Analysis of discretizations of parabolic problems in pairs of spaces

BIT Numerical Mathematics

Volumn 40, Issue 1, 2000, Pages 1-23

  Bakaev, Nikolai Yu ^a  

^a Air Force Engineering Academy   (Russian Federation)

Author keywords

Discretization in time; Pairs of spaces; Parabolic problems; Stability and error estimates

Indexed keywords

EID: 0002645062     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022358016505     Document Type: Article

References (20)

1. N. Yu. Bakaev, Analysis of linear discrete parabolic problems. Part 1. Well-posedness of a Cauchy problem for the evolution equation in equation in discrete time, Applied Mathematics and Computation Reports 1997:1, Universidad de Valladolid, Valladolid.
2. N. Yu. Bakaev, Analysis of linear discrete parabolic problems. Part 2. Well-posedness of single-step time-discrete approximations to abstract parabolic equations, Applied Mathematics and Computation Reports 1998:6, Universidad de Valladolid, Valladolid.
3. N. Yu. Bakaev, Analysis of fully discrete approximations to parabolic problems with rough and distribution-valued initial data, in preparation.
4. N. Yu. Bakaev, S. Larsson, and V. Thomée, Long time behavior of backward difference type methods for parabolic equations with memory in Banach space, Report No. 12, Institut Mittag-Leffler, Stockholm, 1997/98.
5. G. A. Baker, J. H. Bramble, and V. Thomée, Single step Galerkin approximations for parabolic problems, Math. Comp., 31 (1977), pp. 818-847.
6. J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal., 14 (1977), pp. 218-241.
7. Ph. Brenner and V. Thomée, On rational approximations of semigroups, SIAM J. Numer. Anal., 16 (1979), pp. 683-394.
8. M. Crouzeix, S. Larsson, S. Piskarev, and V. Thomée, The stability of rational approximations of analytic semigroups, BIT 33 (1993), pp. 74-84.
9. M. Crouzeix, S. Larsson, and V. Thomée, Resolvent estimates for elliptic finite element operators in one dimension, Math. Comp., 63 (1994), pp. 121-140.
10. H. Komatsu, Fractional powers of operators. II. Interpolation spaces, Pacific J. Maths 21 (1967), pp. 89-111.
11. H. Komatsu, Fractional powers of operators. VI. Interpolation of non-negative operators and imbedding theorems, J. Fac. Sci. Univ. Tokyo, Sec. IA, Math. 19 (1972), 1-63.
12. S. G. Krein, Linear differential equations in Banach space, Nauka, Moscow, 1973 (in Russian).
13. M.-N. Le Roux, Semidiscretization in time for parabolic problems, Math. Comp., 33 (1979), pp. 919-931.
14. C. Lubich and O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991), pp. 293-313.
15. C. Palencia, A stability result for sectorial operators in Banach spaces, SIAM J. Numer. Anal., 30 (1993), pp. 1373-1384.
16. C. Palencia, On the stability of variable stepsize rational approximations of holomorphic semigroups, Math. Comp., 62 (1994), pp. 93-103.
17. A. H. Schatz, V. Thomée, and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math., 33 (1980), pp. 265-304.
18. P. E. Sobolevskii, On the Green's functions of arbitrary (in particular, integral) powers of elliptic operators, Dokl. Akad. Nauk SSSR 142 (1962), pp. 804-807 (in Russian).
19. B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc., 199 (1974), pp. 141-161.
20. V. Thomée and L. B. Wahlbin, Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable, Numer. Math., 41 (1983), pp. 345-371.