Skip to main content
Log in

A mixed finite element for the stokes problem using quadrilateral elements

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Brezzi and M. Fortin,Mixed and Hybrid Finite Element Methods (Springer, 1991).

  2. P.G. Ciarlet,The Finite Element Method for Elliptic Problems (North-Holland, 1978).

  3. M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and the elasticity problems, SIAM J. Num. Anal. 30(1993)971–990.

    Article  MATH  MathSciNet  Google Scholar 

  4. V. Girault and P.A. Raviart,Finite Element for Navier-Stokes Equations, Theory and Algorithms, Springer S.C.M. 5 (Springer, 1986).

  5. R.A. Nicolaides, Flow discretization by complementary volume techniques, AIAA paper 89-1978, in:Proc. 9th AIAA CFD Meeting, Buffalo, NY (1989).

  6. R.A. Nicolaides, Analysis and convergence of the MAC scheme. I. The linear problem, SIAM J. Num. Anal. 29(1992)1579–1591.

    Article  MATH  MathSciNet  Google Scholar 

  7. S.V. Patankar, Numerical heat transfer and fluid flow, Series in Computational Methods in Mechanics and Thermal Sciences (1980).

  8. P.A. Raviart and J.M. Thomas,A Mixed Finite Element Method for 2nd Order Elliptic Problems, Lecture Notes in Mathematics 606 (Springer, New York, 1977) pp. 292–315.

    Google Scholar 

  9. J.M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes, Thèse, Université Pierre et Marie Curie, Paris 6 (1977).

    Google Scholar 

References

  1. G. Chandler, Superconvergence of numerical solutions to second kind integral equations, PhD Thesis, Australian National University (1979).

  2. M. Costabel and E.P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987) 461–478.

    Article  MATH  MathSciNet  Google Scholar 

  3. D.K. Fadeev and V.N. Fadeeva, in:Computational Methods of Linear Algebra (W.H. Freeman. San Francisco and London, 1963) pp. 532–551.

    Google Scholar 

  4. D. Gaier,Lectures on Complex Approximation (Birkhäuser, Boston/Basel/Stuttgart, 1987).

    MATH  Google Scholar 

  5. M.H. Gutknecht, Stationary and almost stationary iterative (k,l)-step methods for linear and non-linear systems of equations, Numer. Math. 56 (1989) 179–213.

    Article  MATH  MathSciNet  Google Scholar 

  6. M.H. Gutknecht, in:Numerical Conformal Mapping, ed. L.N. Trefethen (North-Holland, 1986) pp. 31–77.

  7. M.H. Gutknecht, Numerical experiments on solving Theodorsen's integral equation for conformal maps with the fast Fourier transform and various non-linear iterative methods, SIAM J. Sci. State. Comp. 4 (1983) 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  8. D.M. Hough, The use of splines and singular functions in an integral equation method for conformal mapping, PhD. thesis, Brunel University (1983).

  9. D.M. Hough, Exact formulae for certain integrals arising in potential theory, IMAJNA 1 (1981) 223–228.

    MATH  MathSciNet  Google Scholar 

  10. D.M. Hough, Jacobi polynomial solutions of first kind integral equations for numerical conformal mapping. JCAM 12 & 13 (1985) 359–369.

    MathSciNet  Google Scholar 

  11. R.S. Lehman, Development of the mapping function at an analytic corner, Pacific J. Math 7 (1957) 1437–1449.

    MATH  MathSciNet  Google Scholar 

  12. M.A. Jaswon and G.T. Symm,Integral Equation Methods in Potential Theory and Elastostatics (Academic Press, London, 1977).

    MATH  Google Scholar 

  13. A. Kufner, O. John and S. Fucik,Function Spaces (Noordhoff, Leiden, 1977).

    MATH  Google Scholar 

  14. J. Levesley, A study of Chebyshev weighted approximations to the solutions of Symm's integral equation for numerical conformal mapping, PhD. Thesis, Coventry Polytechnic (1991).

