A posteriori error estimation for acoustic wave propagation problems

Archives of Computational Methods in Engineering

Volumn 12, Issue 4, 2005, Pages 343-389

  Oden, J Tinsley ^a   Prudhomme, Serge ^a   Demkowicz, Leszek ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 32044460438     PISSN: 11343060     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02736190     Document Type: Review

References (102)

1. M. Ainsworth (2004). Discrete dispersion relation for hp-veision finite element approximation at high wave number. SIAM Journal on Numerical Analysis, 42 (2), 553-575.
2. M. Ainsworth, D. W. Kelly, I.H. Sloan and S.L. Wang (1997). Post-processing with computable error bounds for the finite element approximation of a nonlinear heat conduction problem. IMA J. Numer. Anal., 17, 547-561.
3. M. Ainsworth and J.T. Oden (1993). A unified approach to a posteriori error estimation using element residual methods. Numer. Math., 65(1), 23-50.
4. M. Ainsworth and J.T. Oden (1997). A posteriori error estimation in finite element analysis. Comput. Meth. Appl. Mech. Engrg., 142(1-2), 1-88.
5. M. Ainsworth and J.T. Oden (2000). A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons.
6. J.-P. Aubin (1967). Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin and finite difference methods. Ann. Scuola. Norm. Sup. Pisa, 3, 599-637.
7. I. Babuška (1971). Error bounds for finite element method. Numer. Math., 16, 322-333.
8. I. Babuška, F. Ihlenburg, E.T. Paik and S.A. Sauter (1995). A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Methods Appl. Mech. Engrg., 128, 325-359.
9. I. Babuška, F. Ihlenburg, T. Strouboulis, and S.K. Gangaraj (1997). A posteriori error estimation for finite element solutions of Helmholtz' equation. Part I: The quality of local indicators and estimators. Int. J. Numer. Methods Engrg., 40(18), 3443-3462.
10. I. Babuška, F. Ihlenburg, T. Strouboulis and S.K. Gangaraj (1997). A posteriori error estimation for finite element solutions of Helmholtz' equation. Part II: Estimation of the pollution error. Int. J. Numer. Methods Engrg., 40(21), 3883-3900.
11. I. Babuška and J.M. Melenk (1997). Partition of unity method. Int. J. Numer. Methods Engrg., 40(4), 727-758.
12. I. Babuška and A. Miller (1984). 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 Engrg., 20, 1085-1109.
13. I. Babuška and A. Miller (1984). The post-processing approach in the finite element method -Part 2: The calculation of stress intensity factors. Int. J. Numer. Methods Engrg., 20, 1111-1129.
14. I. Babuška and A. Miller (1984). The post-processing approach in the finite element method - Part 3: A-posteriori error estimates and adaptive mesh selection. Int. J. Numer. Methods Engrg., 20, 2311-2324.
15. I. Babuška and A. Miller (1987). A feedback finite element method with a posteriori error estimation Part 1. Comput. Meth. Appl. Mech. Engrg., 61, 1-40.
16. I. Babuška and J.E. Osborn (1983). Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal., 20, 510-536.
17. I. Babuška and W. Rheinboldt (1978). A posteriori error estimates for the finite element method. Int. J. Numer. Methods Engrg., 12, 1597-1615.
18. I. Babuška and W. Rheinboldt (1979). On the reliability and optimality of the finite element method. Computers & Structures, 10, 87-94.
19. I. Babuška and T. Strouboulis (2001). The Finite Element Method and its Reliability. Oxford University Press.
20. I. Babuška, T. Strouboulis and S.K. Gangaraj (1999). Guaranteed computable bounds for the exact error in the finite element solution Part I: One-dimensional model problem. Comput. Meth. Appl. Mech. Engrg., 176, 51-79.
21. I. Babuška, T. Strouboulis, A. Mathur and C.S. Upadhyay (1994). Pollution error in the h-version of the FEM and the local quality of a posteriori error estimators. Finite Elem. Anal. and Design, 273-321.
