Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: Motivation, formulation, and numerical examples

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 45-48, 2008, Pages 3605-3622

  Eyck, Alex Ten ^a   Celiker, Fatih ^a   Lew, Adrian ^a  

^a Stanford University   (United States)

Author keywords

Adaptivity; Discontinuous Galerkin methods; Nonlinear elasticity; Stabilization

Indexed keywords

ELASTICITY; GALERKIN METHODS; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS; STABILIZATION; THREE DIMENSIONAL; TWO DIMENSIONAL;

ADAPTIVE STABILIZATIONS; ADAPTIVITY; COMPUTATIONALLY EFFICIENT; DG METHODS; DISCONTINUOUS GALERKIN METHODS; HEXAHEDRAL ELEMENTS; NONLINEAR ELASTICITY; NUMERICAL EXAMPLES; PENALTY TERM; TRIANGULAR ELEMENTS;

NUMERICAL METHODS;

EID: 50149088362     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2008.02.020     Document Type: Article

References (31)

1. Arnold D.N. An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760
2. Arnold D.N., Brezzi F., Cockburn B., and Marini L.D. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779
3. Babuška I., and Zlámal M. Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 5 (1973) 863-875
4. S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc users manual, Technical Report ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.
5. Bassi F., and Rebay S. A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997) 267-279
6. Brenner P., and Scott L.R. The mathematical theory of finite element methods. Texts in Applied Mathematics vol. 15 (1994), Springer-Verlag
7. Brezzi F., Cockburn B., Marini L.D., and Süli E. Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg. (2005)
8. Brezzi F., Hughes T.J.R., Marini L.D., and Masud A. Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput. 22 1 (2005) 119-145
9. F. Brezzi, G. Manzini, L.D. Marini, P. Pietra, A. Russo, Discontinuous finite elements for diffusion problems, in: Atti Convegno in onore di F. Brioschi (Milan 1997), Istituto Lombardo, Accademia di Scienze e Lettere, 1999, pp. 197-217.
10. Brezzi F., Manzini G., Marini L.D., Pietra P., and Russo A. Discontinuous Galerkin approximations for elliptic problems. Numer. Methods Partial Differ. Eq. 16 (2000) 365-378
11. Castillo P. Performance of discontinuous Galerkin methods for elliptic PDE's. SIAM J. Sci. Comput. 24 (2002) 524-547
12. Castillo P., Cockburn B., Perugia I., and Schötzau D. An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38 (2000) 1676-1706
13. Performance of discontinuous Galerkin methods for elliptic PDEs P.Castillo. SIAM J. Numer. Anal. 24 2 (2002) 524-547
14. Ten Eyck A., Celiker F., and Lew A. Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: analytical estimates. Comput. Methods Appl. Mech. Engrg. (2008)
15. Ciarlet P.G. Mathematical elasticity - Volume I: three-dimensional elasticity. Studies Math. Appl. 20 (1988)
16. Djoko J.K., Ebobisse F., McBride A.T., and Reddy B.D. A discontinuous Galerkin formulation for classical and gradient plasticity. Part 1: formulation and analysis. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3881-3897
17. Engel G., Garikipati K., Hughes T.J.R., Larson M.G., Mazzei L., and Taylor R.L. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, strain gradient elasticity. Comput. Methods Appl. Mech. Engrg. 191 (2002) 3669-3750
18. Ten Eyck A., and Lew A. Discontinuous Galerkin methods for nonlinear elasticity. Int. J. Numer. Methods Engrg. 67 (2006) 1204-1243
19. Hackbusch W. Iterative solution of large sparse systems of equations. Appl. Math. Sci. 95 (2004)
20. Hansbo P., and Larson M.G. Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. Math. Model. Numer. Anal. 37 1 (2003) 63-72
21. Douglas Jr. J., and Dupont T. Interior penalty procedures for elliptic and parabolic Galerkin method. Lecture Notes in Physics vol. 58 (1976), Springer-Verlag, Berlin
22. Lew A., Neff P., Sulsky D., and Ortiz M. Optimal BV estimates for a discontinuous Galerkin method in linear elasticity. Appl. Math. Res. Exp. 3 (2004) 3-106
23. Marsden J.E., and Hughes T.J.R. Mathematical Foundations of Elasticity (1994), Dover, Mineola, NY
24. J. Nitsche, Über ein variationsprinzip zur lösung von Dirichlet problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, in: Abh. Math. Sem., vol. 36, University of Hamburg, 1971, pp. 9-15.
25. Noels L., and Radovitzky R. A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int. J. Numer. Methods Engrg. (2006)
26. Percell P., and Wheeler M.F. A local residual finite element procedure for elliptic equations. SIAM J. Numer. Anal. 15 (1978) 705-714
27. B. Rivière, M.F. Wheeler, Optimal error estimates for discontinuous Galerkin methods applied to linear elasticity problems, Technical Report 00-30, Texas Institute for Computational and Applied Mathematics, 2000.
28. Rivière B., Wheeler M.F., and Girault V. A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 902-931
29. Saad Y. Iterative Methods for Sparse Linear Systems (2003), Society for Industrial Mathematics
30. Wheeler M.F. An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152-161
31. Wihler T.P. Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45-75