh71] H. Cohen, Nonlinear diffusion problems, MAA Studies in Math. 7 (1971), 27-64.
16. 2Δu - f(u), Comm. Pure Appl. Math. 42 (1989), 523-576.
17. [CT85] C. Chen and V. Thomée, The lumped mass finite element method for a parabolic problem, J. Austral. Math. Soc. Ser. B 26 (1985), 329-354.
18. 2-projection onto finite element spaces, Math. Comp. 48 (1987), 521-532.
19. [Dup82] T. Dupont, Mesh modification for evolution equations, Math. Comp. 39 (1982), 85-107.
20. [EEHJ95] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations, Acta Numerica (1995), 105-158.
21. [EEHJ96] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, New York, 1996.
22. [EF94] D. Estep and D. French, Global error control for the continuous Galerkin finite element method for ordinary differential equations, RAIRO Modél. Math. Anal. Numér 28 (1994), 815-852.
23. [EHW99] D. Estep, D. Hodges, and M. Warner, Computational error estimation for a finite element solution of missile trajectory optimization problems, SIAM J. Sci. Computing (1999), to appear.
24. [EJ87] K. Eriksson and C. Johnson, Error estimates and automatic time step control for non-linear parabolic problems, SIAM J. Numer. Anal. 24 (1987), 12-23.
25. [EJ91] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal. 28 (1991), 43-77.
26. [EJ95a] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems V: Long-time integration, SIAM J. Numer. Anal. 32 (1995), 1750-1763.
27. [EJ95b] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems IV: Non-linear problems, SIAM J. Numer. Anal. 32 (1995), 1729-1749.
28. ∞), SIAM J. Numer. Anal. 32 (1995), 706-740.
29. [EJ98] D. Estep and C. Johnson, The computability of the Lorenz system, Math. Models Meth. Appl. Sci. 8 (1998), 1277-1305.
30. [EL] D. Estep and M. Larson, A new approach to estimating the error of numerical methods for ordinary differential equations, in preparation.
31. [EL93] D. Estep and S. Larsson, The discontinuous Galerkin method for semilinear parabolic problems, RAIRO Modél. Math. Anal. Numér. 27 (1993), 35-54.
32. [Eri93] K. Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Preprint #1993-20, Department of Mathematics, Chalmers University of Technology, Göteborg, 1993.
33. [ES93] C. Elliot and A. Stuart, The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal. 30 (1993), 1622-1663.
34. [ES99] D. Estep and A. Stuart, The dynamical behavior of Galerkin finite element methods for ordinary differential equations and related difference schemes, Math. Comp. (1999), submitted.
35. [Est94] D. Estep, An analysis of numerical approximations of metastable solutions of the bistable equation, Nonlinearity 7 (1994), 1445-1462.
36. [Est95] D. Estep, A posteriori error bounds and global error control for approximations of ordinary differential equations, SIAM J. Numer. Anal. 32 (1995), 1-48.
37. [EW96] D. Estep and R. Williams, The structure of an adaptive differential equation solver, Math. Models Meth. Appl. Sci. 6 (1996), 535-568.
38. [EW97] D. Estep and R. Williams, Cards: Concurrent Adaptive Reaction-Diffusion Solver, 2.0, 1997.
39. [FH89] G. Fusco and J. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynam. Differ. Equa. 1 (1989), 75-94.
40. [Fif88] P. Fife, Dynamics of Internal Layers and Diffusive Interfaces, SIAM, Philadelphia, 1988.
41. [FJ95] D. French and S. Jensen, Global dynamics of a discontinuous galerkin approximation to a class of reaction-diffusion equations, Appl. Numer. Math. 18 (1995), 473-487.
42. [FN91] R. Freund and N. Nachtigal, QMR: A quasi-minimal residual method for non-hermitian linear systems, Numer. Math. 60 (1991), 315-339.
43. [FN96] R. Freund and N. Nachtigal, QMRPACK: a package of QMR algorithms, ACM Trans. Math. Software 22 (1996), 46-77.
44. [Has75] S. Hastings, Some mathematical problems from neurobiology, AMS Monthly 82 (1975), 881-895.
45. [Hen62] P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1962.
46. [HH52] A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol. 117 (1952), 500-544.
