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Introduction to Adaptive Methods for Differential Equations

Published online by Cambridge University Press:  07 November 2008

Kenneth Eriksson ,
Don Estep ,
Peter Hansbo  and
Claes Johnson
Kenneth Eriksson
Affiliation:
Mathematics Department, Chalmers University of Technology, 412 96 Göteborgkenneth@math.chalmers.se
Don Estep
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332estep@pmath.gatech. edu
Peter Hansbo
Affiliation:
Mathematics Department, Chalmers University of Technology, 412 96 Göteborghansbo@math.chalmers.se
Claes Johnson
Affiliation:
Mathematics Department, Chalmers University of Technology, 412 96 Göteborgclaes@math.chalmers.se
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Abstract

Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).

When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

Type
Research Article
Information
Acta Numerica , Volume 4 , January 1995 , pp. 105 - 158
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Asadzadeh, M. and Eriksson, K. (1994), ‘An adaptive finite element method for a potential problem’, M3AS, to appear.Google Scholar
Babuška, I. (1986), ‘Feedback, adaptivity and a posteriori estimates in finite el ements: Aims, theory, and experience’, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations, (Babuška, I., Zienkiewicz, O. C., Gago, J. and Oliveira, E. R. de A., eds.), Wiley, New York, 3–23.Google Scholar
Bank, R. (1986), ‘Analysis of local a posteriori error estimate for elliptic equations’, in Accuracy Estimates and Adaptive Refinements in Finite Element Computations (Babuška, I., Zienkiewicz, O. C., Gago, J. and Oliveira, E. R. de A., eds.), Wiley, New York.Google Scholar
Becker, R., Johnson, C. and Rannacher, R. (1994), ‘An error control for multigrid finite element methods’, Preprint #1994–36, Department of Mathematics, Chalmers University of Technology, Göteborg.Google Scholar
Carstensen, C. and Stephan, E. (1993), ‘Adaptive boundary element methods’, Institute für Angewandt Mathematik, Universität Hannover.Google Scholar
Cooper, G. (1971), ‘Error bounds for numerical solutions of ordinary differential equations’, Num. Math. 18, 162–170.CrossRefGoogle Scholar
Eriksson, K. (1994), ‘An adaptive finite element method with efficient maximum norm error control for elliptic problems’, M3AS, to appear.Google Scholar
Eriksson, K., Estep, D., Hansbo, P. and Johnson, C. (1994a), Adaptive Finite Element Methods, North Holland, Amsterdam, in preparation.Google Scholar
Eriksson, , et al. (1994) Eriksson, K., Estep, D., Hansbo, P. and Johnson, C. (1994b), ‘Introduction to Numerical Methods for Differential Equations’, Department of Mathematics, Chalmers University of Technology, Göteborg.Google Scholar
Eriksson, K. and Johnson, C. (1988), ‘An adaptive finite element method for linear elliptic problems’, Math. Comput. 50, 361–383.CrossRefGoogle Scholar
Eriksson, K. and Johnson, C. (1991), ‘Adaptive finite element methods for parabolic problems I: A linear model problem’, SIAM J. Numer. Anal. 28, 43–77.Google Scholar
Eriksson, K. and Johnson, C. (1993), ‘Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems’, Math. Comp. 60, 167–188.Google Scholar
Eriksson, K. and Johnson, C. (1994a), ‘Adaptive finite element methods for parabolic problems II: Optimal error estimates in L∞L2 and L∞’, SIAM J. Numer Anal. to appear.Google Scholar
Eriksson, K. and Johnson, C. (1994b), ‘Adaptive finite element methods for parabolic problems III: Time steps variable in space’. in preparation.Google Scholar
Eriksson, K. and Johnson, C. (1994c), ‘Adaptive finite element methods for parabolic problems IV: Non-linear problems’, SIAM J. Numer Anal, to appear.Google Scholar
Eriksson, K. and Johnson, C. (1994d), ‘Adaptive finite element methods for parabolic problems V: Long-time integration’, SIAM J. Numer Anal. to appear.Google Scholar
Eriksson, K. and Johnson, C. (1994e), ‘Adaptive streamline diffusion finite element methods for time-dependent convection-diffusion problems’, Math. Comp, to appear.Google Scholar
Eriksson, K., Johnson, C. and Larsson, S. (1994), ‘Adaptive finite element methods for parabolic problems VI: Analytic semigroups’, Preprint, Department of Mathematics, Chalmers University of Technology, Göteborg.Google Scholar
