Error estimates for a finite volume element method for parabolic equations in convex polygonal domains

Numerical Methods for Partial Differential Equations

Volumn 20, Issue 5, 2004, Pages 650-674

  Chatzipantelidis, P ^a   Lazarov, R D ^a   Thomee V ^b  

^a TEXAS A AND M UNIVERSITY   (United States)

^b CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

Author keywords

Elliptic projection; Error estimates; Finite volume element method; Parabolic equation

Indexed keywords

EID: 4444381362     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.20006     Document Type: Article

References (24)

1. 1 L. Angermann, The one‐step Θ‐method for spatially stabilized finite volume semidiscretizations of parabolic equations, R. Herbin and D. Kröner, editors, Finite Volumes for Complex Applications III, Hermes Penton Science, 2002, pp 25–39.
2. 2 R. Eymard, T. Gallouët, and R. Herbin, Finite volume methods, Handbook of numerical analysis, VII, North‐Holland, Amsterdam, 2000, pp 713–1020.
3. 3 K. W. Morton, On the analysis of finite volume methods for evolutionary problems, SIAM J Numer Anal 35 (1998), 2195–2222.
4. 4 R. E. Bank and D. J. Rose, Some error estimates for the box method, SIAM J Numer Anal 24 (1987), 777–787.
5. 5 Z. Cai, On the finite volume element method, Numer Math 58 (1991), 713–735.
6. 6 A. Süli, Convergence of finite volume schemes for Poisson equation on nonuniform meshes, SIAM J Numer Anal 28 (1991), 1419–1430.
7. 7 S.‐H. Chou and Q. Li, Error estimates in L 2, H 1 and L ∞ in covolume methods for elliptic and parabolic problems: A unified approach, Math Comp 69 (2000), 103–120.
8. 8 R. E. Ewing, R. D. Lazarov, and Y. Lin, Finite volume element approximations of nonlocal reactive flows in porous media, Numer Methods Partial Differential Eq 16 (2000), 285–311.
9. 9 R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J Numer Anal 39 (2002), 1865–1888.
10. 10 J. Droniou and T. Galouët, Finite volume methods for convection‐diffusion equations with right‐hand side in H −1, M2AN Math Model Numer Anal 36 (2002), 705–724.
11. 11 P. Chatzipantelidis and R. Lazarov, The finite volume element method in nonconvex polygonal domains, R. Herbin and D. Kröner, editors, Finite Volumes for Complex Applications III, Hermes Penton Science, London, 2002, pp 171–178.
12. 12 P. Chatzipantelidis and R. Lazarov, The finite volume element method in nonconvex polygonal domains, Technical Report ISC‐MATH‐03‐04, Texas A&M University.
13. 13 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
14. 14 L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Rhode Island, 1998.
15. 15 V. Thomée, Galerkin finite element methods for parabolic problems, Springer‐Verlag, New York, 1997.
16. 16 P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Massachusetts, 1985.
17. 17 R. Li, Z. Chen, and W. Wu, Generalized difference methods for differential equations, Marcel Dekker, New York, 2000.
18. 18 C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Boston, 1988.
19. 19 P. Chatzipantelidis, Finite volume methods for elliptic PDE's: A new approach, Math Model Numer Anal 36 (2002), 307–324.
20. 20 I. D. Mishev, Finite volume methods for non‐definite problems, Numer Math 83 (1999), 161–175.
21. 21 A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer‐Verlag, New York, 1983.
22. 22 S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer‐Verlag, New York, 1994.
23. 23 A. H. Schatz, A weak discrete maximum principle and stability of the finite element method in L ∞ on plane polygonal domains. I., Math Comp 34 (1980), 77–91.
24. 24 A. H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular meshes: Part III. Local estimates up to the boundary. Under preparation.