22. I. Babuška, T. Strouboulis, C.S. Upadhyay and S.K. Gangaraj (1995). A model study of element residual estimators for linear elliptic problems: The quality of the estimators in the interior of meshes of triangles and quadrilaterals. Computers & Structures, 57(6), 1009-1028.
23. I. Babuška, T. Strouboulis, C.S. Upadhyay and S.K. Gangaraj (1995). A posteriori estimation and adaptive control of the pollution error in the h-version of the finite element method. Int. J. Numer. Methods Engrg., 38, 4207-4235.
24. I. Babuška, T. Strouboulis, C.S. Upadhyay, S.K. Gangaraj and K. Copps (1994). Validation of a posteriori error estimators by numerical approach. Int. J. Numer. Methods Engrg., 37, 1073-1123.
25. I. M. Babuška and S.A. Sauter (1997). Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal., 34(6), 2392-2423.
26. I. M. Babuška and S.A. Sauter (2000). Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Review, 42(3), 451-484.
27. W. Bangerth and R. Rannacher (1999). Finite element approximation of the acoustic wave equation: Error control and mesh refinement. East-West J. Numer. Math., 7(4), 263-282.
28. W. Bangerth and R. Rannacher (2003). Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel.
29. R.E. Bank and A. Weiser (1985). Some a posteriori error estimators for elliptic partial differential equations. Math. Comp., 44(170), 283-301.
30. R. Becker and R. Rannacher (1996). A feedback approach to error control in finite elements methods: Basic analysis and examples. East-West J. Numer. Math., 4, 237-264.
31. R. Becker and R. Rannacher (2001). An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 19, 1-102.
32. P. Bouillard (1999). Admissible fields and error estimation for acoustic FEA with low wavenumbers. Computers and Structures, 73(1-5), 227-237.
33. P. Bouillard (1999). Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation. Int. J. Numer. Methods Engrg., 45(7), 783-800.
34. P. Bouillard and F. Ihlenburg (1998). Error estimation and adaptivity for the finite element method in acoustics. In Ladevèze and Oden, editors, Advances in Adaptive Computational Methods in Mechanics, pp. 477-492. Elsevier, Amsterdam.
35. P. Bouillard and F. Ihlenburg (1999). Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications. Comput. Methods Appl. Mech. Engrg., 176(1), 147-163.
36. A. Buffa, M. Costabel and D. Sheen (2002). On traces for H(curl, Ω) in Lipschitz domains. J. Math. Anal. Appl., 276(2), 845-867.
37. A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab (2003). Boundary element methods for Maxwell transmission problems in Lipschitz domains. Numer. Math., 95 (3), 459-485.
38. T. Cao, D.W. Kelly, and I.H. Sloan (1999). Local error bounds for post-processed finite element calculations. Int. J. Numer. Methods Engrg., 45, 1085-1098.
39. C. Carstensen and S.A. Funken (2000). Fully reliable localized error control in the FEM. SIAM J. Sci. Comput., 21(4), 1465-1484.
40. Y.C. Chang and L. Demkowicz (1998). A stability analysis for vibrating viscoelastic spherical shells. Taiwanese Journal of Mathematics, 2(2), 213-242.
41. Y.C. Chang and L. Demkowicz (1999). Solution of viscoelastic scattering problems using coupled hp BE/FE methods. Int. J. Numer. Methods Engrg., 44(12), 1885-1907.
42. L. Demkowicz (1994). Asymptotic convergence in finite and boundary element methods. Part 2: The LBB constant for rigid and elasctic problems. Computers & Mathematics with Applications, 28(6), 93-109.
43. L. Demkowicz (2003). hp-adaptive finite elements for time-harmonic Maxwell equations. In M. Ainsworth, P. Davies, D. Duncan, P. Martin and B. Rynne (Eds.), Topics in Computational Wave Propagation, Lecture Notes in Computational Science and Engineering. Springer Verlag, Berlin.
