Adjoint and defect error bounding and correction for functional estimates

Journal of Computational Physics

Volumn 200, Issue 2, 2004, Pages 769-794

  Pierce, Niles A ^a   Giles, Michael B ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Adjoint; Defect; Error bounding; Error correction; Functionals

Indexed keywords

DEFECTS; DRAG; ERROR ANALYSIS;

ADJOINT METHODS; ADJOINTS; ENTIRE SOLUTIONS; ERROR BOUNDING; ERRORS CORRECTION; ESTIMATION APPROACHES; EULER FLOWS; FUNCTIONAL ESTIMATE; FUNCTIONALS; RESIDUAL ERROR;

ERROR CORRECTION;

EID: 7244226418     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2004.05.001     Document Type: Article

References (56)

1. Honig B. Nicholls A. Classical electrostatics in biology and chemistry Science 268 1995 1144-1149
2. Monk P. Süli E. The adaptive computation of far field patterns by a posteriori error estimates of linear functionals SIAM J. Numer. Anal. 36 1 1998 251-274
3. M. Giles, N. Pierce, Improved lift and drag estimates using adjoint Euler equations, in: Proc. of the 14th AIAA Computational Fluid Dynamics Conference, 1999, AIAA Paper 99-3293
4. Pierce N. Giles M. Adjoint recovery of superconvergent funtionals from PDE approximations SIAM Rev. 42 2 2000 247-264
5. Giles M. Pierce N. Adjoint error correction for integral outputs Lecture Notes in Computational Science & Engineering: Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics vol. 25 2002 Springer Berlin
6. N. Pierce, M. Giles, Adjoint and defect error bounding and correction for functional estimates, in: Proc. of the 16th AIAA Computational Fluid Dynamics Conference, 2003, AIAA Paper 03-3846
7. Zadunaisky P. On the estimation of errors propagated in the numerical integration of ordinary differential equations Numer. Math. 27 1976 21-40
8. Stetter H. Economical global error estimation in: Willoughby R. (Ed.) Stiff Differential Systems 1974 Plenum Press New York
9. Frank R. The method of iterated defect-correction and its application to two-point boundary value problems Numer. Math. 25 1976 409-419
10. Böhmer K. Discrete Newton methods and iterated defect corrections Numer. Math. 37 1981 167-192
11. Giles M. Defect and adjoint error correction in: Satofuka N. (Ed.) Computational Fluid Dynamics 2000 2001 Springer Berlin
12. Venditti D. Darmofal D. Grid adaptation for functional outputs: application to two-dimensional inviscid flows J. Comput. Phys. 176 2002 40-69
13. Darmofal D. Venditti D. Anisotropic grid adaptation for functional outputs: Application to two-dimensional viscous flows J. Comput. Phys. 187 2003 22-46
14. M. Park, Three-dimensional turbulent RANS error correction, in: Proc. of the 16th AIAA Computational Fluid Dynamics Conference, 2003, AIAA Paper 03-3849
15. Babuška I. Miller A. The post-processing approach in the finite element method - part 1: Calculation of displacements, stresses and other higher derivatives of the displacements Int. J. Numer. Methods Eng. 20 1984 1085-1109
16. Babuška I. Miller A. The post-processing approach in the finite element method - part 2: The calculation of stress intensity factors Int. J. Numer. Methods Eng. 20 1984 1111-1129
17. Barrett J. Elliott C. Total flux estimates for a finite-element approximation of elliptic equations IMA J. Numer. Anal. 7 1987 129-148
18. Johnson C. Rannacher R. Boman M. Numerics and hydrodynamic stability - toward error control in computational fluid dynamics SIAM J. Numer. Anal. 32 4 1995 1058-1079
19. Paraschivoiu M. Peraire J. Patera A. A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations Comput. Methods Appl. Mech. Eng. 150 1-4 1997 289-312
20. Peraire J. Patera A. Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement in: Ladeveze P. Oden J. (Eds.) New Advances in Adaptive Computational Methods in Mechanics 1997 Elsevier Amsterdam
21. Oden J. Prudhomme S. New approaches to error estimation and adaptivity for the Stokes and Oseen equations Int. J. Numer. Methods Fluids 31 1 1999 3-15
22. Houston P. Rannacher R. Süli E. A posteriori error analysis for stabilised finite element approximations of transport problems Comput. Methods Appl. Mech. Eng. 190 11-12 2000 1483-1508
23. Rannacher R. Adaptive Galerkin finite element methods for partial differential equations J. Comput. Appl. Math. 1-2 2000 205-233
