On the finite element method for parabolic equations, I; approximation of holomorphic semi-groups

Hiroshi FUJITA; Akira MIZUTANI

doi:10.2969/jmsj/02840749

Hiroshi FUJITA, Akira MIZUTANI

著者情報

ジャーナルフリー

1976 年 28 巻 4 号 p. 749-771

DOI https://doi.org/10.2969/jmsj/02840749

詳細

Published: 1976 Received: 1976/01/27 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文タイトル訂正内容: Wrong : On the finite element method for parabolic equations, I
Right : On the finite element method for parabolic equations, I; approximation of holomorphic semi-groups

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 論文サブタイトル訂正内容: Wrong : approximation of holomorphic semi-groups

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) I. Babuska and A. K. Aziz, Foundations of the finite element method, pp. 3-359. The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972.
2) J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124.
3) J. H. Bramble and M. Zlámal, Triangular elements in the finite element method, Math. Comp., 24 (1970), 809-820.
4) J. H. Bramble and V. Thomée, Semi-discrete least square methods for a parabolic boundary value problems, Math. Comp., 26 (1972), 633-648.
5) P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer, Berlin-Heidelberg-New York, 1967.
6) J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1970), 575-626.
7) H. Fujii, Finite element schemes: stability and convergence, pp. 201-218. Advances in Computational Methods in Structural Mechanics and Design, UAH Press, Alabama, 1972.
8) H. Fujita, On the finite element approximation for parabolic equations: an operator theoretical approach, Proceedings of the IRIA symposium, 1975, 2nd international symposium on computing methods in applied sciences and engineering, Springer Lecture Note, to appear.
9) H. P. Helfrich, Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen, Manuscripta Math., 13 (1974), 219-235.
10) T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.
11) T. Kato, Fractional powers of dissipative operators, II, J. Math. Soc. Japan, 14 (1962), 242-248.
12) T. Kato, Some mapping theorems for the numerical range, Proc. Japan Acad., 41 (1965), 652-655.
13) T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg-New York, 1966.
14) P. D. Lax and B. Wendroff, Difference schemes for hyperbolic equations with higher order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381-398.
15) J. L. Lions, Equations Différentielles Opérationnelles et Problèmes au Limites, Springer, Berlin-Heidelberg-New York, 1961.
16) J. L. Lions, Espaces d'interpolation et domaines du puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241.
17) J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1 and 2, Dunod, Paris, 1968.
18) G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, 1973.
19) V. Thomée, Spline approximation and difference schemes for the heat equation, pp. 711-736. The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972.
20) T. Ushijima, On the finite element approximation of parabolic equations-Consistency, boundedness and convergence, Mem. Numer. Math., 2 (1975), 21-33.
21) K. Yosida, Functional Analysis, Springer, Berlin-Heidelberg-New York, 1965.
22) M. Zlámal, Curved elements in the finite element method, I, SIAM J. Numer. Anal., 10 (1973), 229-240.
23) M. Zlámal, Curved elements in the finite element method, II, SIAM J. Numer. Anal., 11 (1974), 347-362.
24) M. Zlámal, Finite element methods for parabolic equations, Math. Comp., 28 (1974), 393-404.

Right : [1] I. Babuska and A. K. Aziz, Foundations of the finite element method, pp. 3-359. The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972.
[2] J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM J. Numer. Anal., 7 (1970), 112-124.
[3] J. H. Bramble and M. Zlámal, Triangular elements in the finite element method, Math. Comp., 24 (1970), 809-820.
[4] J. H. Bramble and V. Thomée, Semi-discrete least square methods for a parabolic boundary value problems, Math. Comp., 26 (1972), 633-648.
[5] P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer, Berlin-Heidelberg-New York, 1967.
[6] J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (1970), 575-626.
[7] H. Fujii, Finite element schemes: stability and convergence, pp. 201-218. Advances in Computational Methods in Structural Mechanics and Design, UAH Press, Alabama, 1972.
[8] H. Fujita, On the finite element approximation for parabolic equations: an operator theoretical approach, Proceedings of the IRIA symposium, 1975, 2nd international symposium on computing methods in applied sciences and engineering, Springer Lecture Note, to appear.
[9] H. P. Helfrich, Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen, Manuscripta Math., 13 (1974), 219-235.
[10] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.
[11] T. Kato, Fractional powers of dissipative operators, II, J. Math. Soc. Japan, 14 (1962), 242-248.
[12] T. Kato, Some mapping theorems for the numerical range, Proc. Japan Acad., 41 (1965), 652-655.
[13] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heidelberg-New York, 1966.
[14] P. D. Lax and B. Wendroff, Difference schemes for hyperbolic equations with higher order of accuracy, Comm. Pure Appl. Math., 17 (1964), 381-398.
[15] J. L. Lions, Equations Différentielles Opérationnelles et Problèmes au Limites, Springer, Berlin-Heidelberg-New York, 1961.
[16] J. L. Lions, Espaces d'interpolation et domaines du puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241.
[17] J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Vol. 1 and 2, Dunod, Paris, 1968.
[18] G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, 1973.
[19] V. Thomée, Spline approximation and difference schemes for the heat equation, pp. 711-736. The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972.
[20] T. Ushijima, On the finite element approximation of parabolic equations-Consistency, boundedness and convergence, Mem. Numer. Math., 2 (1975), 21-33.
[21] K. Yosida, Functional Analysis, Springer, Berlin-Heidelberg-New York, 1965.
[22] M. Zlámal, Curved elements in the finite element method, I, SIAM J. Numer. Anal., 10 (1973), 229-240.
[23] M. Zlámal, Curved elements in the finite element method, II, SIAM J. Numer. Anal., 11 (1974), 347-362.
[24] M. Zlámal, Finite element methods for parabolic equations, Math. Comp., 28 (1974), 393-404.

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