Local and parallel finite element algorithms based on two-grid discretizations

Mathematics of Computation

Volumn 69, Issue 231, 2000, Pages 881-909

  Xu, Jinchao ^a   Zhou, Aihui ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

Adaptive; Finite elements; Local a priori and a posteriori error estimates; Nonsymmetric; Parallel algorithm; Two grid method

Indexed keywords

EID: 0040735802     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0025-5718-99-01149-7     Document Type: Article

References (103)

1. Adams R.A.(1975): Sobolev Spaces, Academic Press, New York. MR 56:9247
2. Adams R.A.(1975): Sobolev Spaces, Academic Press, New York. MR 56:9247
3. Ainsworth, M. and Oden, J.T.(1993): A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 23-50. MR 95a:65185
4. Ainsworth, M. and Oden, J.T.(1993): A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 23-50. MR 95a:65185
5. Axelsson, O. and Layton, W.(1996): A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal., 33, 2359-2374. MR 98c:65181
6. Axelsson, O. and Layton, W.(1996): A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal., 33, 2359-2374. MR 98c:65181
7. Babuska, I., Duran, R. and Rodriguez, R.(1992): Analysis of the efficiency of an posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal., 29, 947-946. MR 93d:65096
8. Babuska, I., Duran, R. and Rodriguez, R.(1992): Analysis of the efficiency of an posteriori error estimator for linear triangular finite elements, SIAM J. Numer. Anal., 29, 947-946. MR 93d:65096
9. Babuska, I. and Rheinboldt, C.(1978): Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754. MR 58:3400
10. Babuska, I. and Rheinboldt, C.(1978): Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15, 736-754. MR 58:3400
11. Babuska, I., Zienkiewicz, O.C., Gago, J. and Oliveira, E.R. de A. (eds.)(1986): Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York. MR 87j:65004
12. Babuska, I., Zienkiewicz, O.C., Gago, J. and Oliveira, E.R. de A. (eds.)(1986): Accuracy Estimates and Adaptive Refinements in Finite Element Computations, Wiley, New York. MR 87j:65004
13. Babuska, I., Strouboulis, T. and Gangaraj, S.K.(1997): A posteriori estimation of the error in the recovered derivatives of the finite element solution, Comput. Methods Appl. Mech. Engrg., 150, 369-396. CMP 98:06
14. Babuska, I., Strouboulis, T. and Gangaraj, S.K.(1997): A posteriori estimation of the error in the recovered derivatives of the finite element solution, Comput. Methods Appl. Mech. Engrg., 150, 369-396. CMP 98:06
15. Babuska, I., Strouboulis, T., Gangaraj, S.K. and Upadhyay, C.S.(1997): Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives, Comput. Methods Appl. Mech. Engrg., 140, 1-37. MR 97i:73092
16. Babuska, I., Strouboulis, T., Gangaraj, S.K. and Upadhyay, C.S.(1997): Pollution error in the h-version of the finite element method and the local quality of the recovered derivatives, Comput. Methods Appl. Mech. Engrg., 140, 1-37. MR 97i:73092
17. Babuska, I., Strouboulis, T. and Upadhyay, C.S.(1994): A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles, Comput. Methods Appl.Mech. Engrg., 114, 307-378. MR 95d:65093
18. Babuska, I., Strouboulis, T. and Upadhyay, C.S.(1994): A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles, Comput. Methods Appl.Mech. Engrg., 114, 307-378. MR 95d:65093
19. Bank, R.E.(1996): Hierarchical bases and the finite element method, Acta Numerica, 5, 1-43. CMP 98:14
20. Bank, R.E.(1996): Hierarchical bases and the finite element method, Acta Numerica, 5, 1-43. CMP 98:14
21. Bank, R.E.(1998): A simple analysis of some a posteriori error estimates, Appl. Numer. Math., 26, 153-164. CMP 98:08