44. L. Demkowicz, J.T. Oden, M. Ainsworth and P. Geng (1991). Solution of elastic scattering problems in linear acoustics using h-p boundary element methods. Journal of Computational and Applied Mathematics, 36, 29-63.
45. L. Demkowicz, W. Rachowicz and P. Devloo (2002). A fully automatic hp-adaptivity. Journal of Scientific Computing, 17(1-3), 127-155.
46. A. Deraemaeker, I. Babuška and P. Bouillard (1999). Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int. J. Numer. Methods Engrg., 46(4), 471-499.
47. C. A. Duarte and J. T. Oden (1996). h-p clouds - an h-p meshless method. Numer. Methods Partial Differential Equations, 12(6), 673-705.
48. A. Ern and J.-L. Guermond (2002). Eléments Finis: Théorie, Applications, Mise en Oeuvre, volume 36 of Mathématiques & Applications. Springer, Paris.
49. E. C. Gartland, Jr. (1984). Computable pointwise error bounds and the Ritz method in one dimension. SIAM J. Numer. Anal., 21(1), 84-100.
50. P. Geng, J.T. Oden and L. Demkowicz (1996). Solution of exterior acoustic problems by the boundary element method at high wave number, error estimation and parallel computation. Journal of Acoustical Society of America, 100(1), 335-345.
51. K. Gerdes and F. Ihlenburg (1999). On the pollution effect in FE solutions of the 3D-Helmholtz equation. Comput. Methods Appl. Mech. Engrg., 170(1-2), 155-172.
52. F.M. Guerra (1977). Finite element analysis for the adaptive method of rezoning. PhD thesis, The University of Texas at Austin.
53. I. Harari and D. Avraham (1997). High-order finite element methods for acoustic problems. J. Comput. Acoust., 5(1), 33-51.
54. I. Harari and T.J.R. Hughes (1991). Finite element methods for the Helmholtz equation in an exterior domain. Model problems. Comput. Methods Appl. Mech. Engrg., 87(1), 59-96.
55. I. Harari and T.J.R. Hughes (1992). Galerkin least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains. Comput. Methods Appl. Mech. Engrg., 98(3), 411-454.
56. J. Hoffman and C. Johnson (2003). Adaptive finite element methods for incompressible fluid flow. In T. Barth and H. Deconinck (Eds.), Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Volume 25 of Lecture Notes in Computational Science and Engineering, pp. 97-157, Springer-Verlag, Heidelberg.
57. F. Ihlenburg (1998). Finite Element Analysis of Acoustic Scattering, Volume 132 of Applied Mathematical Sciences. Springer-Verlarg, New York.
58. F. Ihlenburg and I. Babuška (1995). Finite element solution of the helmholtz equation with high wave number Part I: The h-version of the FEM. Computers & Mathematics with Applications, 30(9), 9-37.
59. F. Ihlenburg and I. Babuška (1997). Finite element solution of the helmholtz equation with high wave number Part II: The h-p version of the FEM. SIAM J. Numer. Anal., 34(1), 315-358.
60. S. Irimie and P. Bouillard (2001). A residual a posteriori error estimator for the finite element solution of the Helmholtz equation. Comput. Methods Appl. Mech. Engrg., 190(31), 4027-4042.
61. P. Ladevèze and D. Leguillon (1983). Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal., 20, 485-509.
62. P. Ladevèze and E.A.W. Maunder (1996). A general method for recovering equilibrating element tractions. Comput. Meth. Appl. Mech. Engrg., 137, 111-151.
63. P. Ladevèze and J.-P. Pelle (2001). La Maîtrise du Calcul en Mécanique Linéaire et Non Linéaire. Hermès.
64. P. Ladevèze and P. Rougeot (1997). New advances on a posteriori error on constitutive relation in f.e. analysis. Comput. Meth. Appl. Mech. Engrg., 150(1-4), 239-249.
