Enclosures and semi-analytic discretization of boundary value problems

c. Grossmann

Banach Center Publications (1994)

  • Volume: 29, Issue: 1, page 283-304
  • ISSN: 0137-6934

How to cite


Grossmann, c.. "Enclosures and semi-analytic discretization of boundary value problems." Banach Center Publications 29.1 (1994): 283-304. <http://eudml.org/doc/262668>.

author = {Grossmann, c.},
journal = {Banach Center Publications},
keywords = {enclosures; semi-analytic discretization; monotonic discretization; upper and lower bounds; two-point boundary problems; singularly perturbed problems},
language = {eng},
number = {1},
pages = {283-304},
title = {Enclosures and semi-analytic discretization of boundary value problems},
url = {http://eudml.org/doc/262668},
volume = {29},
year = {1994},

AU - Grossmann, c.
TI - Enclosures and semi-analytic discretization of boundary value problems
JO - Banach Center Publications
PY - 1994
VL - 29
IS - 1
SP - 283
EP - 304
LA - eng
KW - enclosures; semi-analytic discretization; monotonic discretization; upper and lower bounds; two-point boundary problems; singularly perturbed problems
UR - http://eudml.org/doc/262668
ER -


  1. [1] H. Abhari and C. Grossmann, Approximate bases for boundary value problems and applications to MID-techniques, Z. Angew. Math. Mech. 72 (1992), 83-91. Zbl0763.65062
  2. [2] H. Abhari and M. Al-Zanaidi, Computational aspects of monotone iteration discretization algorithm, preprint, TU Dresden, 07-01-88. 
  3. [3] R. A. Adams, Sobolev Spaces, Academic Press, New York 1975. 
  4. [4] E. Adams, Invers-Monotonie, direkte und indirekte Intervallmethoden, Forschungszentrum Graz, Ber. 185 (1982). Zbl0506.65021
  5. [5] E. Adams, R. Ansorge, C. Grossmann and H.-G. Roos (eds.), Discretization in Differential Equations and Enclosures, Akademie-Verlag, Berlin 1987. Zbl0636.00010
  6. [6] E. Adams, D. Cordes and H. Keppler, Enclosure methods as applied to linear periodic ODEs and matrices, Z. Angew. Math. Mech. 70 (1990), 565-578. Zbl0739.65059
  7. [7] E. Adams und H. Spreuer, Konvergente numerische Schrankenkonstruktionen mit Spline-Funktionen für nichtlineare gewöhnliche bzw. parabolische Randwertaufgaben, in: K. Nickel (ed.), Interval Mathematics, Springer, Berlin 1975. Zbl0304.65072
  8. [8] M. Al-Zanaidi and C. Grossmann, Monotone discretization in boundary value problems using PASCAL-SC, in: I. Marek (ed.), Proc. ISNA 87, Teubner, Leipzig 1988, 91-96. Zbl0681.65055
  9. [9] M. Al-Zanaidi and C. Grossmann, Monotone iteration discretization algorithm for BVP's, Computing 31 (1989), 59-74. 
  10. [10] M. Al-Zanaidi and C. Grossmann, 1D-grid generation by monotone iteration discretization, ibid. 34 (1990), 377-390. Zbl0693.65053
  11. [11] N. Anderson and A. M. Arthurs, Variational solutions of the Thomas-Fermi equations, Quart. Appl. Math. 39 (1981), 127-129. Zbl0462.49042
  12. [12] S. Carl, The monotone iterative technique for a parabolic boundary value problem with discontinuous nonlinearity, Nonlinear Anal. 13 (1989), 1399-1407. Zbl0696.35083
  13. [13] S. Carl and C. Grossmann, Iterative spline bounds for systems of boundary value problems, in: J. W. Schmidt and H. Späth (eds.), Splines in Numerical Analysis, Akademie-Verlag, Berlin 1989, 19-30. Zbl0673.65042
  14. [14] S. Carl and C. Grossmann, Monotone enclosure for elliptic and parabolic systems with nonmonotone nonlinearities, J. Math. Anal. Appl. 151 (1990), 190-202. Zbl0778.65074
  15. [15] S. Carl and S. Heikkilä, On extremal solutions of an elliptic boundary value problem involving discontinuous nonlinearities, preprint, Univ. Oulu, 1990; to appear in Differential Integral Equations. Zbl0785.35034
  16. [16] C. Y. Chan and Y. C. Hon, A constructive solution for a generalized Thomas-Fermi theory of ionized atoms, Quart. Appl. Math. 45 (1987), 591-599. Zbl0639.34021
  17. [17] R. C. Duggan and C. Goodmann, Pointwise bounds for a nonlinear heat conduction model of the human head, Bull. Math. Biol. 48 (1986), 229-236. Zbl0585.92011
  18. [18] A. Felgenhauer, Monotone discretization of the Poisson equation in the plane, to appear. 
  19. [19] R. C. Flagg, C. D. Luning and W. L. Perry, Implementation of new iterative techniques for solutions of Thomas-Fermi and Emden-Fowler equations, J. Comput. Phys. 38 (1980), 396-405. Zbl0456.65054
  20. [20] E. C. Gartland, On the uniform convergence of the Scharfetter-Gummel discretization, preprint, Kent State Univ., Feb. 1991. Zbl0765.65112
  21. [21] E. C. Gartland and C. Grossmann, Semianalytic solution of boundary value problems by using approximated operators, Z. Angew. Math. Mech. 72 (1992), 615-619. Zbl0771.65045
  22. [22] C. Grossmann, Monotone Einschließung höherer Ordnung für 2-Punkt-Randwertauf- gaben, Z. Angew. Math. Mech. 67 (1987), T475-T477. 
