A general output bound result: Application to discretization and iteration error estimation and control

Mathematical Models and Methods in Applied Sciences

Volumn 11, Issue 4, 2001, Pages 685-712

  Patera, Anthony T ^a   Rønquist, Einar M ^b  

^a USA   (United States)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

A posteriori error estimation; Conjugate gradient iteration; Finite element methods; Output bounds

Indexed keywords

EID: 0035588785     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202501001057     Document Type: Article

References (29)

1. R. A. Adams, Sobolev Spaces (Academic Press, 1975).
2. M. Ainsworth and J. T. Oden, A unified approach to a posteriori error estimation using element residual methods, Numer. Math. 65 (1993) 23-50.
3. S. F. Ashby, T. A. Manteuffel and P. E. Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal. 27(6) (1990) 1542-1568.
4. R. E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44(170) (1985) 283-301.
5. R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing 55 (1995) 271-288.
6. R. Becker and R. Rannacher, "Weighted a posteriori error control in finite element methods", IWR Preprint 96-1 (SFB 359) (Heidelberg, 1996).
7. R. Becker and R. Rannacher, A feedback approach to error control in finite element methods: basic analysis and examples, IWR Preprint 96-52 (SFB 359) (Heidelberg, 1996).
8. T. F. Chan and T. P. Mathew, Domain decomposition algorithms, Acta Numer. (1994) 61-143.
9. J. Elliott and J. Peraire, Practical 3D aerodynamic design and optimization using unstructured meshes, AIAA J. 35(9) (1997) 1479-1485.
10. M. B. Giles and N. A. Pierce, Superconvergent lift estimates through adjoint error analysis, Technical Memorandum (Oxford University Computing Laboratory, 1998).
11. G. H. Golub and G. Meurant, Matrices, moments and quadrature II; how to compute the norm of the error in iterative methods, BIT 37 (1997) 687-705.
12. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins Press, 1989), 2nd edition.
13. P. Ladeveze and D. Leguillon, Error estimation procedures in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983) 485-509.
14. R. M. Lewis, Numerical computation of sensitivites and the adjoint approach, in Proceedings Second International Workshop on Optimal Design and Control, eds. J. Borggaard, J. Burns, E. Cliff, and S. Schreck (Birkhäuser, 1998), pp. 285-302.
15. R. M. Lewis, A. T. Patera and J. Peraire, A posteriori finite element bounds for sensitivity derivatives of partial-differential-equation outputs, Finite Elements in Design, to appear.
16. L. Machiels, A. T. Patera, J. Peraire and Y. Maday, A general framework for finite element a posteriori error control: application to linear and nonlinear convection-dominated problems, in Proceedings ICFD Conference on Numerical Methods for Fluid Dynamics, to appear.
17. L. Machiels, J. Peraire and A. T. Patera, Output bound approximations for partial differential equations; application to the incompressible Navier-Stokes equations, in Proceedings Industrial and Environmental Applications of Direct and Large Eddy Numerical Simulation, eds. S. Biringen, H. Örs, A. Tezel and J. H. Ferziger (Springer-Verlag, 1998), pp. 93-108.
18. Y. Maday and A. T. Patera, Numerical analysis of a posteriori finite element bounds for linear-functional outputs, Math. Models Methods Appl. Sci., to appear.
19. Y. Maday, A. T. Patera and J. Peraire, A general formulation for a posteriori bounds for output functional of partial differential equations: application to the eigenvalue problem, C.R. Acad. Sci. Paris A328 (1999) 823-828.
20. J. Mandel and M. Brezina, Balancing domain decomposition: theory and performance in two and three dimensions, Technical Memorandum, Center for Computational Mathematics, University of Colorado at Denver, 1993.
21. G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numer. Alg. 16 (1997) 77-87.
22. M. Paraschivoiu and A. T. Patera, A hierarchical duality approach to bounds for the outputs of partial differential equations, Comp. Methods. Appl. Mech. Engrg. 158 (1998) 389-407.
23. M. Paraschivoiu and A. T. Patera, A posteriori bounds for linear-functional outputs of Crouzeix-Raviart finite element discretizations of the incompressible Stokes problem, Internat. J. Numer. Methods Fluids 32 (2000) 823-849.
24. M. Paraschivoiu, J. Peraire and A. T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comp. Methods Appl. Mech. Engrg. 150 (1997) 289-312.
25. J. Peraire and A. T. Patera, Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement, in Proceedings of the Workshop on New Advances in Adaptive Computational Methods in Mechanics, eds. P. Ladeveze and J. T. Oden (Elsevier, 1997).
26. J. Peraire and A. T. Patera, Asymptotic a posteriori finite element bounds for the outputs of noncoercive problems: the Helmholtz and Burgers equations, Comp. Methods Appl. Mech. Engrg. 171 (1999) 77-86.
27. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer-Verlag, 1994).
28. E. M. Rønquist, A domain decomposition solver for the steady Navier-Stokes equations, in Proceedings 3rd International Conference on Spectral and High-Order Methods for Partial Differential Equations, eds. A. Ilin and R. Scott (1996).
29. J. Sarrate, J. Peraire and A. T. Patera, A posteriori finite element error bounds for nonlinear outputs of the Helmholtz equation, J. Numer. Methods in Fluids, to appear.