65. L. Machiels, Y. Maday and A.T. Patera (2000). A "flux-free" nodal Neumann subproblem approach to output bounds for partial differential equations. C. R. Acad. Sci. Paris Sér. I Math., 330(3), 249-254.
66. P. Morin, R. H. Nochetto and K.G. Siebert (2003). Local problems on stars: A posteriori error estimators, convergence, and performance. Math. Comp., 72(243), 1067-1097.
67. J. Nitsche (1968). Ein kriterium für die quasi-optimalität des ritzschen Verfahrens. Numer. Math., 11, 346-348.
68. Q. Niu and M.S. Shephard (1993). Superconvergent extraction techniques for finite element analysis. Int. J. Numer. Methods Engrg., 36, 811-836.
69. Q. Niu and M.S. Shephard (1994). Superconvergent boundary stress extraction and some experiments with adaptive pointwise error control. Int. J. Numer. Methods Engrg., 37, 877-891.
70. J.T. Oden and L. Demkowicz (1989). A survey of adaptive finite element methods in computational mechanics. In A. K. Noor and J. T. Oden (Eds.), State-of-the-Art Surveys on Computational Mechanics. The American Society of Mechanical Engineers, New York.
71. J.T. Oden, L. Demkowicz, T. Westermann and W. Rachowicz (1989). Toward a universal h-p adaptive finite element strategy. Part 2. A posteriori error estimates. Comput. Meth. Appl. Mech. Engrg., 77, 79-120.
72. J.T. Oden, C.A.M. Duarte, and O.C. Zienkiewicz (1998). A new cloud-based hp finite element method. Comput. Methods Appl. Mech. Engrg., 153(1-2), 117-126.
73. J.T. Oden and S. Prudhomme (2001). Goal-oriented error estimation and adaptivity for the finite element method. Computers Math. Applic., 41(5-6), 735-756.
74. J.T. Oden and S. Prudhomme (2002). Estimation of modeling error in computational mechanics. Journal of Computational Physics, 182, 496-515.
75. J.T. Oden, S. Prudhomme, T. Westermann, J. Bass and M.E. Botkin (2003). Estimation of eigenfrequencies for elasticity and shell problems. Math. Models Methods Appl. Sci., 13(3), 323-344.
76. J.T. Oden and J.N. Reddy (1976). An Introduction to the Mathematical Theory of Finite Elements. John Wiley & Sons, New-York.
77. M. Paraschivoiu and A.T. Patera (1998). A hierarchical duality approach to bounds for the outputs of partial differential equations. Comput. Meth. Appl. Mech. Engrg., 158, 389-407.
78. M. Paraschivoiu and A.T. Patera (2000). A posteriori bounds for linear functional outputs of Crouzeix-Raviart finite element discretizations of the incompressible Stokes problem. Int. J. Numer. Methods Fluids, 32(7), 823-849.
79. M. Paraschivoiu, J. Peraire, and A.T. Patera (1997). A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comput. Meth. Appl. Mech. Engrg., 150, 289-312.
80. D. Pardo and L. Demkowicz (2002). Integration of hp-adaptivity and multigrid. I. A two grid solver for hp finite elements. TICAM Report 02-33, The University of Texas at Austin.
81. A.T. Patera and J. Peraire (2003). A general Lagrangian formulation for the computation of a posteriori finite element bounds. In T. Barth and H. Deconinck (Eds.), Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Volume 25 of Lecture Notes in Computational Science and Engineering, pp. 159-206, Springer-Verlag, Heidelberg.
82. J. Peraire and A.T. Patera (1998). Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In P. Ladevèze and J. T. Oden (Eds.), Advances in Adaptive Computational Methods in Mechanics, pp. 199-215, Elsevier, Amsterdam.
83. J. Peraire and A.T. Patera (1999). Asymptotic a posteriori finite element bounds for the outputs of noncoercive problems: The Helmholtz and Burgers equations. Comput. Methods Appl. Mech. Engrg., 171(1), 77-86.
