Abstract
We present our general framework for implicit A posteriori finite element procedures to compute upper and lower bounds for functional outputs of finite element solutions. The approach is based on a finite element domain decomposition technique and the construction of an augmented Lagrangian, in which the objective is a “quadratic” energy re-formulation of the desired output, and the constraints are the finite element equilibrium equations and inter-subdomain continuity requirements. Bounds for the output, on a suitably refined “truth” mesh, are then obtained by appealing to a dual min-max relaxation, evaluated for Lagrange multipliers computed on a coarse “working” mesh. For coercive problems, the computed bounds are uniformly valid regardless of the size of the underlying coarse discretization. An extension of the basic method to deal with non-coercive problems is also presented. In this case, the bounds computed can only be proven to be asymmptotic although in practice, they are also found to be valid for very coarse meshes. Adaptive mesh procedures can be incorporated naturally into the bounding framework to produce optimized meshes that meet a target bound gap. The methods presented are illustrated with several application examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Ainsworth and T.J. Oden, A unified approach to a posteriori error estimation based on element residual methods, Numer. Math., 65:23–50, 1993.
M. Ainsworth and J.T. Oden, A posteriori Error Estimation in Finite ELement Analysis, Wiley-Interscience, 2000,
I. Babuška and W.C. Rheinboltd, A posteriori error estimates for the finite element method, Int. J. Num. Meth. in Egngr., 12:1597–1615, 1978.
R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44:283–301, 1985.
R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods, IWR Preprint 96–1 (SFB359), Heildelberg, 1996.
D. Bertsekas, Nonlinear Programming, Athena Scientific, 1995.
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer-Verlag, New York, 1991.
C. Johnson, U. Nävert and J. Pitkäranta, Finite Element Methods for Linear Hyperbolic Problems, Comp. Meth. in Appl. Mech. and Engrg., 45 (1984), pp. 285–312.
P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20:485–509, 1983.
L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Course of Theoretical Physiscs, vol. 6, Pergamon Press 1982.
L. Machiels, Y. Maday and A.T. Patera, A“Flux-Free” nodal Neumann subproblem approach to output bounds for partial differential equations, to appear in C.. Acad.Sci. Paris, 1999.
L. Machiels, A.T. Patera, J. Peraire and Y. Maday, A general framework for finitie element A posteriori error control: application to linear and nonlinear convection-dominated problems, ICFD Conference on Numerical Methods for Fluid Dynamics, Oxford, 1998.
L. Machiels, J. Peraire and A.T. Patera, A posteriori finite element output bounds for the incompressible Navier-Stokes equations; Application to a natural convection problem, J. Comp. Phys., 172, 401–425, 2001.
Y. Maday, A.T. Patera and J. Peraire, A general formulation for A posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem, C.. Acad.Sci. paris, t. 328, Serie I, p. 823–828, 1999.
M. Paraschivoiu and A.T. Patera, A hierarchy duality approach to bounds for the outputs of partial differential equations, Comput. Methods Appl. Mech. Engrg., 15 Ladevèze and J.T. Oden editors. Elsevier, 1998.
M. Paraschivoiu, J. Peraire and A.T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg., 150:289–312 (1997).
A.T. Patera and E. Ronquist, A general output bound result: application to discretization and iteration error estimation and control, Mathematical Models and Methods in Applied Science, to appear.
J. Peraire and A.T. Patera, Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adpative refinement, in Advances in Adaptive Computational Methods in Mechanics, P. Ladevèze and J.T. Oden editors. Elsevier, 1998.
J. Peraire and A.T. Patera, Asymmptotic a posteriori finite element bounds for the outputs of non-coercive problems: the Helmholtz and Burgers equations, Comp. Meth. Appl. Mech. Engrg., 171, 77–86, 1999.
C. Prud’homme, D. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bond methods, Journal of Fluids Engineering, 2002 (to appear) .
J. Sarrate, J. Peraire and A.T. Patera, A posteriori bounds for non-linear outputs of solutions of the Helmholtz equation, Int. J. Num. Meth. Fluids, 31, 17–36, 1999.
G. Strang, Introduction to applied mathematics, Wellesley-Cambridge Press, 1986.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Patera, A.T., Peraire, J. (2003). A General Lagrangian Formulation for the Computation of A Posteriori Finite Element Bounds. In: Barth, T.J., Deconinck, H. (eds) Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. Lecture Notes in Computational Science and Engineering, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05189-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-662-05189-4_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07841-5
Online ISBN: 978-3-662-05189-4
eBook Packages: Springer Book Archive