Determination of the modulus of quadrilaterals by finite element methods

Heihachiro HARA; Hisao MIZUMOTO

doi:10.2969/jmsj/04220295

Heihachiro HARA, Hisao MIZUMOTO

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JOURNAL FREE ACCESS

1990 Volume 42 Issue 2 Pages 295-326

DOI https://doi.org/10.2969/jmsj/04220295

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Published: 1990 Received: March 28, 1989 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: CITATION Details: Wrong : 1) J.E. Akin, Elements for the analysis of line singularities, J.R. Whiteman (ed.). The Mathematics of Finite Elements and Applications III, Academic Press, London, 1979, pp. 65-75.
2) I. Babuška, The selfadaptive approach in the finite element method, J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications II, Academic Press, London, 1976, pp. 125-142.
3) I. Babuška and M.B. Rosenzweig, A finite element scheme for domains with corners, Numer. Math., 20 (1972), 1-21.
4) I. Babuška, B.A. Szabo and I.N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515-545.
5) R.E. Barnhill and J.R. Whiteman, Error analysis of finite element methods with triangles for elliptic boundary value problems, J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications, Academic Press, London, 1973, pp. 83-112.
6) W.S. Blackburn, Calculation of stress intensity factors at crack tips using special finite elements, J.R. Whiteman (ed.), The Mathematics of Finite Elements and. Applications, Academic Press, London, 1973, pp. 327-336.
7) J.H. Bramble and M. Zlámal, Triangular elements in the finite element method, Math. Comp., 24 (1970), 809-820.
8) A.W. Craig, J.Z. Zhu and O.C. Zienkiewicz, A-posteriori error estimation, adaptive mesh refinement and multigrid methods using hierarchical finite element bases, J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V, Academic. Press, London, 1985, pp. 587-594.
9) D. Gaier, Ermittlung des konformen Moduls von Vierecken mit Differenzenmethoden, Numer. Math., 19 (1972), 179-194.
10) H. Mizumoto, An application of Green's formula of a discrete function: Determination of periodicity moduli, I, Kodai Math. Sem. Rep., 22 (1970), 231-243.
11) H. Mizumoto, An application of Green's formula of a discrete function: Determination of periodicity moduli, II, Kodai Math. Sem. Rep., 22 (1970), 244-249.
12) H. Mizumoto, A finite-difference method on a Riemann surface, Hiroshima Math J., 3(1973), 277-332.
13) H. Mizumoto and H. Hara, Finite element method in engineering science: Theory (in Japanese), Morikita, Tokyo, 1983.
14) H. Mizumoto and H. Hara, Finite element method in engineering science: Programs (in Japanese), Morikita, Tokyo, 1983.
15) H. Mizumoto and H. Hara, Finite element approximations of harmonic differentials on a Riemann surface, Hiroshima Math. J., 18(1988), 617-654.
16) G. Opfer, Untere, beliebig verbesserbare Schranken für den Modul eines zweifach zusammenhängenden Gebietes mit Hilfe von Differenzenverfahren, Dissertation, Hamburg, 1967, pp. 1-65.
17) G. Opfer, Die Bestimmung des Moduls zweifach zusammenhängender Gebiete mit Hilfe von Differenzenverfahren, Arch. Rational Mech. Anal., 32 (1969), 281-297.
18) G. Opfer and M.L. Puri, Complex planar splines, J. Approx. Theory, 31 (1981), 383-402.
19) M.C. Rivara, Dynamic implementation of the h-version of the finite element method, J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V, Academic Press, London, 1985, pp. 595-602.
20) A.H. Schatz and L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part I, Math. Comp., 32 (1978), 73-109.
21) A.H. Schatz and L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part II, Math. Comp., 33 (1979), 465-492.
22) G. Strang and G.J. Fix, An analysis of the finite element method, Prentice-Hall, Englewood Cliffs, 1973.
23) R.W. Thatcher, The use of infinite grid refinements at singularities in the solution of Laplace's equation, Numer. Math., 25 (1976), 163-178.
24) G. Tsamasphyros, Singular element construction using a mapping technique, Internat. J. Numer. Methods Engrg., 24 (1987), 1305-1316.
25) J. Weisel, Numerische Ermittlung quasikonformer Abbildungen mit finiten Elementen, Numer. Math., 35 (1980), 201-222.
26) J.R. Whiteman and J.E. Akin, Finite elements, singularities and fracture, J.R. Whiteman (ed.), The Mathematics of Finite Elements and Applications III, Academic Press, London, 1979, pp. 35-54.
27) H. Yserentant, On the multi-level spliting of finite element spaces, Numer. Math., 49 (1986), 379-412.

