An adaptive finite element method for solving a double well problem describing crystalline microstructure
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 33 (1999) no. 4, p. 781-796
@article{M2AN_1999__33_4_781_0,
     author = {Prohl, Andreas},
     title = {An adaptive finite element method for solving a double well problem describing crystalline microstructure},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {33},
     number = {4},
     year = {1999},
     pages = {781-796},
     zbl = {0956.74064},
     mrnumber = {1726485},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1999__33_4_781_0}
}
Prohl, Andreas. An adaptive finite element method for solving a double well problem describing crystalline microstructure. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 33 (1999) no. 4, pp. 781-796. http://www.numdam.org/item/M2AN_1999__33_4_781_0/

[1] R.E. Bank, PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. User's Guide 6.0. SIAM Philadelphia (1990). | MR 1052151 | Zbl 0717.68001

[2] J. Ball and R. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | MR 906132 | Zbl 0629.49020

[3] J. Ball and R. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450. | Zbl 0758.73009

[4] C. Carstensen and P. Plechac, Numerical solution of the scalar double-well problem allowing microstructure. Math. Comp. 66 (1997) 997-1026. | MR 1415798 | Zbl 0870.65055

[5] M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math. 56 (1991) 747-767. | MR 1128031 | Zbl 0712.65063

[6] M. Chipot and C. Collins, Numerical approximations in variational problems with potential wells. SIAM J. Numer. Anal. 29 (1992) 1002-1019. | MR 1173182 | Zbl 0763.65049

[7] M. Chipot, C. Collins and D. Kinderlehrer, Numerical analysis of oscillations in multiple well problems. Numer. Math. 70 (1995) 259-282. | MR 1330864 | Zbl 0824.65045

[8] C. Collins, Computation and Analysis of Twinning in Crystalline Solids, Ph.D. thesis, University of Minnesota, USA (1990).

[9] P. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[10] C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential. SIAM J. Numer. Anal. 28 (1991) 321-332. | MR 1087507 | Zbl 0725.65067

[11] C. Collins and M. Luskin, Optimal order estimates for the finite element approximation of the solution of a nonconvex variational problem. Math. Comp. 57 (1991) 621-637. | MR 1094944 | Zbl 0735.65042

[12] J. Ericksen, Constitutive theory for some constrained elastic crystals. J. Solids and Structures 22 (1986) 951-964. | Zbl 0595.73001

[13] J. Ericksen, Some constrained elastic crystals, in Material Instabilites in Continuum Mechanics and Related Problems, J. Ball Ed., Oxford University Press, Oxford (1987) 119-137. | MR 970522 | Zbl 0655.73022

[14] J. Ericksen, Twinning of crystals I, in Metastability and Incompletely Posed Problems, S. Antman, J. Ericksen, D. Kinderlehrer and I. Muller Eds., Springer-Verlag, New York (1987) 77-96; IMA Volumes in Mathematics and Its Applications, Vol 3. | MR 870011 | Zbl 0638.73006

[15] M. Gobbert and A. Prohl, A discontinuous finite element method for solving a multi-well problem. Technical Report 1539, IMA (1998) and SIAM J. Numer. Anal. (to be published). | Zbl 0957.49019

[16] M. Gobbert and A. Prohl, A survey of classical and new finite element methods for the computation of crystalline microstructure. Technical Report 1576, IMA (1998).

[17] P. Gremaud, Numerical analysis of a nonconvex variational problem related to solid-solid phase transitions. SIAM J. Numer. Anal. 31 (1994) 111-127. | MR 1259968 | Zbl 0797.65052

[18] P. Kloucek, B. Li and M. Luskin, Analysis of a class of nonconforming finite elements for crystalline microstructure. Math. Comp. 67 (1996) 1111-1125. | MR 1344616 | Zbl 0903.65081

[19] P. Kloucek and M. Luskin, The computation of the dynamics of martensitic microstructure. Contin. Mech. Thermodyn. 6 (1994) 209-240. | MR 1285922 | Zbl 0825.73047

[20] M. Kruzik, Numerical approach to double well problems. SIAM J. Numer. Anal. 35 (1998) 1833-1849. | MR 1639950 | Zbl 0929.49016

[21] M. Kruzik, Oscillations, Concentrations and Microstructure Modeling, Ph.D. thesis, Charles University, Prague, Czech Republic (1996).

[22] B. Li, Analysis and Computation of Martensitic Microstructure, Ph.D. thesis, University of Minnesota, USA (1996).

[23] B. Li and M. Luskin, Finite element analysis of microstructure for the cubic to tetragonal transformation. SIAM J. Numer. Anal. 35 (1998) 376-392. | MR 1618484 | Zbl 0919.49020

[24] B. Li and M. Luskin, Nonconforming Finite element approximation of crystalline microstructure. Math. Comp. 67 (1998) 917-946. | MR 1459391 | Zbl 0901.73076

[25] B. Li and M. Luskin, Approximation of a martensitic laminate with varying volume fractions. RAIRO ModéL Math. Anal. Numér. 33 (1999) 67-87. | Numdam | MR 1685744 | Zbl 0928.74012

[26] M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density. Numer. Math. 75 (1997) 205-221. | MR 1421987 | Zbl 0874.73060

[27] M. Luskin, On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. | MR 1624603 | Zbl 0867.65033