Approximation of a martensitic laminate with varying volume fractions
ESAIM: Modélisation mathématique et analyse numérique, Tome 33 (1999) no. 1, pp. 67-87. 
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     author = {Li, Bo and Luskin, Mitchell},
     title = {Approximation of a martensitic laminate with varying volume fractions},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {67--87},
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     volume = {33},
     number = {1},
     year = {1999},
     mrnumber = {1685744},
     zbl = {0928.74012},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1999__33_1_67_0/}
}
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Li, Bo; Luskin, Mitchell. Approximation of a martensitic laminate with varying volume fractions. ESAIM: Modélisation mathématique et analyse numérique, Tome 33 (1999) no. 1, pp. 67-87. http://www.numdam.org/item/M2AN_1999__33_1_67_0/

[1] R. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl

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

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

[4] Z. S. Basinski and J. W. Christian, Experiments on the martensitic transformation in single crystals of indium-thallium alloys.Acta Metall. 2 (1954) 148-166.

[5] M. W. Burkart and T. A. Read, Diffusionless phase changes in the indium-thallium System. Trans. AIME J. Metals 197 (1953)1516-1524.

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

[7] M. Chipot, Numerical analysis of oscillations in nonconvex problems. Numer. Math. 59 (1991) 747-767. | MR | Zbl

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

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

[10] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rat Mech. Anal 103 (1988) 237-277. | MR | Zbl

[11] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[12] 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 | Zbl

[13] 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 | Zbl

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

[15] J. Ericksen, Stable equilibrium configurations of elastic crystals. Arch. Rat. Mech. Anal. (1986) 1-14. | MR | Zbl

[16] D. French, On the convergence of finite element approximations of a relaxed variational problem. SIAM J. Numer. Anal. 28, (1991) 419-436. | MR | Zbl

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

[18] R. James, Minimizing sequences and the microstructure of crystals in Proceedings of the-Society of Metals Conference on Phase Transformations. Cambridge University Press (1989).

[19] D. Kinderlehrer, Twinning in crystals II. In Metastability and Incompletely Posed Problems. S. Antman, J.L. Ericksen, D. Kinderlehrer and I. Muller Eds. IMA Volumes in Mathematics and Its Applications. Springer-Verlag, New York 3 (1987) 185-212. | MR | Zbl

[20] D. Kinderlehrer and P. Pedregal, Characterizations of gradient Young measures. Arch. Rat Mech. Anal 115, (1991) 329-365. | MR | Zbl

[21] R. Kohn, Relaxation of a double-well energy. Cont Mech. Thermodyn. 3 (1991) 193-236. | MR | Zbl

[22] 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 | Zbl

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

[24] M. Luskin, On the computation of crystalline microstructure. Acta Numerica (1996) 191-257. | MR | Zbl

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

[26] M. Luskin and L. Ma, Analysis of the finite element approximation of microstructure in micromagnetics. SIAM J. Numer. Anal. 29 (1992) 320-331. | MR | Zbl

[27] R. A. Nicolaides and N. Walkington, Strong convergence of numerical solutions to degenerate variational problems. Math. Comp. 64 (1995) 117-127. | MR | Zbl

[28] P. Pedregal, Numerical approximation of parametrized measures. Num. Funct. Anal. Opt. 16, (1995) 1049-1066. | MR | Zbl

[29] P. Pedregal, On the numerical analysis of non-convex variational problems. Numer. Math. 74 (1996) 325-336. | MR | Zbl

[30] A. Quarteronia and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin (1994). | MR | Zbl

[31] T. Roubíček, Numerical approximation of relaxed variational problems. J. Convex Analysis 3, (1996) 329-347. | MR | Zbl

[32] V. Šverák, Lower-semicontinuity of variational integrals and compensated compactness in Proceedings ICM 94 Birkhäuser. Zürich (1995). | MR | Zbl

[33] L. Tartar, Compensated compactness and applications to partial differential equations in Nonlinear analysis and mechanics. Pitman Research Notes in Mathematics. R. Knops Eds. Pitman, London (1978). | MR | Zbl

[34] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR | Zbl