Right : [1] J. E. Akin, Elements for the analysis of line singularities, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications III, Academic Press, London, 1979, pp. 65-75.
[2] I. Babuška, The selfadaptive approach in the finite element method, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications II, Academic Press, London, 1976, pp. 125-142.
[3] I. Babuška and M. B. Rosenzweig, A finite element scheme for domains with corners, Numer. Math., 20 (1972), 1-21.
[4] I. Babuška, B. A. Szabo and I. N. Katz, The p-version of the finite element method, SIAM J. Numer. Anal., 18 (1981), 515-545.
[5] R. E. Barnhill and J. R. Whiteman, Error analysis of finite element methods with triangles for elliptic boundary value problems, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications, Academic Press, London, 1973, pp. 83-112.
[6] W. S. Blackburn, Calculation of stress intensity factors at crack tips using special finite elements, J.R. Whiteman (ed.), The Mathematics of Finite Elements and. Applications, Academic Press, London, 1973, pp. 327-336.
[7] J. H. Bramble and M. Zlámal, Triangular elements in the finite element method, Math. Comp., 24 (1970), 809-820.
[8] A. W. Craig, J. Z. Zhu and O. C. Zienkiewicz, A-posteriori error estimation, adaptive mesh refinement and multigrid methods using hierarchical finite element bases, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V, Academic. Press, London, 1985, pp. 587-594.
[9] D. Gaier, Ermittlung des konformen Moduls von Vierecken mit Differenzenmethoden, Numer. Math., 19 (1972), 179-194.
[10] H. Mizumoto, An application of Green's formula of a discrete function: Determination of periodicity moduli, I, Kodai Math. Sem. Rep., 22 (1970), 231-243.
[11] H. Mizumoto, An application of Green's formula of a discrete function: Determination of periodicity moduli, II, Kodai Math. Sem. Rep., 22 (1970), 244-249.
[12] H. Mizumoto, A finite-difference method on a Riemann surface, Hiroshima Math J., 3(1973), 277-332.
[13] H. Mizumoto and H. Hara, Finite element method in engineering science: Theory (in Japanese), Morikita, Tokyo, 1983.
[14] H. Mizumoto and H. Hara, Finite element method in engineering science: Programs (in Japanese), Morikita, Tokyo, 1983.
[15] H. Mizumoto and H. Hara, Finite element approximations of harmonic differentials on a Riemann surface, Hiroshima Math. J., 18 (1988), 617-654.
[16] G. Opfer, Untere, beliebig verbesserbare Schranken für den Modul eines zweifach zusammenhängenden Gebietes mit Hilfe von Differenzenverfahren, Dissertation, Hamburg, 1967, pp. 1-65.
[17] G. Opfer, Die Bestimmung des Moduls zweifach zusammenhängender Gebiete mit Hilfe von Differenzenverfahren, Arch. Rational Mech. Anal., 32 (1969), 281-297.
[18] G. Opfer and M.L. Puri, Complex planar splines, J. Approx. Theory, 31 (1981), 383-402.
[19] M. C. Rivara, Dynamic implementation of the h-version of the finite element method, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications V, Academic Press, London, 1985, pp. 595-602.
[20] A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part I, Math. Comp., 32 (1978), 73-109.
[21] A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains, Part II, Math. Comp., 33 (1979), 465-492.
[22] G. Strang and G.J. Fix, An analysis of the finite element method, Prentice-Hall, Englewood Cliffs, 1973.
[23] R. W. Thatcher, The use of infinite grid refinements at singularities in the solution of Laplace's equation, Numer. Math., 25 (1976), 163-178.
[24] G. Tsamasphyros, Singular element construction using a mapping technique, Internat. J. Numer. Methods Engrg., 24 (1987), 1305-1316.
[25] J. Weisel, Numerische Ermittlung quasikonformer Abbildungen mit finiten Elementen, Numer. Math., 35 (1980), 201-222.
[26] J. R. Whiteman and J. E. Akin, Finite elements, singularities and fracture, J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications III, Academic Press, London, 1979, pp. 35-54.
[27] H. Yserentant, On the multi-level spliting of finite element spaces, Numer. Math., 49 (1986), 379-412.

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