Discontinuous galerkin method for an evolution equation with a memory term of positive type

Mathematics of Computation

Volumn 78, Issue 268, 2009, Pages 1975-1995

  Mustapha, Kassem ^a   McLean, William ^b  

^a KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS   (Saudi Arabia)

^b UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

A priori error estimates; Discontinuous Galerkin method; Memory term; Nite element method; Non uniform time steps

Indexed keywords

EID: 70349849810     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-09-02234-0     Document Type: Article

References (18)

1. K. Adolfsson, M. Enelund and S. Larsson, Adaptive discretization ofan integro-differential equation with a weakly singluar kernel, Comput. Methods Appl. Mech. Engrg., 192, 5285-5304 (2003). MR2023899 (2004j:65218).
2. B. Cockburn, G. E. Karniadakis and C.-W. Shu (Eds.), Discontinuous Galerkin Methods: Theory, Computation and Algorithms, Lecture Notes in Computational Science and Engineering 11, Springer 2000. MR1842160 (2002b:65004).
3. E. Cuesta, C. H. Lubich and C. Palencia, Convolution quadrature time discretization offrac-tional diffusion-wave equations, Math. Comp., 75, 673-696 (2006). MR2196986 (2006j:65404).
4. K. Eriksson and C. Johnson, Adaptive nite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28, 199-208 (1991). MR1083324 (91m:65274).
5. K. Eriksson, C. Johnson and Thomée, Time discretization ofparabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér., 19, 611-643 (1985). MR826227 (87e:65073).
6. S. Larsson, V. Thomée and L. B. Wahlbin, Numerical solution ofparabolic integro-differential equations by the discontinuous Galerkin method, Math. Comp., 67, 45-71 (1998). MR1432129 (98d:65168).
7. M. López-Fernández and C. Palencia, On the numerical inversion of the Laplace transform ofcertain holomorphic functions, Appl. Numer. Math., 51, 289-303 (2004). MR2091405 (2005e:65210).
8. M. López-Fernandez, C. Palencia and A. Schädle, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal. 44, 1332-1350 (2006). MR2231867 (2007e:65138).
9. J.-C. López Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27, 20-31 (1990). MR1034918 (91e:65160).
10. C. H. Lubich, I. H. Sloan and V. Thomée, Nonsmooth data error estimates for approximations ofan evolution equation with a positive-type memory term, Math. Comp., 65, 1-17 (1996). MR1322891 (96d:65207).
11. W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105, 481-510 (2007). MR2266834 (2008d:65097).
12. W. McLean and V. Thomée, Numerical solution ofan evolution equation with a positive-type memory term, J. Austral. Math. Soc. Ser. B, 35, 23-70 (1993). MR1225703 (94e:65094).
13. W. McLean and V. Thomée, Time discretization ofan evolution equation via Laplace trans-forms, IMA. J. Numer. Anal., 24, 439-463 (2004). MR2068831 (2005d:47072).
14. W. McLean and V. Thomée, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equations Appl.,toappear.
15. W. McLean, V. Thomée and L. B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math., 69, 49-69 (1996). MR1391611 (97b:65073).
16. M. J. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25, 319-327 (1988). MR933727 (89d:65113).
17. Achim Schädle, María López-Fernández and Christian Lubich, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput., 28, 421-438 (2006). MR2231714 (2007b:65142).
18. W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989). MR974464 (89m:45017).