22. Bank, R.E.(1998): A simple analysis of some a posteriori error estimates, Appl. Numer. Math., 26, 153-164. CMP 98:08
23. Bank, R.E. and Holst, M.(1998): A new paradigm for parallel adaptive meshing algorithms (manuscript).
24. Bank, R.E. and Smith, R.K.(1993): A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., 30, 921-935. MR 95f:65212
25. Bank, R.E. and Smith, R.K.(1993): A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal., 30, 921-935. MR 95f:65212
26. Bank, R.E. and Smith, R.K.(1997): Mesh smoothing using a posteriori error estimates, SIAM J. Numer. Anal., 34, 979-997. MR 98M:65162
27. Bank, R.E. and Smith, R.K.(1997): Mesh smoothing using a posteriori error estimates, SIAM J. Numer. Anal., 34, 979-997. MR 98M:65162
28. Bank, R.E. and Weiser, A.(1985): Some a posteriori error estimates for elliptic partial differential equations, Math. Comp., 44, 283-301. MR 86g:65207
29. Bank, R.E. and Weiser, A.(1985): Some a posteriori error estimates for elliptic partial differential equations, Math. Comp., 44, 283-301. MR 86g:65207
30. Bedivan, D.M.(1995): A two-grid method for solving elliptic problems with inhomogeneous boundary conditions, Comput. Math. Appl., 29, 59-66. MR 95k:65103
31. Bedivan, D.M.(1995): A two-grid method for solving elliptic problems with inhomogeneous boundary conditions, Comput. Math. Appl., 29, 59-66. MR 95k:65103
32. Blum, H., Lin, Q. and Rannacher, R.(1986): Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math., 49, 11-38. MR 87m:65172
33. Blum, H., Lin, Q. and Rannacher, R.(1986): Asymptotic error expansion and Richardson extrapolation for linear finite elements, Numer. Math., 49, 11-38. MR 87m:65172
34. Bornemann, F.A., Erdmann, B. and Kornhuber, R.(1996): A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal., 33, 1188-1204. MR 98a:65161
35. Bornemann, F.A., Erdmann, B. and Kornhuber, R.(1996): A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal., 33, 1188-1204. MR 98a:65161
36. Bramble, J.H.(1993): Multigrid Methods, Pitman Research Notes in Mathematics, 294, London Co-published in the USA with Wiley, New York. MR 95b:65002
37. Bramble, J.H.(1993): Multigrid Methods, Pitman Research Notes in Mathematics, 294, London Co-published in the USA with Wiley, New York. MR 95b:65002
38. Bramble, J.H., Ewing, R.E., Parashkevov, R.R. and Pasciak, J.E.(1992): Domain decomposition methods for problems with partial refinement, SIAM J. Sci. Stat. Comp., 13, 397-410. MR 92i:65179
39. Bramble, J.H., Ewing, R.E., Parashkevov, R.R. and Pasciak, J.E.(1992): Domain decomposition methods for problems with partial refinement, SIAM J. Sci. Stat. Comp., 13, 397-410. MR 92i:65179
40. Bramble, J.H., Ewing, R.E., Pasciak, J.E. and Schatz, A.H.(1988): A preconditioning technique for the efficient solution of problems with local grid refinement, Comp. Meth. Appl. Mech. Eng., 67, 149-159.
41. Chan, T. and Mathew, T.(1994): Domain decomposition algorithms, Acta Numerica, 3, 61-143. MR 95f:65214
42. Chan, T. and Mathew, T.(1994): Domain decomposition algorithms, Acta Numerica, 3, 61-143. MR 95f:65214
43. Ciarlet, P.G. and Lions J.L.(1991): Handbook of Numerical Analysis, Vol.II, Finite Element Methods (Part I), North-Holland. MR 92f:65001
44. Ciarlet, P.G. and Lions J.L.(1991): Handbook of Numerical Analysis, Vol.II, Finite Element Methods (Part I), North-Holland. MR 92f:65001
45. Dawson, C.N. and Wheeler, M.F.(1994): Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. Math., 180, 191-203. MR 95j:65117
46. Dawson, C.N. and Wheeler, M.F.(1994): Two-grid methods for mixed finite element approximations of nonlinear parabolic equations, Contemp. Math., 180, 191-203. MR 95j:65117
47. Dawson, C.N., Wheeler, M.F. and Woodward, C.S.(1998): A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal., 35, 435-452. MR 99b:65097
