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

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JOURNAL FREE ACCESS

1976 Volume 28 Issue 4 Pages 749-771

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

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Published: 1976 Received: January 27, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: 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

Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : approximation of holomorphic semi-groups

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: 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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