Advances in Computational Mathematics

, Volume 15, Issue 1–4, pp 139–191 | Cite as

Adaptive Numerical Treatment of Elliptic Systems on Manifolds

  • M. HolstEmail author


Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established in [55] are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented; more detailed examples using MC for this application may be found in [26].


Elliptic System Posteriori Error Local Residual Posteriori Error Estimation Galerkin Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    R.A. Adams, Sobolev Spaces (Academic Press, San Diego, CA, 1978).Google Scholar
  2. [2]
    B. Aksoylu, Adaptive multilevel numerical methods, Ph.D. thesis, Department of Mathematics, UC San Diego (2001).Google Scholar
  3. [3]
    D. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput. 22(2) (1997) 431–448.Google Scholar
  4. [4]
    T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampére Equations (Springer, New York, 1982).Google Scholar
  5. [5]
    I. Babuška and J.M.Melenk, The partition of unity finite element method, Internat. J. Numer.Methods Engrg. 40 (1997) 727–758.Google Scholar
  6. [6]
    I. Babuška and W. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736–754.Google Scholar
  7. [7]
    I. Babuška and W. Rheinboldt, A posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978) 1597–1615.Google Scholar
  8. [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.Google Scholar
  9. [9]
    R. Bank and M. Holst, A new paradigm for parallel adaptive mesh refinement, SIAM J. Sci. Statist. Comput. 22(4) (2000) 1411–1443.Google Scholar
  10. [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).Google Scholar
  11. [11]
    R.E. Bank and T.F. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36(153) (1981) 35–51.Google Scholar
  12. [12]
    R.E. Bank, T.F. Dupont and H. Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988) 427–458.Google Scholar
  13. [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.Google Scholar
  14. [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.Google Scholar
  15. [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.Google Scholar
  16. [16]
    R.E. Bank and D.J. Rose, Global approximate Newton methods, Numer. Math. 37 (1981) 279–295.Google Scholar
  17. [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.Google Scholar
  18. [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.Google Scholar
  19. [19]
    R.E. Bank and R.K. Smith, Mesh smoothing using a posteriori error estimates, SIAM J. Numer. Anal. 34 (1997) 979–997.Google Scholar
  20. [20]
    R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44(170) (1985) 283–301.Google Scholar
  21. [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.Google Scholar
  22. [22]
    E. Bänsch, Local mesh refinement in 2 and 3 dimensions, Impact of Computing in Science and Engineering 3 (1991) 181–191.Google Scholar
  23. [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).Google Scholar
  24. [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).Google Scholar
  25. [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).Google Scholar
  26. [26]
    J. Bey, Adaptive Grid Manager: AGM3D Manual, Technical Report 50, SFB 382, Math. Inst., University of Tubingen, 1996.Google Scholar
  27. [27]
    S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).Google Scholar
  28. [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.Google Scholar
  29. [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.Google Scholar
  30. [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).Google Scholar
  31. [31]
    T.F. Chan, B. Smith and J. Zou, Overlapping Schwarz methods on unstructured meshes using nonmatching coarse grids, Technical Report CAM 94-8, Department of Mathematics, UCLA (1994).Google Scholar
  32. [32]
    Y. Choquet-Bruhat, J. Isenberg and V. Moncrief, Solutions of constraints for Einstein equations, C. R. Acad. Sci. Paris 315 (1992) 349–355.Google Scholar
  33. [33]
    Y. Choquet-Bruhat and J.W. York, Jr., The Cauchy problem, in: General Relativity and Gravitation, ed. A. Held (Plenum, New York, 1980).Google Scholar
  34. [34]
    P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, New York, 1978).Google Scholar
  35. [35]
    P. Clèment, Approximation by finite element functions using local regularization, RAIRO 2 (1975) 77–84.Google Scholar
  36. [36]
    W. Dahmen, Smooth piecewise quadratic surfaces, in: Mathematical Methods in Computer-Aided Geometric Design (1989) pp. 181–193.Google Scholar
  37. [37]
    W. Dahmen and C. Micchelli, Subdivision algorithms for the generation of box spline surfaces, Computer-Aided Geom. Design 18(2) (1984) 115–129.Google Scholar
  38. [38]
    P.J. Davis, Interpolation and Approximation (Dover Publications, New York, 1963).Google Scholar
  39. [39]
    R.S. Dembo, S.C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19(2) (1982) 400–408.Google Scholar
  40. [40]
    R.A. DeVore and G.G. Lorentz, Constructive Approximation (Springer, New York, 1993).Google Scholar
  41. [41]
    S.C. Eisenstat and H.F. Walker, Globally convergent inexact Newton methods, Technical Report, Department of Mathematics and Statistics, Utah State University (1992).Google Scholar
  42. [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.Google Scholar
  43. [43]
    R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974) 963–971.Google Scholar
  44. [44]
    S. Fucik and A. Kufner, Nonlinear Differential Equations (Elsevier Science, New York, 1980).Google Scholar
  45. [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.Google Scholar
  46. [46]
    C. Grimm, Modeling surfaces of arbitrary topology using manifolds, Ph.D. thesis, Department of Computer Science, Brown University (1996).Google Scholar
  47. [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.Google Scholar