  15. D.M. Hough, J. Levesley and S.N. Chandler-Wilde, Numerical conformal mapping via Chebyshev weighted solutions of Symm's integral equation, J. Comp. Appl. Math. 46 (1993) 29–48.

    Article  MATH  MathSciNet  Google Scholar 

  16. Z. Nehari,Conformal Mapping (McGraw-Hill, New York, 1952).

    MATH  Google Scholar 

  17. J. Hoschek, Detecting regions with undesirable curvature, Comp. Aided Geom. Design 1 (1984) 183–192.

    Article  MATH  Google Scholar 

  18. J. Hoschek and D. Lasser,Grundlagen der geometrischen Datenverarbeitung, 2nd ed. (Teubner, Stuttgart, 1992).

    MATH  Google Scholar 

  19. W. Lü, Rational offsets by reparametrizations, preprint (1992).

  20. W. Lü, Rationality of the offsets to algebraic curves and surfaces, preprint (1993).

  21. B. Pham, Offset curves and surfaces: a brief survey, Comp. Aided Design 24 (1992) 223–229.

    Article  Google Scholar 

  22. H. Pottmann, Rational curves and surfaces with rational offsets, Comp. Aided Geom. Design (1995), to appear.

  23. H. Pottmann, Applications of the dual Bézier representation of rational curves and surfaces, in:Curves and Surfaces in Geometric Design, eds. P. J. Laurent, A. L. Méhauté and L. L. Schumaker (AK Peters, Wellesley, MA, 1994) pp. 377–384.

    Google Scholar 

  24. W. Wunderlich, Algebraische Böschungslinien dritter und vierter Ordnung, Sitzungsberichte der Österreichischen Akademie der Wissenschaften 181 (1973) 353–376.

    MATH  MathSciNet  Google Scholar 

References

  1. H. Arndt, Numerical solution of retarded initial value problems: local and global error and stepsize control, Numer. Math. 43 (1984) 343–360.

    Article  MATH  MathSciNet  Google Scholar 

  2. H. Arndt, P.J. van der Houwen and B.P. Sommeijer, Numerical interpolation of retarded differential equations,Delay Equations: Approximation, Theory and Applications, ISNM 72 (Birkhäuser, 1985) pp. 41–51.

  3. M. Artola, Equations paraboliques à retardment, C.R. Acad. Sci. Paris 264 (1967) 668–671.

    MATH  MathSciNet  Google Scholar 

  4. S. Arunsawatwong and V. Zakian,I MN recursions for long sequences of linear delay differential equations, Control Systems Centre report 791, UMIST, Manchester (1993).

    Google Scholar 

  5. U.M. Ascher and L.R. Petzold, The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. Numer, Anal., to appear.

  6. G. Bader, Ein Mehrschrittverfahren mit variabler Schrittweite und variabler Ordnung zur Intergration von Systemen retardierter Differentialgleichungen mit zustandsabhängiger Verzögerung, Diplomarbeit Universität Heidelberg (1983).

  7. C.T.H. Baker, Propositions on the robustness of multistep formulae, Technical Report No. 244, University of Manchester (1994). J. Numer, Func. Anal. Optim., to appear.

  8. C.T.H. Baker, Dynamics of discretized equations for DDEs,Proc. HERMIS '94 Conf., Athens University of Economics and Business (Sept. 1994).

  9. C.T.H. Baker and C.A.H. Paul, Computing stability regions — Runge — Kutta methods for delay differential equations, IMA J. Numer. Anal. 14 (1994) 347–362.

    Article  MATH  MathSciNet  Google Scholar 

  10. C.T.H. Baker and C.A.H. Paul, Parallel continuous Runge—Kutta methods and vanishing lag delay differential equations, Adv. Comp. Math. 1 (1993) 367–394.

    Article  MATH  MathSciNet  Google Scholar 

  11. C.T.H. Baker and C.A.H. Paul, A global convergence theorem for a class of parallel continuous explicit Runge—Kutta methods and vanishing lag delay differential equations, Technical Report No. 229, University of Manchester (1993); SIAM J. Numer. Anal. (1996), to appear.