48. Dawson, C.N., Wheeler, M.F. and Woodward, C.S.(1998): A two-grid finite difference scheme for nonlinear parabolic equations, SIAM J. Numer. Anal., 35, 435-452. MR 99b:65097
49. Eriksson, K., Estep, D., Hansbo, P. and Johnson, C.(1995): Introduction to adaptive methods for differential equations, Acta Numerica, 105-158. MR 96k:65057
50. Eriksson, K., Estep, D., Hansbo, P. and Johnson, C.(1995): Introduction to adaptive methods for differential equations, Acta Numerica, 105-158. MR 96k:65057
51. Eriksson, K., Estep, D., Hansbo, P. and Johnson, C.(1996): Computational Differential Equations, Cambridge University Press. MR 97m:65006
52. Eriksson, K., Estep, D., Hansbo, P. and Johnson, C.(1996): Computational Differential Equations, Cambridge University Press. MR 97m:65006
53. Eriksson, K. and Johnson, C.(1991): Adaptive finite element methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28, 43-77. MR 91m:65274
54. Eriksson, K. and Johnson, C.(1991): Adaptive finite element methods for parabolic problems I: a linear model problem, SIAM J. Numer. Anal., 28, 43-77. MR 91m:65274
55. Eriksson, K. and Johnson, C.(1995): Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Numer. Anal., 32, 1729-1749. MR 96i:65081
56. Eriksson, K. and Johnson, C.(1995): Adaptive finite element methods for parabolic problems IV: Nonlinear problems, SIAM J. Numer. Anal., 32, 1729-1749. MR 96i:65081
57. Grisvard, P.(1985): Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA. MR 86m:35044
58. Grisvard, P.(1985): Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA. MR 86m:35044
59. Hackbusch, W.(1985): Multigrid Methods and Applications, Springer, New York. MR 87e:65082
60. Hackbusch, W.(1985): Multigrid Methods and Applications, Springer, New York. MR 87e:65082
61. Johnson, C.(1990): Adaptive finite element methods for diffusion and convection problems, Comp. Methods Appl. Mech. Engrg., 82, 301-322. MR 91k:65134
62. Johnson, C.(1990): Adaptive finite element methods for diffusion and convection problems, Comp. Methods Appl. Mech. Engrg., 82, 301-322. MR 91k:65134
63. Layton, W. and Lenferink, W.(1995): Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comp., 69, 263-274. MR 95m:65191
64. Layton, W. and Lenferink, W.(1995): Two-level Picard and modified Picard methods for the Navier-Stokes equations, Appl. Math. Comp., 69, 263-274. MR 95m:65191
65. Marion, M. and Xu, J.(1995): Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal., 32, 1170-1184. MR 96f:65136
66. Marion, M. and Xu, J.(1995): Error estimates on a new nonlinear Galerkin method based on two-grid finite elements, SIAM J. Numer. Anal., 32, 1170-1184. MR 96f:65136
67. Nitsche, J. and Schatz, A.H.(1974): Interior estimates for Ritz-Galerkin methods, Math. Comp., 28, 937-955. MR 51:9525
68. Nitsche, J. and Schatz, A.H.(1974): Interior estimates for Ritz-Galerkin methods, Math. Comp., 28, 937-955. MR 51:9525
69. Nochetto, R. H.(1995): Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp., 64, 1-22. MR 95c:65172
70. Nochetto, R. H.(1995): Pointwise a posteriori error estimates for elliptic problems on highly graded meshes, Math. Comp., 64, 1-22. MR 95c:65172
71. Rannacher, R. and Scott, R.(1982): Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38, 437-445. MR 83e:65180
72. Rannacher, R. and Scott, R.(1982): Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38, 437-445. MR 83e:65180
73. Schatz, A.H.(1998): Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates, Math. Comp., 67, 877-899. MR 98j:65082
74. Schatz, A.H.(1998): Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates, Math. Comp., 67, 877-899. MR 98j:65082