47. [Hof78] D. Hoff, Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations, SIAM J. Numer. Anal. 15 (1978), 1161-1177.
48. [Joh88] C. Johnson, Error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations, SIAM J. Numer. Anal. 25 (1988), 908-926.
49. [Lar97] M. Larson, Analysis of adaptive finite element methods, Ph.D. thesis, Chalmers University of Technology, S-412 96 Göteborg, Sweden, 1997.
50. [LSU68] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'cev, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, 1968.
51. [Mas90] M. Mascagni, The backward euler method for numerical solution of the hodgkin-huxley equations of nerve conduction, SIAM J. Numer. Anal. 27 (1990), 941-962.
52. [Mat79] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ. 15 (1979), 401-454.
53. [MNY79] M. Mimura, Y. Nishiura, and M. Yamaguti, Some diffusive predator-prey systems and their bifurcation problems, Ann. New York Acad. Sci. 316 (1979), 490-510.
54. [Moo94] P. Moore, A posteriori error estimation with finite element semi-and fully discrete methods for nonlinear parabolic equations in one space dimension, SIAM J. Numer. Anal. 31 (1994), 149-169.
55. [Mur91] J. D. Murray, Mathematical Biology, Springer-Verlag, New York, 1991.
56. [NPV91a] R. Nochetto, M. Paolini, and C. Verdi, An adaptive finite-element method for 2-phase Stefan-problems in 2-space dimensions 1: stability and error estimates, Math. Comp. 57 (1991), 73-108.
57. [NPV91b] R. Nochetto, M. Paolini, and C. Verdi, An adaptive finite-element method for 2-phase Stefan-problems in 2-space dimensions 2: implementation and numerical experiments, SIAM J. Sci. Stat. Comput. 12 (1991), 1207-1244.
58. [NPV93] R. Nochetto, M. Paolini, and C. Verdi, Towards a unified approach for the adaptive solution of evolution phase changes, Variational Problems: IMA Series in Mathematics and its Applications (New York), Springer-Verlag, New York, 1993, pp. 1-15.
59. [NSVa] R. Nochetto, G. Savaré, and C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., submitted.
60. [NSVb] R. Nochetto, A. Schmidt, and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems, Math. Comp., to appear.
61. [NSV97] R. Nochetto, A. Schmidt, and C. Verdi, Adaptive solution of phase change problems over unstructured tetrahedral meshes, Grid Generation and Adaptive Algorithms, IMA Series (New York), Springer-Verlag, New York, 1997.
62. [Pea93] J. E. Pearson, Complex patterns in a simple system, Science 261 (1993), 189.
63. [Rac92] R. Racke, Lectures on Nonlinear Evolution Equations, Max Planck Institüt für Mathematik, Bonn, 1992, Aspects of Mathematics: E, Vol. 19.
64. [Ran97] R. Rannacher, A posteriori error estimates in least-squares stabilized finite element schemes, Comp. Meth. Appl. Mech. Engrg. (1997), to appear.
65. [Rav73] P. Raviart, The use of numerical integration in finite element methods for solving parabolic equations, Topics in Numerical Analysis I (New York) (J. Miller, ed.), Academic Press, New York, 1973.
66. [RS78] J. Rauch and J. Smoller, Qualitative theory of the Fitz-Hugh-Nagumo equations, Adv. in Math. 27 (1978), 12-44.
67. [RS98] R. Rannacher and F.-T. Suttmeier, A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity, Comp. Meth. Appl. Mech. Engrg. (1998), to appear.
68. [San96] R. Sandboge, Adaptive finite element methods for reactive flow problems, Ph.D. thesis, Chalmers University of Technology, S-412 96 Göteborg, Sweden, 1996.
69. [Smo83] J. A. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
70. [Tho80] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, New York, 1980.
71. [Tro77] W. Troy, A threshold phenomenon in the Field-Noyes model of the Belousov-Zhabotinsky reaction, J. Math. Anal. Appl. 58 (1977), 233-248.
72. [Yse85] H. Yserentant, Hierarchical bases of finite-element spaces in the discretization of non-symmetric elliptic boundary value problems. Computing 35 (1985), 39-49.
73. [Yse86] H. Yserentant, On the multi-level splitting of finite element spaces, Numer. Math. 49 (1986), 379-412.