Adaptive numerical treatment of elliptic systems on manifolds

Advances in Computational Mathematics

Volumn 15, Issue 1-4, 2001, Pages 139-191

  Holst, M ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035566962     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1014246117321     Document Type: Article

References (97)

1. R.A. Adams, Sobolev Spaces (Academic Press, San Diego, CA, 1978).
2. B. Aksoylu, Adaptive multilevel numerical methods, Ph.D. thesis, Department of Mathematics, UC San Diego (2001).
3. D. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput. 22(2) (1997) 431-448.
4. T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations (Springer, New York, 1982).
5. I. Babuška and J.M. Melenk, The partition of unity finite element method, Internat. J. Numer. Methods Engrg. 40 (1997) 727-758.
6. I. Babuška and W. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736-754.
7. I. Babuška and W. Rheinboldt, A posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978) 1597-1615.
8. N. Baker, M. Holst and F. Wang, Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II: Refinement at solvent accessible surfaces in biomolecular systems, J. Comput. Chem. 21 (2000) 1343-1352.
9. R. Bank and M. Holst, A new paradigm for parallel adaptive mesh refinement, SIAM J. Sci. Statist. Comput. 22(4) (2000) 1411-1443.
10. R.E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations, Users' Guide 8.0, Software, Environments and Tools, Vol. 5. (SIAM, Philadelphia, PA, 1998).
11. R.E. Bank and T.F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36(153) (1981) 35-51.
12. R.E. Bank, T.F. Dupont and H. Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988) 427-458.
13. R.E. Bank, M. Holst, B. Mantel, J. Periaux and C.H. Zhou, CFD PPLTMG: Using a posteriori error estimates and domain decomposition, in: ECCOMAS 98, New York, 1998.
14. R.E. Bank and H.D. Mittelmann, Stepsize selection in continuation procedures and damped Newton's method, J. Comput. Appl. Math. 26 (1989) 67-77.
15. R.E. Bank and D.J. Rose, Parameter selection for Newton-like methods applicable to nonlinear partial differential equations, SIAM J. Numer. Anal. 17(6) (1980) 806-822.
16. R.E. Bank and D.J. Rose, Global approximate Newton methods, Numer. Math. 37 (1981) 279-295.
17. R.E. Bank and D.J. Rose, Analysis of a multilevel iterative method for nonlinear finite element equations, Math. Comp. 39(160) (1982) 453-465.
18. R.E. Bank and R.K. Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30(4) (1993) 921-935.
19. R.E. Bank and R.K. Smith, Mesh smoothing using a posteriori error estimates, SIAM J. Numer. Anal. 34 (1997) 979-997.
20. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44(170) (1985) 283-301.
21. E. Bänsch, An adaptive finite-element strategy for the three-dimsional time-dependent Navier-Stokes equations, J. Comput. Appl. Math. 36 (1991) 3-28.
22. E. Bänsch, Local mesh refinement in 2 and 3 dimensions, Impact of Computing in Science and Engineering 3 (1991) 181-191.
23. E. Bänsch and K.G. Siebert, A posteriori error estimation for nonlinear problems by duality techniques, Technical Report, Institut für Angewandte Mathematik, Freiburg, Germany (1995).
24. P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert and C. Wieners, UG - A flexible software toolbox for solving partial differential equations (1998).
25. R. Beck, B. Erdmann and R. Roitzsch, KASKADE 3.0: An object-oriented adaptive finite element code, Technical Report TR95-4, Konrad-Zuse-Zentrum for Informationstechnik, Berlin (1995).
26. J. Bey, Adaptive Grid Manager: AGM3D Manual, Technical Report 50, SFB 382, Math. Inst., University of Tubingen, 1996.
27. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).
28. F. Brezzi, Mathematical theory of finite elements, in: State-of-the-Art Surveys on Finite Element Technology, eds. A.K. Noor and W.D. Pilkey (New York, 1985) pp. 1-25.
29. F. Brezzi, W.W. Hager and P.A. Raviart, Error estimates for the finite element solution of variational inequalities, Numer. Math. 28 (1977) 431-443.