  23. [23] C. Grossmann, Monotone discretization of two-point boundary value problems and related numerical methods, in [5], 99-122. 
  24. [24] C. Grossmann, Semianalytic discretization of weakly nonlinear boundary value problems with variable coefficients. Z. Anal. Anwendungen 10 (1991), 513-523. Zbl0760.65077
  25. [25] C. Grossmann, Enclosures of the solution of the Thomas-Fermi equation by monotone discretization, J. Comput. Phys. (1992), 26-38. 
  26. [26] C. Grossmann und M. Krätzschmar, Monotone Diskretisierung und adaptive Gittergenerierung für Zwei-Punkt-Randwertaufgaben, Z. Angew. Math. Mech. 65 (1985), T264-T266. Zbl0621.65093
  27. [27] C. Grossmann, M. Krätzschmar und H.-G. Roos, Gleichmäßig einschließende Diskretisierungsverfahren für schwach nichtlineare Randwertaufgaben, Numer. Math. 49 (1986), 95-110. Zbl0621.65092
  28. [28] C. Grossmann and H.-G. Roos, Feedback grid generation via monotone discretization for two-point boundary-value problems, IMA J. Numer. Anal. 6 (1986), 421-432. Zbl0655.65105
  29. [29] C. Grossmann and H.-G. Roos, Uniform enclosure of high order for boundary value problems by monotone discretization, Math. Comp. 53 (1989), 609-617. Zbl0673.34025
  30. [30] C. Grossmann and H.-G. Roos, Convergence analysis of higher order monotone discretization, Wiss. Z. Tech. Univ. Dresden 38 (1989), 155-168. Zbl0675.65081
  31. [31] C. Grossmann and H.-G. Roos, Enclosing discretization for singular and singularly perturbed boundary value problems, in: H.-G. Roos, A. Felgenhauer and L. Angermann (eds.), Numerical Methods in Singular Perturbed Problems, TU Dresden, 1991, 71-82. 
  32. [32] G. Koeppe, H.-G. Roos and L. Tobiska, An enclosure generating modification of the method of discretization in time, Comment. Math. Univ. Carolin. 28 (1987), 447-453. Zbl0635.65122
  33. [33] M. Krätzschmar, Iterationsverfahren zur Lösung schwach nichtlinearer elliptischer Randwertaufgaben mit monotoner Lösungseinschließung, Dissertation, TU Dresden, 1983. 
  34. [34] U. Kulisch (ed.), PASCAL-SC, Pascal Extension for Scientific Computation, Wiley, New York and Teubner, Stuttgart 1987. 
  35. [35] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, 1985. Zbl0658.35003
  36. [36] R. Lohner, Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Dissertation, Univ. Karlsruhe, 1988. Zbl0663.65074
  37. [37] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York 1970. Zbl0241.65046
  38. [38] M. Plum, Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems, to appear. 
  39. [39] H.-G. Roos, The Rothe method and monotone discretization for parabolic equations, in [5], 155-166. 
  40. [40] H.-G. Roos, Uniformly enclosing discretization methods and grid generation for semilinear boundary value problems with first order terms, Apl. Mat. 34 (1989), 274-284. Zbl0685.65069
  41. [41] H.-G. Roos, A uniformly convergent discretization method for singularly perturbed boundary value problems of the fourth order, Zb. Rad. Prirod.-Mat. Fak. Novi Sad Ser. Mat. 19 (1989), 51-64. Zbl0719.65067
  42. [42] H.-G. Roos, Global uniformly convergent schemes for a singularly perturbed boundary-value problem using patched base spline functions, J. Comput. Appl. Math. 29 (1990), 69-77. Zbl0686.65048
  43. [43] H.-G. Roos, Uniform enclosing discretization methods for semilinear boundary value problems, in: Nonlinear Analysis and Mathematical Modelling, Banach Center Publ. 24, PWN, Warszawa 1990, 257-268. 
  44. [44] J. W. Schmidt, Ein Einschließungsverfahren für Lösungen fehlerbehafteter linearer Gleichungen, Period. Math. Hungar. 13 (1982), 29-37. Zbl0496.65014
  45. [45] J. Schröder, Operator Inequalities, Academic Press, New York 1980. 
  46. [46] J. Schröder, A method for producing verified results for two-point boundary value problems, in: Comput. Suppl. 6, Springer, Vienna 1988, 9-22. 
  47. [47] J. Schröder, Operator inequalities and applications, in: E. W. Norrie (ed.), Inequalities, Marcel Dekker, New York 1991, 163-210. 
  48. [48] J. Sprekels and H. Voss, Pointwise inclusions of fixed points by finite dimensional iteration schemes, Numer. Math. 32 (1979), 381-392. Zbl0393.65052
  49. [49] G. Vanden Berghe and H. De Meyer, Accurate computation of higher Sturm-Liouville eigenvalues, to appear. Zbl0716.65079
  50. [50] R. Voller, Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation, Computing 42 (1989), 245-258. Zbl0681.65076
  51. [51] W. Walter, Differential and Integral Inequalities, Springer, Berlin 1970. 
  52. [52] J. Weissinger, A kind of difference methods for enclosing solutions of ordinary linear boundary value problems, in: Comput. Suppl. 6, Springer, Vienna 1988, 23-32. Zbl0658.65072

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