# Adaptive Numerical Treatment of Elliptic Systems on Manifolds

## Abstract

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].

## Keywords

Elliptic System Posteriori Error Local Residual Posteriori Error Estimation Galerkin Approximation## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R.A. Adams,
*Sobolev Spaces*(Academic Press, San Diego, CA, 1978).Google Scholar - [2]B. Aksoylu, Adaptive multilevel numerical methods, Ph.D. thesis, Department of Mathematics, UC San Diego (2001).Google Scholar
- [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]T. Aubin,
*Nonlinear Analysis on Manifolds. Monge-Ampére Equations*(Springer, New York, 1982).Google Scholar - [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]I. Babuška and W. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736–754.Google Scholar
- [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]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]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]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]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]R.E. Bank, T.F. Dupont and H. Yserentant, The hierarchical basis multigrid method, Numer. Math. 52 (1988) 427–458.Google Scholar
- [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]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]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]R.E. Bank and D.J. Rose, Global approximate Newton methods, Numer. Math. 37 (1981) 279–295.Google Scholar
- [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]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]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]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]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]E. Bänsch, Local mesh refinement in 2 and 3 dimensions, Impact of Computing in Science and Engineering 3 (1991) 181–191.Google Scholar
- [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]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]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]J. Bey, Adaptive Grid Manager: AGM3D Manual, Technical Report 50, SFB 382, Math. Inst., University of Tubingen, 1996.Google Scholar
- [27]S.C. Brenner and L.R. Scott,
*The Mathematical Theory of Finite Element Methods*(Springer, New York, 1994).Google Scholar - [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]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]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]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]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]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]P.G. Ciarlet,
*The Finite Element Method for Elliptic Problems*(North-Holland, New York, 1978).Google Scholar - [35]P. Clèment, Approximation by finite element functions using local regularization, RAIRO 2 (1975) 77–84.Google Scholar
- [36]W. Dahmen, Smooth piecewise quadratic surfaces, in:
*Mathematical Methods in Computer-Aided Geometric Design*(1989) pp. 181–193.Google Scholar - [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]P.J. Davis,
*Interpolation and Approximation*(Dover Publications, New York, 1963).Google Scholar - [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]R.A. DeVore and G.G. Lorentz,
*Constructive Approximation*(Springer, New York, 1993).Google Scholar - [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]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]R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974) 963–971.Google Scholar
- [44]S. Fucik and A. Kufner,
*Nonlinear Differential Equations*(Elsevier Science, New York, 1980).Google Scholar - [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]C. Grimm, Modeling surfaces of arbitrary topology using manifolds, Ph.D. thesis, Department of Computer Science, Brown University (1996).Google Scholar
- [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]W. Hackbusch,
*Multi-grid Methods and Applications*(Springer, Berlin, 1985).Google Scholar - [49]W. Hackbusch,
*Iterative Solution of Large Sparse Systems of Equations*(Springer, Berlin, 1994).Google Scholar - [50]S.W. Hawking and G.F.R. Ellis,
*The Large Scale Structure of Space-Time*(Cambridge Univ. Press, Cambridge, 1973).Google Scholar - [51]E. Hebey,
*Sobolev Spaces on Riemannian Manifolds*(Springer, Berlin, 1991).Google Scholar - [52]M. Holst, Finite element approximation theory on Riemannian manifolds (in preparation).Google Scholar
- [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]M. Holst and D. Bernstein, Some results on non-constant mean curvature solutions to the Einstein constraint equations (in preparation).Google Scholar
- [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]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]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]J. Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 13 (1996) 1819–1847.Google Scholar
- [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]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]H.B. Keller,
*Numerical Methods in Bifurcation Problems*(Tata Institute of Fundamental Research, Bombay, India, 1987).Google Scholar - [62]H.B. Keller,
*Numerical Methods for Two-Point Boundary-Value Problems*(Dover, New York, 1992).Google Scholar - [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]J.M. Lee,
*Riemannian Manifolds*(Springer, New York, 1997).Google Scholar - [65]A. Liu and B. Joe, Relationship between tetrahedron shape measures, BIT 34 (1994) 268–287.Google Scholar
- [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]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]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]J.E. Marsden and T.J.R. Hughes,
*Mathematical Foundations of Elasticity*(Dover, New York, 1994).Google Scholar - [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]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]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]J.A. Nitsche and A.H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974) 937–958.Google Scholar
- [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]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]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]J.M. Ortega and W.C. Rheinboldt,
*Iterative Solution of Nonlinear Equations in Several Variables*(Academic Press, New York, 1970).Google Scholar - [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]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]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]S. Rosenberg,
*The Laplacian on a Riemannian Manifold*(Cambridge Univ. Press, Cambridge, 1997).Google Scholar - [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]A.H. Schatz, An Oberservation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28(128) (1974) 959–962.Google Scholar
- [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]G. Schwarz,
*Hodge Decomposition: A Method for Solving Boundary Value Problems*(Springer, New York, 1991).Google Scholar - [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]M. Stynes, On faster convergence of the bisection method for all triangles, Math. Comp. 35 (1980) 1195–1201.Google Scholar
- [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]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]R. Verfürth,
*A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques*(Wiley, New York, 1996).Google Scholar - [91]R.M. Wald,
*General Relativity*(Univ. of Chicago Press, Chicago, IL, 1984).Google Scholar - [92]J. Wloka,
*Partial Differential Equations*(Cambridge Univ. Press, Cambridge, 1992).Google Scholar - [93]J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review 34(4) (1992) 581–613.Google Scholar
- [94]J. Xu, A novel two-grid method for semilinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).Google Scholar
- [95]J. Xu, Two-grid finite element discretization for nonlinear elliptic equations, Technical Report, Department of Mathematics, Penn State University (1992).Google Scholar
- [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]K. Yosida,
*Functional Analysis*(Springer, Berlin, 1980).Google Scholar