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Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems. (English) Zbl 0957.65085
The authors 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. Moreover, based on this error bound, the authors designed an adaptive algorithm to ensure global control of the error with respect to a predetermined tolerance.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
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[1] 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.
[2] 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. · Zbl 0968.65083
[3] 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. · Zbl 0508.76007
[4] 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. · Zbl 0804.65101
[5] E. Burman. Adaptive Finite Element Methods for Compressible Two-Phase Flow. PhD thesis, Chalmers University of Technology, Göteborg, 1998. · Zbl 1018.76024
[6] 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. · Zbl 0492.65051 · doi:10.1137/0719063 · doi.org
[7] 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. · Zbl 0732.65093 · doi:10.1137/0728003 · doi.org
[8] 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. · Zbl 0795.65074
[9] 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. · Zbl 0829.65122 · doi:10.1017/S0962492900002531 · doi.org
[10] P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. · Zbl 0766.35001
[11] P. Hansbo and C. Johnson. Streamline diffusion finite element methods for fluid flow. von Karman Institute Lectures, 1995.
[12] P. Houston. Lagrange-Galerkin Methods for Unsteady Convection-Diffusion Problems: A Posteriori Error Analysis and Adaptivity. PhD thesis, University of Oxford, 1996.
[13] 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. CMP 99:14 · Zbl 0935.65096
[14] 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). · Zbl 0970.65115
[15] 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).
[16] 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).
[17] 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).
[18] 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.
[19] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309 – 332. · Zbl 0505.76100 · doi:10.1007/BF01396435 · doi.org
[20] R. Sandboge. Adaptive Finite Element Methods for Reactive Flow Problems. PhD thesis, Chalmers University of Technology, Göteborg, 1996. · Zbl 0944.76038
[21] L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483 – 493. · Zbl 0696.65007
[22] 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.
[23] 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. · Zbl 0886.65104
[24] R. Verfürth. Error estimates for some quasi-interpolation operators. \(\mbox{M}_2\)AN, 33:695-713, 1999. · Zbl 0938.65125
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