  12. C.T.H. Baker, C.A.H. Paul and D.R. Willé, A bibliography on the numerical solution of delay differential equations, Technical Report, University of Manchester, in preparation.

  13. C.T.H. Baker, J.C. Butcher and C.A.H. Paul, Experience of STRIDE applied to delay differential equations, Technical Report No. 208, University of Manchester (1992).

  14. H.T. Banks and F. Kappel, Spline approximations for functional differential equations, J. Diff. Eqns. 34 (1979) 496–522.

    Article  MATH  MathSciNet  Google Scholar 

  15. V.K. Barwell, Special stability problems for functional differential equations, BIT 15 (1975) 130–135.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Bellen, One-step collocation for delay differential equations, J. Comp. Appl. Math. 10 (1984) 275–283.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Bellen and M. Zennaro, Numerical solution of delay differential equations by uniform corrections to an implicit Runge-Kutta method, Numer. Math. 47 (1985) 301–316.

    Article  MATH  MathSciNet  Google Scholar 

  18. R.E. Bellman and K.L. Cooke,Differential-Difference Equations, Mathematics in Science and Engineering 6 (Academic Press, 1963).

  19. R.E. Bellman and J.M. Danskin, A survey of the mathematical theory of time-lag, retarded control, and hereditary processes, R-256, The Rand Corporation, Santa Monica (1954).

    Google Scholar 

  20. R.E. Bellman, J.D. Buell and R.E. Kalaba, Mathematical experimentation in time-lag modulation, Commun. ACM 9 (1966) 752–753.

    Article  Google Scholar 

  21. R.E. Bellman, J.D. Buell and R.E. Kalaba, Numerical integration of a differential-difference equation with a decreasing time-lag. Commun. ACM 8 (1965) 227–228.

    Article  MATH  MathSciNet  Google Scholar 

  22. H.G. Bock and J.P. Schlöder, Numerical solution of retarded differential equations with state-dependent time lags. Z. angew. Math. Mech. 61 (1981) 269–271.

    Google Scholar 

  23. U. Buchacker and S. Filippi, Stepsize control for delay differential equations using a pair of Runge-Kutta formulae. J. Comp. Appl. Math. 26 (1989) 339–343.

    Article  MATH  MathSciNet  Google Scholar 

  24. K. Burrage,Parallel and Sequential Methods for Ordinary Differential Equations (OUP, 1995).

  25. K. Burrage, J.C. Butcher and F.H. Chipman, An implementation of singly-implicit Runge-Kutta methods, BIT 20 (1980) 326–340.

    Article  MATH  MathSciNet  Google Scholar 

  26. J.C. Butcher, The adaptation of STRIDE to delay differential equations, Appl. Numer. Math. 9 (1992) 415–425.

    Article  MATH  MathSciNet  Google Scholar 

  27. J. Carr, The asymptotic behaviour of the solutions of some linear functional differential equations, D. Phil., University of Oxford (1974).

  28. N.H. Chosky, Time-lag controls: a bibliography, IRE Trans. Auto. Cont. AC-5 (1966) 66–70.

    Google Scholar 

  29. C.W. Cryer, Numerical methods for functional differential equations,Delay and Functional Differential Equations, ed. K. Schmitt (Academic Press, 1972).

  30. G. Dahlquist, Recent work of stiff differential equations, Technical Report No. TRITA-NA-7512, Royal Institute of Technology, Stockholm (1975).

    Google Scholar 

  31. R.D. Driver,Ordinary and Delay Differential Equations, Applied Mathematics Series 20 (Springer, 1977).

  32. E.L. El'sgol'ts,Qualitative Methods in Mathematical Analysis, Transl. Math. Monog. 12 (Academic Press, 1964).

  33. E.L. El'sgol'ts and S.B. Norkin,Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Mathematics in Science and Engineering 105 (Academic Press, 1973).

  34. W.H. Enright, Continuous numerical methods for ODEs and the implication for delay differential equations,Proc. HERMIS '94 Conf., Athens University of Economics and Business (Sept. 1994).