84. S. Prudhomme, F. Nobile, L. Chamoin, and J.T. Oden (2004). Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numerical Methods for Partial Differential Equations, 20(2), 165-326.
85. S. Prudhomme and J.T. Oden (1999). On goal-oriented error estimation for elliptic problems: Application to the control of pointwise errors. Comput. Meth. Appl. Mech. Engrg., 176, 313-331.
86. S. Prudhomme and J.T. Oden (2003). Computable error estimators and adaptive techniques for fluid flow problems. In T. Earth and H. Deconinck (Eds.), Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Volume 25 of Lecture Notes in Computational Science and Engineering, pp. 207-268, Springer-Verlag, Heidelberg.
87. S. Prudhomme, J.T. Oden, T. Westermann, J. Bass and M.E. Botkin (2003). Practical methods for a posteriori error estimation in engineering applications. Int. J. Numer. Methods Engrg., 56(8), 1193-1224.
88. J. Sarrate, J. Peraire, and A.T. Patera (1999). A posteriori finite element error bounds for non-linear outputs of the Helmholtz equation. International Journal for Numerical Methods in Fluids, 31(1), 17-36.
89. H. Schulz, C. Schwab and W. Wendland (1998). The computation of potentials near and on the boundary by an extraction technique for BEM. Comput. Methods Appl. Mech. Engrg., 157(3-4), 225-238.
90. P. Solin and L. Demkowicz (2002). Goal-oriented hp-adaptivity for elliptic problems. TICAM Report 02-32, The University of Texas at Austin.
91. J. R. Stewart and T. J.R. Hughes (1996). Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation. Computation of the constant and new measures of error estimator quality. Comput. Methods Appl. Mech. Engrg., 131(3-4), 335-363.
92. T. Strouboulis, I. Babuška, D.K. Datta, K. Copps and S.K. Gangaraj (2000). A posteriori estimation and adaptive control of the error in the quantity of interest. Part I: A posteriori estimation of the error in the von Mises stress and the stress intensity factor. Comput. Meth. Appl. Mech. Engrg., 181, 261-294.
93. T. Strouboulis, I. Babuška and S.K. Gangaraj (2000). Guaranteed computable bounds for the exact error in the finite element solution - Part II: bounds for the energy norm of the error in two dimensions. Int. J. Numer. Methods Engrg., 47, 427-475.
94. E. Süli and P. Houston (2003). Adaptive finite element approximation of hyperbolic problems. In T. Barth and H. Deconinck, editors, Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Volume 25 of Lecture Notes in Computational Science and Engineering, pp. 269-344, Springer-Verlag, Heidelberg.
95. L.L. Thompson and P.M. Pinsky (1994). Complex wavenumber Fourier analysis of the p-version finite element method. Comput. Mech., 13, 255-275.
96. R. Verfürth (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-refinement Techniques. Wiley-Teubner, Stuttgart.
97. T. Walsh and L. Demkowicz (2003). hp boundary element modeling of the external human auditory system - Goal oriented adaptivity with multiple load vectors. Comput. Meth. Appl. Mech. Engrg., 192(1-2), 125-146.
98. S. Wang, I.H. Sloan, and D.W. Kelly (1998). Computable error bounds for pointwise derivatives of a neumann problem. IMA J. Numer. Anal., 18, 251-271.
99. J.D. Yang, D.W. Kelly, and J.D. Isles (1993). A posteriori pointwise upper bound error estimates in the finite element method. Int. J. Numer. Methods Engrg., 36, 1279-1298.
100. O.C. Zienkiewicz and J.Z. Zhu (1987). A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Engrg., 24, 337-357.
101. O.C. Zienkiewicz and J.Z. Zhu (1992). The superconvergent patch recovery and a posteriori error estimates. Part 1: the recovery technique. Int. J. Numer. Methods Engrg., 33, 1331-1364.
102. O.C. Zienkiewicz and J.Z. Zhu (1992). The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. Int. J. Numer. Methods Engrg., 33, 1365-1382.