  48. [48]
    W. Hackbusch, Multi-grid Methods and Applications (Springer, Berlin, 1985).Google Scholar
  49. [49]
    W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations (Springer, Berlin, 1994).Google Scholar
  50. [50]
    S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (Cambridge Univ. Press, Cambridge, 1973).Google Scholar
  51. [51]
    E. Hebey, Sobolev Spaces on Riemannian Manifolds (Springer, Berlin, 1991).Google Scholar
  52. [52]
    M. Holst, Finite element approximation theory on Riemannian manifolds (in preparation).Google Scholar
  53. [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.Google Scholar
  54. [54]
    M. Holst and D. Bernstein, Some results on non-constant mean curvature solutions to the Einstein constraint equations (in preparation).Google Scholar
  55. [55]
    M. Holst and D. Bernstein, Adaptive finite element solution of the initial value problem in general relativity I. Algorithms (in preparation).Google Scholar
  56. [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).Google Scholar
  57. [57]
    M. Holst and S. Vandewalle, Schwarz methods: to symmetrize or not to symmetrize, SIAM J. Numer. Anal. 34(2) (1997) 699–722.Google Scholar
  58. [58]
    J. Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 13 (1996) 1819–1847.Google Scholar
  59. [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.Google Scholar
  60. [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.Google Scholar
  61. [61]
    H.B. Keller, Numerical Methods in Bifurcation Problems (Tata Institute of Fundamental Research, Bombay, India, 1987).Google Scholar
  62. [62]
    H.B. Keller, Numerical Methods for Two-Point Boundary-Value Problems (Dover, New York, 1992).Google Scholar
  63. [63]
    T. Kerkhoven and J.W. Jerome, L∞ stability of finite element approximations of elliptic gradient equations, Numer. Math. 57 (1990) 561–575.Google Scholar
  64. [64]
    J.M. Lee, Riemannian Manifolds (Springer, New York, 1997).Google Scholar
  65. [65]
    A. Liu and B. Joe, Relationship between tetrahedron shape measures, BIT 34 (1994) 268–287.Google Scholar
  66. [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.Google Scholar
  67. [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.Google Scholar
  68. [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.Google Scholar
  69. [69]
    J.E. Marsden and T.J.R. Hughes, Mathematical Foundations of Elasticity (Dover, New York, 1994).Google Scholar
  70. [70]
    J. Maubach, Local bisection refinement for N-simplicial grids generated by relection, SIAM J. Sci. Statist. Comput. 16(1) (1995) 210–277.Google Scholar
  71. [71]
    E. Mucke, Shapes and implementations in three-dimensional geometry, Ph.D. thesis, Department of Computer Science, University of Illinois at Urbana-Champaign (1993).Google Scholar
  72. [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).Google Scholar
  73. [73]
    J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974) 937–958.Google Scholar
  74. [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.Google Scholar
  75. [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.Google Scholar
  76. [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.Google Scholar
  77. [77]
    J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).Google Scholar
  78. [78]
    M. Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Internat. J. Numer. Methods Engrg. 20 (1984) 745–756.Google Scholar
  79. [79]
    M. Rivara, Local modification of meshes for adaptive and/or multigrid finite-element methods, J. Comput. Appl. Math. 36 (1991) 79–89.Google Scholar
  80. [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.Google Scholar
  81. [81]
    S. Rosenberg, The Laplacian on a Riemannian Manifold (Cambridge Univ. Press, Cambridge, 1997).Google Scholar
  82. [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.Google Scholar
  83. [83]
    A.H. Schatz, An Oberservation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28(128) (1974) 959–962.Google Scholar
  84. [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.Google Scholar
  85. [85]
    G. Schwarz, Hodge Decomposition: A Method for Solving Boundary Value Problems (Springer, New York, 1991).Google Scholar
  86. [86]
    L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54(190) (1990) 483–493.Google Scholar
  87. [87]
    M. Stynes, On faster convergence of the bisection method for all triangles, Math. Comp. 35 (1980) 1195–1201.Google Scholar
  88. [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).Google Scholar
  89. [89]
    R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations, Math. Comp. 62(206) (1994) 445–475.Google Scholar
  90. [90]
    R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (Wiley, New York, 1996).Google Scholar
  91. [91]
    R.M. Wald, General Relativity (Univ. of Chicago Press, Chicago, IL, 1984).Google Scholar
  92. [92]
    J. Wloka, Partial Differential Equations (Cambridge Univ. Press, Cambridge, 1992).Google Scholar
  93. [93]
    J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34(4) (1992) 581–613.Google Scholar
  94. [94]
    J. Xu, A novel two-grid method for semilinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).Google Scholar
  95. [95]
    J. Xu, Two-grid finite element discretization for nonlinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).Google Scholar
  96. [96]
    J. Xu and A. Zhou, Local and parallel finite element algorithms based on two-grid discretizations, Math. Comp. 69 (2000) 881–909.Google Scholar
  97. [97]
    K. Yosida, Functional Analysis (Springer, Berlin, 1980).Google Scholar

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© Kluwer Academic Publishers 2001

Authors and Affiliations

  1. 1.Department of MathematicsUCSan Diego, La JollaUSA

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