  35. W.H. Enright, The relative efficiency of alternative defect control schemes for high-order continuous Runge-Kutta formulas, SIAM J. Numer. Anal. 30 (1993) 1419–1445.

    Article  MATH  MathSciNet  Google Scholar 

  36. W.H. Enright, A new error-control for initial-value solvers, Appl. Math. Comp. 31 (1989) 288–301.

    Article  MathSciNet  Google Scholar 

  37. W.H. Enright, Analysis of error control strategies for continuous Runge-Kutta methods, SIAM J. Numer. Anal. 26 (1989) 588–599.

    Article  MATH  MathSciNet  Google Scholar 

  38. W.H. Enright and M. Hu, Interpolating Runge-Kutta methods for vanishing lag delay differential equations, Computer Science Technical Report No. 292/94, University of Toronto (1994).

  39. W.H. Enright, K.R. Jackson, S.P. Nørsett and P.G. Thomsen, Interpolants for Runge-Kutta formulas, ACM Trans. Math. Soft. 12 (1986) 193–218.

    Article  MATH  Google Scholar 

  40. M.A. Feldstein, Discretization methods for retarded ordinary differential equations, Doctoral thesis, Department of Mathematics, UCLA (1964).

  41. M.A. Feldstein and K.W. Neves, High-order methods for state-dependent delay differential equations with nonsmooth solutions, SIAM J. Numer. Anal. 21 (1984) 844–863.

    Article  MATH  MathSciNet  Google Scholar 

  42. M.A. Feldstein and J. Sopka, Numerical methods for nonlinear Volterra integro-differential equations. SIAM J. Numer. Anal. 11 (1974) 826–846.

    Article  MATH  MathSciNet  Google Scholar 

  43. K. Gopalsamy,Stability and Oscillations in Delay Differential Equations of Population Dynamics (Kluwer, Dordrecht, 1992).

    MATH  Google Scholar 

  44. E. Hairer, G. Wanner and S.P. Nørsett,Solving Ordinary Differential Equations 1, Springer Series in Computational Mathematics 8 (Springer, 1983).

  45. A. Halanay,Differential Equations: Stability, Oscillations, Time Lags, Mathematics in Science and Engineering 23 (Academic Press, 1966).

  46. H. Hayashi and W.H. Enright, A new algorithm for vanishing delay problems, Canadian Applied Mathematics Society, annual meeting, University of York (1993).

  47. D.J. Higham, Highly continuous Runge-Kutta interpolants, ACM Trans. Math. Soft. 17 (1991) 368–386.

    Article  MATH  MathSciNet  Google Scholar 

  48. D.R. Hill, A new class of one-step methods for the solution of Volterra functional differential equations, BIT 14 (1974), 298–305.

    Article  MATH  Google Scholar 

  49. K.J. in't Hout, Runge—Kutta methods in the numerical solution of delay differential equations, Ph.D. thesis, Department of Mathematics and Computer Science, University of Leiden (1992).

  50. K.J. in't Hout, A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT 32 (1992) 634–649.

    Article  MATH  MathSciNet  Google Scholar 

  51. K.J. in't Hout and M.N. Spijker, Stability analysis of numerical methods for delay differential equations. Numer. Math. 59 (1991) 807–814.

    Article  MATH  MathSciNet  Google Scholar 

  52. P.J. van der Houwen and B.P. Sommeijer, Linear multistep methods with reduced truncation error for periodic initial value problems, IMA J. Num. Anal. 4 (1984) 479–489.

    Article  MATH  Google Scholar 

  53. P.J. van der Houwen and B.P. Sommeijer, Stability in linear multistep methods for pure delay equations, J. Comp. Appl. Math. 10 (1984) 55–63.

    Article  MATH  Google Scholar 

  54. P.J. van der Houwen, B.P. Sommeijer and C.T.H. Baker, On, the stability of predictor-corrector methods for parabolic equations with delay, IMA J. Num. Anal. 6 (1986) 1–23.