30. T. Chan, S. Go, and L. Zikatanov, Lecture notes on multilevel methods for elliptic problems on unstructured meshes, Technical Report, Department of Mathematics, UCLA (1997).
31. T.F. Chan, B. Smith and J. Zou, Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids, Technical Report CAM 94-8, Department of Mathematics, UCLA (1994).
32. Y. Choquet-Bruhat, J. Isenberg and V. Moncrief, Solutions of constraints for Einstein equations, C. R. Acad. Sci. Paris 315 (1992) 349-355.
33. Y. Choquet-Bruhat and J. W. York, Jr., The Cauchy problem, in: General Relativity and Gravitation, ed. A. Held (Plenum, New York, 1980).
34. P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, New York, 1978).
35. P. Clèment, Approximation by finite element functions using local regularization, RAIRO 2 (1975) 77-84.
36. W. Dahmen, Smooth piecewise quadratic surfaces, in: Mathematical Methods in Computer-Aided Geometric Design (1989) pp. 181-193.
37. W. Dahmen and C. Micchelli, Subdivision algorithms for the generation of box spline surfaces, Computer-Aided Geom. Design 18(2) (1984) 115-129.
38. P.J. Davis, Interpolation and Approximation (Dover Publications, New York, 1963).
39. R.S. Dembo, S.C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19(2) (1982) 400-408.
40. R.A. DeVore and G.G. Lorentz, Constructive Approximation (Springer, New York, 1993).
41. S.C. Eisenstat and H.F. Walker, Globally convergent inexact Newton methods, Technical Report, Department of Mathematics and Statistics, Utah State University (1992).
42. D. Estep, M. Holst and D. Mikulencak, Accounting for stability: A posteriori error estimates for finite element methods based on residuals and variational analysis, to appear, in Communications in Numerical Methods in Engineering.
43. R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974) 963-971.
44. S. Fucik and A. Kufner, Nonlinear Differential Equations (Elsevier Science, New York, 1980).
45. M. Griebel and M.A. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs, SIAM J. Sci. Statist. Comput. 22(3) (2000) 853-890.
46. C. Grimm, Modeling surfaces of arbitrary topology using manifolds, Ph.D. thesis, Department of Computer Science, Brown University (1996).
47. C. Grimm and J. Hughes, Modeling surfaces of arbitrary topology using manifolds, in: Graphics (Proceedings of SIGGRAPH 95), SIGGRAPH 29(4) (1995) 359-369.
48. W. Hackbusch, Multi-grid Methods and Applications (Springer, Berlin, 1985).
49. W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations (Springer, Berlin, 1994).
50. S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, Cambridge, 1973).
51. E. Hebey, Sobolev Spaces on Riemannian Manifolds (Springer, Berlin, 1991).
52. M. Holst, Finite element approximation theory on Riemannian manifolds (in preparation).
53. M. Holst, N. Baker and F. Wang, Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I: Algorithms and examples, J. Comput. Chem. 21 (2000) 1319-1342.
54. M. Holst and D. Bernstein, Some results on non-constant mean curvature solutions to the Einstein constraint equations (in preparation).
55. M. Holst and D. Bernstein, Adaptive finite element solution of the initial value problem in general relativity I. Algorithms (in preparation).
56. M. Holst and E. Titi, Determining projections and functionals for weak solutions of the Navier-Stokes equations, in: Recent Developments in Optimization Theory and Nonlinear Analysis, eds. Y. Censor and S. Reich, Contemporary Mathematics, Vol. 204 (Amer. Math. Soc., Providence, RI, 1997).
57. M. Holst and S. Vandewalle, Schwarz methods: to symmetrize or not to symmetrize, SIAM J. Numer. Anal. 34(2) (1997) 699-722.
58. J. Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 13 (1996) 1819-1847.
59. J. Isenberg and V. Moncrief, A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 13 (1996) 1819-1847.
60. J.W. Jerome and T. Kerkhoven, A finite element approximation theory for the drift diffusion semiconductor model, SIAM J. Numer. Anal. 28(2) (1991) 403-422.
61. H.B. Keller, Numerical Methods in Bifurcation Problems (Tata Institute of Fundamental Research, Bombay, India, 1987).
62. H.B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Dover, New York, 1992).