24. Becker R. Rannacher R. An optimal control approach to error control and mesh adaptation in: Iserles A. (Ed.) Acta Numerica 2001 2001 Cambridge University Press London
25. Giles M. Süli E. Adjoints methods for PDEs: A posteriori error analysis and postprocessing by duality in: Iserles A. (Ed.) Acta Numerica 2002 2002 145-236 Cambridge University Press London
26. Patera A. Peraire J. A general Lagrangian formulation for the computation of a posteriori finite element bounds Adaptive Finite Element Approximation of Hyperbolic Problems vol. 25 2002 Springer Berlin
27. Prudhomme S. Oden J. Computable error estimators and adaptive techniques for fluid flow problems Adaptive Finite Element Approximation of Hyperbolic Problems vol. 25 2002 Springer Berlin
28. Süli E. Houston P. Adjoint error correction for integral outputs Adaptive Finite Element Approximation of Hyperbolic Problems vol. 25 2002 Springer Berlin
29. Giles M. Pierce N. Analytic adjoint solutions for the quasi-one-dimensional Euler equations J. Fluid Mech. 426 2001 327-345
30. Giles M. Pierce N. On the properties of solutions of the adjoint Euler equations in: Baines M. (Ed.) Numerical Methods for Fluid Dynamics vol. VI 1998 ICFD
31. Ulbrich S. A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms SIAM J. Control and Optim. 41 3 2002 740-797
32. Ulbrich S. Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws Systems Control Lett. 48 3-4 2003 313-328
33. Giles M. Discrete adjoint approximations with shocks in: Hou T. Tadmor E. (Eds.) Proc. of the 9th International Conference on Hyperbolic Problems 2003 Springer Berlin
34. M. Giles, N. Pierce, Analysis of adjoint error correction for superconvergent functional estimates, Oxford University Computing Laboratory Report NA 01/14, 2001
35. Fox L. Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations Proc. Roy. Soc. London A 190 1947 31-59
36. Pereyra V. On improving an approximate solution of a functional equation by deferred corrections Numer. Math. 8 1966 376-391
37. Pereyra V. Iterated deferred corrections for nonlinear boundary value problems Numer. Math. 11 1968 111-125
38. Lindberg B. Error estimation and iterative improvement for discretization algorithms BIT 20 1980 486-500
39. Stetter H. The defect correction principle and discretization methods Numer. Math. 29 1978 425-443
40. Skeel R. A theoretical framework for proving accuracy results for deferred corrections SIAM J. Numer. Anal. 19 1981 171-196
41. Hackbusch W. Local defect correction method and domain decomposition techniques in: Böhmer K. Stetter H. (Eds.) Defect Correction Methods: Theory and Applications 1984 89-113 Springer Berlin
42. Böhmer K. Hemker P. Stetter H. The defect correction approach in: Böhmer K. Stetter H. (Eds.) Defect Correction Methods: Theory and Applications 1984 1-32 Springer Berlin
43. Jameson A. Optimum aerodynamic design using control theory Comput. Fluid Dyn. Rev. 3 1995 495-528
44. M. Giles, N. Pierce, Adjoint equations in CFD: Duality, boundary conditions and solution behavior, in: Proc. of the 13th Computational Fluid Dynamics Conference, Snowmass, CO, 1997, AIAA Paper 97-1850
45. Jameson A. Martinelli L. Pierce N. Optimum aerodynamic design using the Navier-Stokes equations Theoret. Comput. Fluid Dyn. 10 1998 213-237
46. Bender C. Orszag S. Advanced Mathematical Methods for Scientists and Engineers 1978 McGraw-Hill New York
47. Kevorkian J. Cole J. Perturbation Methods in Applied Mathetics Applied Math Series 34 1981 Springer Berlin
48. Dahlquist G. Björck A. Numerical Methods 1974 Prentice-Hall Englewood cliffs, NJ
49. Liepmann H. Roshko A. Elements of gas dynamics 1957 Wiley New York
50. A. Jameson, W. Schmidt, E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes, 1981, AIAA Paper 81-1259
51. Jameson A. Solution of the Euler equations by a multigrid method Appl. Math. Comput. 13 1983 327-356
52. Jameson A. Aerodynamic design via control theory J. Sci. Comput. 3 1988 233-260
53. Elliott J. Peraire J. Practical three-dimensional aerodynamic design and optimization using unstructured grids AIAA J. 35 1997 1479-1485
54. Nielsen E. Anderson W. Aerodynamic design optimization on unstructured meshes using the Navier-Stokes equations AIAA J. 37 1999 957-964
55. deBoor C. A Practical Guide to Splines 2001 Springer Berlin
56. Batchelor G. An Introduction to Fluid Dynamics 1967 Cambridge University Press London