A mimetic discretization method for linear elasticity

Lourenco Beirão Da Veiga

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 2, page 231-250
  • ISSN: 0764-583X

Abstract

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A Mimetic Discretization method for the linear elasticity problem in mixed weakly symmetric form is developed. The scheme is shown to converge linearly in the mesh size, independently of the incompressibility parameter λ, provided the discrete scalar product satisfies two given conditions. Finally, a family of algebraic scalar products which respect the above conditions is detailed.

How to cite

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Beirão Da Veiga, Lourenco. "A mimetic discretization method for linear elasticity." ESAIM: Mathematical Modelling and Numerical Analysis 44.2 (2010): 231-250. <http://eudml.org/doc/250765>.

@article{BeirãoDaVeiga2010,
abstract = { A Mimetic Discretization method for the linear elasticity problem in mixed weakly symmetric form is developed. The scheme is shown to converge linearly in the mesh size, independently of the incompressibility parameter λ, provided the discrete scalar product satisfies two given conditions. Finally, a family of algebraic scalar products which respect the above conditions is detailed. },
author = {Beirão Da Veiga, Lourenco},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Mimetic finite difference methods; linear elasticity; finite element methods; mixed formulation; mimetic finite difference methods; linear elasticity},
language = {eng},
month = {3},
number = {2},
pages = {231-250},
publisher = {EDP Sciences},
title = {A mimetic discretization method for linear elasticity},
url = {http://eudml.org/doc/250765},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Beirão Da Veiga, Lourenco
TI - A mimetic discretization method for linear elasticity
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 2
SP - 231
EP - 250
AB - A Mimetic Discretization method for the linear elasticity problem in mixed weakly symmetric form is developed. The scheme is shown to converge linearly in the mesh size, independently of the incompressibility parameter λ, provided the discrete scalar product satisfies two given conditions. Finally, a family of algebraic scalar products which respect the above conditions is detailed.
LA - eng
KW - Mimetic finite difference methods; linear elasticity; finite element methods; mixed formulation; mimetic finite difference methods; linear elasticity
UR - http://eudml.org/doc/250765
ER -

References

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  1. S. Agmon, Lectures on Elliptic Boundary Value Problems. Van Nostrand, USA (1965).  Zbl0142.37401
  2. M. Amara and J.M. Thomas, Equilibrium finite elements for the linear elastic problem. Numer. Math.33 (1979) 367–383.  Zbl0401.73079
  3. D.N. Arnold, F. Brezzi and J. Douglas Jr., PEERS: A new mixed finite element for plane elasticity. Japan J. Appl. Math.1 (1984) 347–367.  Zbl0633.73074
  4. D.N. Arnold, R.S. Falk and R. Winther, Differential complexes and stability of finite element methods II: the elasticity complex, in Compatible Spatial Discretizations, D. Arnold, P. Botchev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., IMA Volumes in Mathematics and its Applications142, Springer-Verlag (2005) 47–67.  Zbl1119.65399
  5. D.N. Arnold, R.S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp.76 (2007) 1699–1723.  Zbl1118.74046
  6. L. Beirão da Veiga, A residual based error estimator for the Mimetic Finite Difference method. Numer. Math.108 (2008) 387–406.  Zbl1144.65067
  7. L. Beirão da Veiga and G. Manzini, An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems with general diffusion tensors. Int. J. Num. Meth. Engrg.76 (2008) 1696–1723.  Zbl1195.65146
  8. L. Beirão da Veiga and G. Manzini, A higher-order formulation of the Mimetic Finite Difference method. SIAM J. Sci. Comput.31 (2008) 732–760.  Zbl1185.65201
  9. L. Beirão da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math.113 (2009) 325–356.  Zbl1183.65132
  10. L. Beirão da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, A mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys.228 (2009) 7215–7232.  Zbl1172.76032
  11. M. Berndt, K. Lipnikov, J.D. Moulton and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math.9 (2001) 253–284.  Zbl1014.65114
  12. M. Berndt, K. Lipnikov, M. Shashkov, M.F. Wheeler and I. Yotov, Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals. SIAM J. Numer. Anal.43 (2005) 1728–1749.  Zbl1096.76030
  13. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, USA (1991).  Zbl0788.73002
  14. F. Brezzi, J. Douglas Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.  Zbl0599.65072
  15. F. Brezzi, D. Boffi and M. Fortin, Reduced symmetry elements in linear elasticity. Comm. Pure Appl. Anal.8 (2009) 95–121.  Zbl1154.74041
  16. F. Brezzi, K. Lipnikov and M. Shashkov, Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal.43 (2005) 1872–1896.  Zbl1108.65102
  17. F. Brezzi, K. Lipnikov and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci.15 (2005) 1533–1553.  Zbl1083.65099
  18. F. Brezzi, K. Lipnikov and V. Simoncini, Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci.16 (2006) 275–298.  Zbl1094.65111
  19. F. Brezzi, K. Lipnikov, M. Shashkov and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comp. Meth. Appl. Mech. Engrg.196 (2007) 3682–3692.  Zbl1173.76370
  20. A. Cangiani and G. Manzini, Flux recontruction and pressure post-processing in mimetic finite difference methods. Comput. Meth. Appl. Mech. Engrg.197 (2008) 933–945.  Zbl1169.76404
  21. P.G. Ciarlet, Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications20. Amsterdam, North Holland (1988).  
  22. B.X. Fraejis de Vebeuke, Stress function approach, in World Congress on the Finite Element Method in Structural Mechanics, Bornemouth (1975).  
  23. V. Gryrya and K. Lipnikov, High-order mimetic finite difference method for the diffusion problems on polygonal meshes. J. Comput. Phys.227 (2008) 8841–8854.  Zbl1152.65101
  24. J. Hyman, M. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneus non-isotropic materials. J. Comput. Phys.132 (1997) 130–148.  Zbl0881.65093
  25. J. Hyman, J. Morel, M. Shashkov and S. Steinberg, Mimetic finite difference methods for diffusion equations. Comput. Geosci.6 (2002) 333–352.  Zbl1023.76033
  26. Y. Kuznetsov, K. Lipnikov and M. Shashkov, The mimetic finite difference method on polygonal meshes for diffusion-type problems. Comput. Geosci.8 (2005) 301–324.  Zbl1088.76046
  27. K. Lipnikov, J. Morel and M. Shashkov, Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys.199 (2004) 589–597.  Zbl1057.65071
  28. K. Lipnikov, M. Shashkov and D. Svyatskiy, The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. J. Comput. Phys.211 (2006) 473–491.  Zbl1120.65332
  29. J. Morel, M. Hall and M. Shaskov, A local support-operators diffusion discretization scheme for hexahedral meshes. J. Comput. Phys.170 (2001) 338–372.  Zbl0983.65096

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