63. ∞ stability of finite element approximations of elliptic gradient equations, Numer. Math. 57 (1990) 561-575.
64. J.M. Lee, Riemannian Manifolds (Springer, New York, 1997).
65. A. Liu and B. Joe, Relationship between tetrahedron shape measures, BIT 34 (1994) 268-287.
66. A. Liu and B. Joe, Quality local refinement of tetrahedral meshes based on bisection, SIAM J. Sci. Statist. Comput. 16(6) (1995) 1269-1291.
67. J. Liu and W. Rheinboldt, A posteriori finite element error estimators for indefinite elliptic boundary value problems, Numer. Functional Anal. Optimiz. 15(3) (1994) 335-356.
68. J. Liu and W. Rheinboldt, A posteriori finite element error estimators for parametrized nonlinear boundary value problems, Numer. Functional Anal. Optimiz. 17(5) (1996) 605-637.
69. J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover, New York, 1994).
70. J. Maubach, Local bisection refinement for N-simplicial grids generated by relection, SIAM J. Sci. Statist. Comput. 16(1) (1995) 210-277.
71. E. Mucke, Shapes and implementations in three-dimensional geometry, Ph.D. thesis, Department of Computer Science, University of Illinois at Urbana-Champaign (1993).
72. A. Mukherjee, An adaptive finite element code for elliptic boundary value problems in three dimensions with applications in numerical relativity, Ph.D. thesis, Department of Mathematics, Pennsylvania State University (1996).
73. J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974) 937-958.
74. N. O. Murchadha and J. York, Jr., Initial-value problem of general relativity. I. General formulation and physical interpretation, Phys. Rev. D 10(2) (1974) 428-436.
75. N. O. Murchadha and J. York, Jr., Initial-value problem of general relativity. II. Stability of solutions of the initial-value equations, Phys. Rev. D 10(2) (1974) 437-446.
76. N. O. Murchadha and J. York, Jr., Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys. 14(11) (1973) 1551-1557.
77. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
78. M. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Internat. J. Numer. Methods Engrg. 20 (1984) 745-756.
79. M. Rivara, Local modification of meshes for adaptive and/or multigrid finite-element methods, J. Comput. Appl. Math. 36 (1991) 79-89.
80. I. Rosenberg and F. Stenger, A lower bound on the angles of triangles constructed by bisecting the longest side, Math. Comp. 29 (1975) 390-395.
81. S. Rosenberg, The Laplacian on a Riemannian Manifold (Cambridge Univ. Press, Cambridge, 1997).
82. J.W. Ruge and K. Stüben, Algebraic Multigrid (AMG), in: Multigrid Methods, Frontiers in Applied Mathematics, Vol. 3, ed. S.F. McCormick (SIAM, Philadelphia, PA, 1987) pp. 73-130.
83. A.H. Schatz, An Oberservation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28(128) (1974) 959-962.
84. A.H. Schatz and J. Wang, Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions, Math. Comp. 62 (2000) 445-475.
85. G. Schwarz, Hodge Decomposition: A Method for Solving Boundary Value Problems (Springer, New York, 1991).
86. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54(190) (1990) 483-493.
87. M. Stynes, On faster convergence of the bisection method for all triangles, Math. Comp. 35 (1980) 1195-1201.
88. P. Vanek, J. Mandel and M. Brezina, Algebraic multigrid on unstructured meshes, Technical Report UCD/CCM 34, Center for Computational Mathematics, University of Colorado at Denver (1994).
89. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62(206) (1994) 445-475.
90. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley, New York, 1996).
91. R.M. Wald, General Relativity (Univ. of Chicago Press, Chicago, IL, 1984).
92. J. Wloka, Partial Differential Equations (Cambridge Univ. Press, Cambridge, 1992).
93. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34(4) (1992) 581-613.
94. J. Xu, A novel two-grid method for semilinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).
95. J. Xu, Two-grid finite element discretization for nonlinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).
96. J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000) 881-909.
97. K. Yosida, Functional Analysis (Springer, Berlin, 1980).