    Article  MATH  Google Scholar 

  55. A. Iserles, Stability and dynamics of numerical methods for nonlinear ordinary differential equations, IMA J. Num Anal. 10 (1990) 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  56. A. Iserles and M. Buhmann, Stability of the discretized pantograph differential-equation, Math. Comp. 60 (1993) 575–589.

    Article  MATH  MathSciNet  Google Scholar 

  57. Z. Jackiewicz, Waveform relaxation methods for functional differential systems of neutral type,Proc. HERMIS ′94 Conf., Athens University of Economics and Business (Sept. 1994).

  58. Z. Jackiewicz and E. Lo, The numerical integration of neutral functional-differential equations by fully implicit one-step methods, Technical Report No. 129, Arizona State University (1991).

  59. F. Kappel and K. Kunisch Spline approximations for neutral functional-differential equations, SIAM J. Numer. Anal. 18 (1981) 1058–1080.

    Article  MATH  MathSciNet  Google Scholar 

  60. T. Kato, Asymptotic behaviour of solutions of the functional differential equationsý (x)=ay(λx)+by(x), in:Delay and Functional Differential Equations and Their Applications, ed. K. Schmitt (Academic Press, 1972).

  61. T. Kato and J.B. McLeod, The functional differential equationý(x)=ay(λx)+by(x) Bull. Amer. Math. Soc. 77 (1971) 891–937.

    Article  MATH  MathSciNet  Google Scholar 

  62. V.B. Kolmanovskii and A. Myshkis,Applied Theory of Functional Differential Equations, Mathematics and its Applications 85 (Kluwer, 1992).

  63. V.B. Kolmanovskii and V.R. Nosov,Stability of Functional Differential Equations, Mathematics in Science and Engineering 180 (Academic Press, 1986).

  64. P. Linz, Linear multistep methods for Volterra integro-differential equations, J. ACM 16 (1969) 295–301.

    Article  MATH  MathSciNet  Google Scholar 

  65. Y. Kuang,Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering 191 (Academic Press, 1993).

  66. M.Z. Liu and M.N. Spijker, The stability of σ-methods in the numerical solution of delay differential equations, IMA J. Num. Anal. 10 (1990) 31–48.

    Article  MATH  MathSciNet  Google Scholar 

  67. Matlab, The Mathworks Inc., Natick, MA 01760.

  68. K.W. Neves, Automatic integration of functional differential equations: an approach, ACM Trans. Math. Soft. 1 (1975) 357–368.

    Article  MATH  MathSciNet  Google Scholar 

  69. K.W. Neves and S. Thompson, Software for the numerical-solution of systems of functional-differential equations with state-dependent delay, Appl. Numer. Math. 9 (1992) 385–401.

    Article  MATH  MathSciNet  Google Scholar 

  70. K.W. Neves and M.A. Feldstein, Characterisation of jump discontinuities for state-dependent delay differential equations, J. Math. Anal. Appl. 56 (1976) 689–707.

    Article  MATH  MathSciNet  Google Scholar 

  71. H.J. Oberle and H.J. Pesch, Numerical treatment of delay differential equations by Hermite interpolation, Num. Math. 37 (1981) 235–255.

    Article  MATH  MathSciNet  Google Scholar 

  72. J. Oppelstrup, The RKFHB4 method for delay differential equations, Lecture Notes in Mathematics 631 (1978) pp. 133–146.

    Article  MathSciNet  Google Scholar 

  73. B. Owren and M. Zennaro, Derivation of efficient, continuous explicit Runge-Kutta methods, SIAM J. Sci. Stat. Comp. 13 (1992) 1488–1501.

    Article  MATH  MathSciNet  Google Scholar 

  74. C.A.H. Paul, Performance and properties of a class of parallel continuous explicit Runge-Kutta methods for ordinary and delay differential equations,Proc. HERMIS ′94 Conf., Athens University of Economics and Business (Sept. 1994).

  75. C.A.H. Paul, Runge-Kutta methods for functional differential equations, Ph.D. thesis, Mathematics Department, University of Manchester (1992).

