Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with C1 and C2 coefficients

Electronic Journal of Differential Equations

Volumn 2005, Issue , 2005, Pages

  Dong, Hongjie ^a   Krylov, Nicolai V ^b  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Finite difference approximations; Linear elliptic and parabolic equations

Indexed keywords

EID: 25444514222     PISSN: 10726691     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

1. G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, Math. Modelling and Numerical Analysis, ESAIM, M2AM, Vol. 36, No. 1 (2002), 33-54.
2. G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations, SIAM J. Numer. Anal., Vol. 43 (2005), No. 2, 540-558.
3. L. Bers, F. John, and M. Schechter, "Partial differential equations", Interscience Publishers, New York-London-Sydney, 1964.
4. p norms for solutions of parabolic and elliptic difference equations, Ann. Mat. Pura Appl. (4), Vol. 95 (1973), 1-43.
5. R. Bürger and K. H. Karlsen, On some upwind difference schemes for the phenomenological sedimentation-consolidation model, J. Engineering Math., Vol., 41 (2001), 145-166.
6. Hongjie Dong, About smoothness of solutions of the heat equations in closed smooth space-time domains, Comm. Pure Appl. Math., Vol. 58 (2005) No. 6, 799-820.
7. Hongjie Dong, N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equation with constant coefficients, Algebra i Analis (St. Petersburg Math. J.) Vol. 17 (2005), No 2.
8. W. Fleming and M. Soner, "Controlled Markov processes and viscosity solutions", Springer Verlag, 1993.
9. A. Friedman, "Partial differential equations of parabolic type", Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.
10. D. W. Holtby, Higher-order estimates for fully nonlinear difference equations, II, Proc. Edinb. Math. Soc. (2) 44 (2001), No. 1, 87-102.
11. E. R. Jakobsen, On the rate of convergence of approximation schemes for Bellman equations associated with optimal stopping time problems, Math. Models Methods Appl. Sci., Vol. 13 (2003), No. 5, 613-644.
12. K. H. Karlsen and N. H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients, M2AN Math. Model. Numer. Anal., Vol. 35 (2001), No. 2, 239-269.
13. N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equation with Lipschitz coefficients, to appear in Appl Math Optim.
14. N. V. Krylov, Adapting some idea from stochastic control theory to studying the heat equation in closed smooth domain, Appl Math Optim 46:231-261 (2002).
15. N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Relat. Fields, Vol. 117, No. 1 (2000), 1-16.
16. N. V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electronic Journal of Probability, Vol. 4 (1999), paper No. 2, 1-19, http://www.math.washington.edu/ ejpecp/EjpVol4/paper2.abs.html
17. N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equation, St. Petersburg Math. J., Vol. 9, No. 3 (1997), 245-256.
18. N. V. Krylov, "Controlled diffusion processes", Nauka, Moscow, 1977; English transl., Springer, 1980.
19. N. V. Krylov, Introduction to the theory of diffusion processes, American Mathematical Society, Providence, RI, 1995.
20. H. J. Kuo and N. S. Trudinger, Evolving monotone difference operators on general space-time meshes, Duke Math. J., Vol. 91 (1998), No. 3, 587-607.
21. H. J. Kuo and N. S. Trudinger, Schauder estimates for fully nonlinear elliptic difference operators, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), No. 6, 1395-1406.
22. O. A. Ladyzhenskaya, "Kraevye zadachi matematicheskoj fiziki", Nauka, Moscow, 1973 in Russian;
23. English translation: "The boundary value problems of mathematical physics", Applied Math. Sciences, Vol. 49, Springer-Verlag, New York etc, 1985.
24. S. Larsson and V. Thomée, "Partial differential equations with numerical methods", Texts in Applied Mathematics, 45. Springer-Verlag, Berlin, 2003.
25. R. D. Richtmyer and K. W. Morton, "Difference methods for initial-value problems", Interscience Publishers, New York, 1967.
26. p for elliptic difference operators, SIAM J. Numer. Anal., Vol. 10 (1973), No. 1. 69-80.
27. R. Stevenson, Discrete Sobolev spaces and regularity of elliptic difference schemes, Mathematical Modelling and Numerical Analysis, Vol. 25 (1991), No. 5, 607-640.
28. O. B. Widlund, On the rate of convergence for parabolic difference schemes. II, Comm. Pure Appl. Math., Vol. 23 (1970), 79-96.
29. Jianfeng Zhang, Rate of convergence of finite difference approximations for degenerate ODEs, to appear in Math. Computation.