Discontinuous galerkin finite element methods for dynamic linear solid viscoelasticity problems

Numerical Methods for Partial Differential Equations

Volumn 23, Issue 5, 2007, Pages 1149-1166

  Rlvière, Béatrice ^a   Shaw, Simon ^b   Whiteman, J R ^b  

^a UNIVERSITY OF PITTSBURGH   (United States)

^b BRUNEL UNIVERSITY   (United Kingdom)

Author keywords

A priori error estimates; Discontinuous Galerkin method; Finite element method; Viscoelasticity

Indexed keywords

EID: 34548544577     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20215     Document Type: Article

References (16)

1. 1 B. Rivière and S Shaw, Discontinuous Galerkin finite element approximation of nonlinear non‐Fickian diffusion in viscoelastic polymers, SIAM J Numer Anal ( 2005), to appear. See also Technical Report 05/6 at www.brunel.ac.uk/bicom.
2. 2 B. Rivière, S. Shaw, M. F. Wheeler, and J. R. Whiteman, Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity, Numer Math 95 ( 2003), 347–376.
3. 3 A. R. Johnson, Modeling viscoelastic materials using internal variables, Shock Vibration Dig 31 ( 1999), 91–100.
4. 4 A. R. Johnson, A. Tessler, and M. Dambach, Dynamics of thick viscoelastic beams, J Eng Mater Technol 119 ( 1997), 273–278.
5. 5 V. Girault, B. Rivière, and M. Wheeler, A discontinuous Galerkin method with non‐overlapping domain decomposition for the Stokes and Navier–Stokes problems, Math Computat ( 2002), to appear. TICAM Report 02‐08.
6. 6 B. Rivière, M. F. Wheeler, and V. Girault, A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J Numer Anal 39 ( 2001), 902–931.
7. 7 B. Rivière and M.F. Wheeler, A discontinuous Galerkin method applied to nonlinear parabolic equations, B. Cockburn, G.E. Karniadakis, and C.‐W. Shu, editors, Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer, 1999, pp. 231–244.
8. 8 B. Rivière and M.F. Wheeler, Discontinuous finite element methods for acoustic and elastic wave problems, Contemp Math 329 ( 2003), 271–282.
9. 9 B. Rivière, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Computat Geosci 3 ( 1999), 337–360.
10. 10 B. Rivière, S. Shaw, and J. R. Whiteman, Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems, Technical report, BICOM, Brunel University, 2004. Available at www.brunel.ac.uk/bicom.
11. 11 C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J Diff Eqns 7 ( 1970), 554–569.
12. 12 J. Brilla, Error analysis for Laplace transform—finite element solution of hyperbolic equations, Numer Math 41 ( 1983), 55–62.
13. 13 G. M. Hulbert and T. J. R. Hughes, Space–time finite element methods for second‐order hyperbolic equations, Comptat Methods Appl Mech Eng 84 ( 1990), 327–348.
14. 14 A. K. Pani, V. Thomée, and L. B. Wahlbin, Numerical methods for hyperbolic and parabolic integro‐differential equations, J Integral Equat Appl 4 ( 1992), 533–584.
15. 15 S. Shaw, M. K. Warby, and J. R. Whiteman, An error bound via the Ritz–Volterra projection for a fully discrete approximation to a hyperbolic integrodifferential equation, Technical report, 94/3, BICOM, Brunel University, Uxbridge, UK, 1994. Available at www.brunel.ac.uk/bicom.
16. 16 E. G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro‐differential equations, Nonlinear Anal, Theory, Methods Applic 12 ( 1988), 785–809.