Estep, D. (1994), ‘A posteriori error bounds and global error control for approxima tions of ordinary differential equations’, SIAM J. Numer. Anal. to appear.Google Scholar
Estep, D. and French, D. (1994), ‘Global error control for the continuous Galerkin finite element method for ordinary differential equations’, RAIRO M.M.A.N. to appear.Google Scholar
Estep, D. and Johnson, C. (1994a), ‘The computability of the Lorenz system’, Preprint #1994–33, Department of Mathematics, Chalmers University of Technology, Göteborg.Google Scholar
Estep, D. and Johnson, C. (1994b), ‘An analysis of quadrature in Galerkin finite element methods for ordinary differential equations’, in preparation.Google Scholar
Estep, D. and Larsson, S. (1993), ‘The discontinuous Galerkin method for semilinear parabolic problems’, RAIRO M.M.A.N. 27, 611–643.Google Scholar
Estep, D. and Stuart, A. (1994), ‘The dynamical behavior of Galerkin methods for ordinary differential equations and related quadrature schemes’, in preparation.Google Scholar
Estep, D. and Williams, R. (1994), ‘The structure of an adaptive differential equation solver’, in preparation.Google Scholar
Hansbo, P. and Johnson, C. (1991), ‘Adaptive streamline diffusion finite element methods for compressible flow using conservation variables’, Comput. Methods Appl. Mech. Engrg. 87, 267–280.Google Scholar
Johnson, C. (1988), ‘Error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations’, SIAM J. Numer. Anal. 25, 908–926.CrossRefGoogle Scholar
Johnson, C. (1990), ‘Adaptive finite element methods for diffusion and convection problems’, Comput. Methods Appl. Mech. Engrg. 82, 301–322.Google Scholar
Johnson, C. (1992a), ‘Adaptive finite element methods for the obstacle problem, Math. Models Methods Appl. Sci. 2, 483–487.CrossRefGoogle Scholar
Johnson, C. (1992b), ‘A new approach to algorithms for convection problems based on exact transport + projection’, Comput. Methods Appl. Mech. Engrg. 100, 45–62.CrossRefGoogle Scholar
Johnson, C. (1993a), ‘Discontinuous Galerkin finite element methods for second order hyperbolic problems’, Comput. Methods Appl. Mech. Engrg. 107, 117–129.CrossRefGoogle Scholar
Johnson, C. (1993b), ‘A new paradigm for adaptive finite element methods’, in Proc. Mafelap 93, Brunel Univ., Wiley Comput. Methods Appl. Mech. Engrg. vol. 107, Wiley, New York, 117–129.Google Scholar
Johnson, C. and Hansbo, P. (1992a), ‘Adaptive finite element methods for small strain elasto-plasticity’, in Finite Inelastic Deformations - Theory and Applications (Besdo, D. and Stein, E., eds.), Springer, Berlin, 273–288.Google Scholar
Johnson, C. and Hansbo, P. (1992b), ‘Adaptive finite element methods in computational mechanics’, Comput. Methods Appl. Mech. Engrg. 101, 143–181.CrossRefGoogle Scholar
Johnson, C. and Rannacher, R. (1994), ‘On error control in CFD’, in Proc from Conf. on Navier-Stokes Equations Oct 93, Vieweg, to appear.Google Scholar
Johnson, C., Rannacher, R. and Boman, M. (1994a), ‘Numerics and hydrodynamic stability: Towards error control in CFD’, SIAM J. Numer. Anal, to appear.Google Scholar
Johnson, C., Rannacher, R. and Boman, M. (1994b), ‘On transition to turbulence and error control in CFD’, Preprint #1994–26, Department of Mathematics, Chalmers University of Technology, Göteborg.Google Scholar
Johnson, C. and Szepessy, A. (1994), ‘Adaptive finite element methods for conservation laws based on a posteriori error estimates’, Comm. Pure Appl. Math. to appear.Google Scholar
Lippold, G. (1988), ‘Error estimates and time step control for the approximate solution of a first order evolution equation’, preprint, Akademie der Wissenschaften der Karl Weierstrass Institut für Mathematik, Berlin.Google Scholar
Nystedt, C. (1994), ‘Adaptive finite element methods for eigenvalue problems’, Licentiate Thesis, Department of Mathematics, Chalmers University of Technology, Göteborg in preparation.Google Scholar
Verfürth, R. (1989), ‘A posteriori error estimators for the Stokes equations’, Numer. Math. 55, 309–325.CrossRefGoogle Scholar
Zadunaisky, P. (1976), ‘On the estimation of errors propagated in the numerical integration of ordinary differential equations’, Numer. Math. 27, 21–39.CrossRefGoogle Scholar