  76. C.A.H. Paul, Developing a delay differential equation solver, Appl. Num. Math. 9 (1992) 403–414.

    Article  MATH  Google Scholar 

  77. T.L. Saaty,Modern Nonlinear Mathematics, (Dover, New York, 1967).

    Google Scholar 

  78. L.F. Shampine, Interpolation for Runge-Kutta methods, SIAM J. Numer. Anal. 22 (1985) 1014–1026.

    Article  MATH  MathSciNet  Google Scholar 

  79. L.F. Shampine and P. Bogacki, The effect of changing the stepsize in linear multistep codes, SIAM J. Sci. Stat. Comp. 10 (1989) 1010–1023.

    Article  MATH  MathSciNet  Google Scholar 

  80. H.L. Smith and Y. Kuang, Slowly oscillating periodic-solutions of autonomous state-dependent delay equations, Nonlinear Anal. — Theory, Meth. Appl. 19 (1992) 855–872.

    Article  MATH  MathSciNet  Google Scholar 

  81. H.J. Stetter, Numerische Lösung von Differentialgleichungen mit nacheilendem Argument, ZAMM 45 (1965) 79–80.

    Google Scholar 

  82. L. Tavernini, One-step methods for the numerical solution of Volterra functional differential equations, SIAM J. Numer. Anal. 8 (1971) 786–795.

    Article  MATH  MathSciNet  Google Scholar 

  83. L. Torelli, Stability of numerical methods for delay differential equations, J. Comp. Appl. Math. 25 (1989) 15–26.

    Article  MATH  MathSciNet  Google Scholar 

  84. L. Wiederholt, Stability of multistep methods for delay differential equations, Math. Comp. 30 (1976) 283–290.

    Article  MATH  MathSciNet  Google Scholar 

  85. L. Wiederholt, Numerical solution of delay differential equations, Ph.D. thesis, University of Wisconsin (1970).

  86. D.R. Willé, Experiments in stepsize control for Adams linear multistep methods, Technical Report No. 94-11, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, University of Heidelberg (1994).

  87. D.R. Willé, New stepsize estimators for linear, multistep methods, Technical Report No. 93-47. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, University of Heidelberg (1993). [Note erratum.]

  88. D.R. Willé and C.T.H. Baker, Some issues in the detection and location of derivative discontinuities in delay-differential equations, Technical Report No. 238, University of Manchester (1994).

  89. D.R. Willé and C.T.H. Baker, Stepsize control and continuity consistency for state-dependent delay differential equations, J. Comp. Appl. Math 53 (1994) 163–170.

    Article  MATH  Google Scholar 

  90. D.R. Willé and C.T.H. Baker, The tracking of derivative discontinuities in systems of delay differential equations, Appl. Num. Math. 9 (1992) 209–222.

    Article  MATH  Google Scholar 

  91. D.R. Willé and C.T.H. Baker, DELSOL — a numerical code for the solution of systems of delay differential equations, Appl. Numer. Math. 9 (1992) 223–234.

    Article  MATH  Google Scholar 

  92. D.R. Willé and C.T.H. Baker, A short note on the propagation of derivative discontinuities in Volterra-delay integro-differential equations, Technical Report No. 187, University of Manchester (1990).

  93. E.M. Wright, On a sequence defined by a non-linear recurrence formula, J. London Math. Soc. 20 (1945) 68–73.

    Article  MATH  MathSciNet  Google Scholar 

  94. M. Zennaro, Natural continuous extensions of Runge-Kutta methods. Math. Comp. 46 (1986) 119–133.

    Article  MATH  MathSciNet  Google Scholar 

  95. M. Zennaro,P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math. 49 (1986) 305–318.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by D.N. Arnold

Rights and permissions

Reprints and permissions

About this article

Cite this article

Farhloul, M., Fortin, M. A mixed finite element for the stokes problem using quadrilateral elements. Adv Comput Math 3, 101–113 (1995). https://doi.org/10.1007/BF02431998

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02431998

Key words

Navigation