Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Adaptive Lagrange–Galerkin methods for unsteady convection-diffusion problems
HTML articles powered by AMS MathViewer

by Paul Houston and Endre Süli;
Math. Comp. 70 (2001), 77-106
DOI: https://doi.org/10.1090/S0025-5718-00-01187-X
Published electronically: March 3, 2000

Abstract:

In this paper we derive an a posteriori error bound for the Lagrange–Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this a posteriori bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.
References
  • A.M. Baptista, E.E. Adams and P. Gresho. Benchmarks for the transport equation: the convection-diffusion forum and beyond. In, Lynch and Davies, editors, Quantitative Skill Assessment for Coastal Ocean Models, AGU Coastal and Estuarine Studies, 47:241–268, 1995.
  • R. Becker and R. Rannacher. Weighted a posteriori error control in FE methods. Technical Report $96$-$1$, Institut für Angewandte Mathematik, Universität Heidelberg, Heidelberg, Germany, 1996.
  • M. Bercovier and O. Pironneau. Characteristics and the finite element method. In T. Kawai, editor, Proceedings of the Fourth International Symposium on Finite Element Methods in Flow Problems, pp 67–73. North–Holland, 1982.
  • Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
  • E. Burman. Adaptive Finite Element Methods for Compressible Two-Phase Flow. PhD thesis, Chalmers University of Technology, Göteborg, 1998.
  • Jim Douglas Jr. and Thomas F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), no. 5, 871–885. MR 672564, DOI 10.1137/0719063
  • Kenneth Eriksson and Claes Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), no. 1, 43–77. MR 1083324, DOI 10.1137/0728003
  • K. Eriksson and C. Johnson. Adaptive streamline diffusion finite element methods for time dependent convection diffusion problems. Technical Report 1993-23, Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden, 1993.
  • Kenneth Eriksson, Don Estep, Peter Hansbo, and Claes Johnson, Introduction to adaptive methods for differential equations, Acta numerica, 1995, Acta Numer., Cambridge Univ. Press, Cambridge, 1995, pp. 105–158. MR 1352472, DOI 10.1017/S0962492900002531
  • P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
  • P. Hansbo and C. Johnson. Streamline diffusion finite element methods for fluid flow. von Karman Institute Lectures, 1995.
  • P. Houston. Lagrange–Galerkin Methods for Unsteady Convection-Diffusion Problems: A Posteriori Error Analysis and Adaptivity. PhD thesis, University of Oxford, 1996.
  • P. Houston, J. Mackenzie, E. Süli and G. Warnecke. A posteriori error analysis for numerical approximations of Friedrichs systems. Numer. Math. 82:433–470, 1999.
  • P. Houston, R. Rannacher and E. Süli. A posteriori error analysis for stabilised finite element approximations of transport problems. Comput. Methods Appl. Mech. Engrg. (to appear).
  • P. Houston and E. Süli. Adaptive Lagrange–Galerkin methods for unsteady convection-dominated diffusion problems. Oxford University Computing Laboratory Technical Report NA95/24, 1995 (http://www.comlab.ox.ac.uk/oucl/publications/natr/NA-95-24.html).
  • P. Houston and E. Süli. On the design of an artificial diffusion model for the Lagrange–Galerkin method on unstructured triangular grids. Oxford University Computing Laboratory Technical Report NA96/07, 1996 (http://www.comlab.ox.ac.uk/oucl/publications/natr/NA-96-07.html).
  • P. Houston and E. Süli. A posteriori error analysis for linear convection-diffusion problems under weak mesh regularity assumptions. Oxford University Computing Laboratory Technical Report NA97/03, 1997 (http://www.comlab.ox.ac.uk/oucl/publications/natr/NA-97-03.html).
  • P. Houston and E. Süli. Local mesh design for the numerical solution of hyperbolic problems. In M. Baines, editor, Numerical Methods for Fluid Dynamics VI, pp 17–30. ICFD, 1998.
  • O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309–332. MR 654100, DOI 10.1007/BF01396435
  • R. Sandboge. Adaptive Finite Element Methods for Reactive Flow Problems. PhD thesis, Chalmers University of Technology, Göteborg, 1996.
  • L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
  • E. Süli. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In D. Kröner, M. Ohlberger and C. Rohde, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Volume 5 of Lecture notes in Computational Science and Engineering, pp. 123–144. Springer–Verlag, 1998.
  • Endre Süli and Paul Houston, Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity, The state of the art in numerical analysis (York, 1996) Inst. Math. Appl. Conf. Ser. New Ser., vol. 63, Oxford Univ. Press, New York, 1997, pp. 441–471. MR 1628356
  • R. Verfürth. Error estimates for some quasi-interpolation operators. $\mbox {M}_2$AN, 33:695–713, 1999.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2000): 65M15, 65M25, 65M60
  • Retrieve articles in all journals with MSC (2000): 65M15, 65M25, 65M60
Bibliographic Information
  • Paul Houston
  • Affiliation: Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
  • MR Author ID: 635107
  • Email: Paul.Houston@mcs.le.ac.uk
  • Endre Süli
  • Affiliation: Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom
  • Email: Endre.Suli@comlab.ox.ac.uk
  • Received by editor(s): December 16, 1997
  • Received by editor(s) in revised form: January 4, 1999
  • Published electronically: March 3, 2000
  • Additional Notes: We acknowledge the financial support of the EPSRC (Grant GR/K76221).
  • © Copyright 2000 American Mathematical Society
  • Journal: Math. Comp. 70 (2001), 77-106
  • MSC (2000): Primary 65M15; Secondary 65M25, 65M60
  • DOI: https://doi.org/10.1090/S0025-5718-00-01187-X
  • MathSciNet review: 1681108