75. Schatz, A.H. and Wahlbin, L.B.(1977): Interior maximum-norm estimates for finite element methods, Math. Comp., 31, 414-442. MR 55:4748
76. Schatz, A.H. and Wahlbin, L.B.(1977): Interior maximum-norm estimates for finite element methods, Math. Comp., 31, 414-442. MR 55:4748
77. Schatz, A.H. and Wahlbin, L.B.(1995): Interior maximum-norm estimates for finite element methods, Part II, Math. Comp., 64, 907-928. MR 95j:65143
78. Schatz, A.H. and Wahlbin, L.B.(1995): Interior maximum-norm estimates for finite element methods, Part II, Math. Comp., 64, 907-928. MR 95j:65143
79. Schatz, A.H. and Wang, J.(1996): Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions, Math. Comp., 65, 19-27. MR 96d:65190
80. Schatz, A.H. and Wang, J.(1996): Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions, Math. Comp., 65, 19-27. MR 96d:65190
81. Utnes, T.(1997): Two-grid finite element formulations of the incompressible Navier-Stokes equations, Comm. Numer. Methods Engrg., 34, 675-684. MR 98d:76110
82. Utnes, T.(1997): Two-grid finite element formulations of the incompressible Navier-Stokes equations, Comm. Numer. Methods Engrg., 34, 675-684. MR 98d:76110
83. Verfürth, R,.(1994): A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp., 62, 445-475. MR 94j:65136
84. Verfürth, R,.(1994): A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp., 62, 445-475. MR 94j:65136
85. Verfürth, R.(1995): A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations, Bericht Nr. 180, Fakultät für Mathematik, RuhrUniversität Bochum.
86. Verfürth, R.(1996): A Review of A-Posteriori Error Estimation and Adaptive Mesh Refinement, Wiley-Teubner.
87. Wahlbin, L.B.(1991): Local behavior in finite element methods, in [23], pp. 355-522. MR 92f:65001
88. Wahlbin, L.B.(1991): Local behavior in finite element methods, in [23], pp. 355-522. MR 92f:65001
89. Wahlbin, L.B.(1995): Superconvergence in Galerkin Finite Element Methods, Vol. 1605, Lecture Notes in Math., Springer. MR 98j:65083
90. Wahlbin, L.B.(1995): Superconvergence in Galerkin Finite Element Methods, Vol. 1605, Lecture Notes in Math., Springer. MR 98j:65083
91. Xu, J.(1992): A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29, 303-319. MR 92k:65063
92. Xu, J.(1992): A new class of iterative methods for nonselfadjoint or indefinite problems, SIAM J. Numer. Anal., 29, 303-319. MR 92k:65063
93. Xu, J.(1992): Iterative methods by space decomposition and subspace correction, SIAM Review, 34, 4, 581-613. MR 93k:65029
94. Xu, J.(1992): Iterative methods by space decomposition and subspace correction, SIAM Review, 34, 4, 581-613. MR 93k:65029
95. Xu, J.(1994): A novel two-grid method for semilinear equations, SIAM J. Sci. Comput., 15, 231-237. MR 94m:65178
96. Xu, J.(1994): A novel two-grid method for semilinear equations, SIAM J. Sci. Comput., 15, 231-237. MR 94m:65178
97. Xu, J.(1996): Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33, 1759-1777. MR 971:65169
98. Xu, J.(1996): Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33, 1759-1777. MR 971:65169
99. Xu, J. and Zou, J.(1998): Some non-overlapping domain decomposition methods, SIAM Review 40, 4, 857-914.
100. Yserentant, H.(1993): Old and new proofs for multigrid algorithms, Acta Numerica, 2, 285-326. MR 94i:65128
101. Yserentant, H.(1993): Old and new proofs for multigrid algorithms, Acta Numerica, 2, 285-326. MR 94i:65128
102. Zhou, A., Liem, C.L., Shih, T.M. and Lü, T.(1998): Error analysis on bi-parameter finite elements, Comput. Methods Appl. Mech. Engrg., 158, 329-339. CMP 98:14
103. Zhou, A., Liem, C.L., Shih, T.M. and Lü, T.(1998): Error analysis on bi-parameter finite elements, Comput. Methods Appl. Mech. Engrg., 158, 329-339. CMP 98:14