In this paper, we use the finite element approximation for the solution of a multidimensional nonlinear system describing the evolution of materials which can pass through several phase transitions with dissipation. The finite element error estimates are obtained.
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References
1.
Blanchard, D. and H. Guidouche. 1990. "A Nonlinear System for Irreversible Phase Changes", European J. Appl. Math., 1:91-100.
2.
Ciarlet, P. G.1978. The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland.
3.
Colli, P. and K.-H. Hoffmann. 1993. "A Nonlinear Evolution Problem Describing Multi-Component Phase Changes with Dissipation", Numer. Funct. Anal. and Optimiz., 14:275-297.
4.
Colli, P. , M. Fremond and A. Visintin. 1990. "Thermo-Mechanical Evolution Shape Memory Alloys", Quart. Appl. Math., 48:31-47.
5.
Fremond, M.1990. "Shape Memory Alloys. A Thermomechanical Model", in Free Boundary Problems: Theory and Applications, Vol. I, Pitman Res. Notes Math. Ser. 185, K.-H. Hoffmann and J. Sprekels, eds., London: Longman, pp. 295-306.
6.
Glowinski, R.1984. Numerical Methods for Nonlinear Variational Problems. New York: Springer-Verlag.
7.
Hoffmann, K.-H. and J. Zou. 1993a. "Finite Element Approximations of a Landau-Ginzburg Model for Structural Phase Transitions in Shape Memory Alloys", Report No. 473 (1993), SPP-DFG, "Anwendungsbezogene Optimierung und Steuerung", Institute for Applied Mathematics, Technical University of Munich.
8.
Hoffmann, K.-H. and J. Zou. 1993b. "The Existence and Uniqueness of Global Solutions to a Mathematical Model for Shape Memory Alloys", Report No. 461 (1993), SPP-DFG, "Anwendungsbezogene Optimierung und Steuerung", Institute for Applied Mathematics, Technical University of Munich.
9.
Hoffmann, K.-H. , M. Niezgodka and S. M. Zheng. 1990. "Existence and Uniqueness of Global Solutions to an Extended Model of the Dynamical Developments in Shape Memory Allcys", Nonlinear Anal., Theory, Methods and Applications, 15:977-990.
10.
Johnson, C.1976. "A Convergence Estimate for an Approximation of a Parabolic Variational Inequality", SIAM J. Numer. Anal., 13:599-606.
11.
Kinderlehrer, D. and G. Stampacchia. 1980. An Introduction to Variational Inequalities and Their Applications. New York: Academic Press.
12.
Lions, J. L. and E. Magenes. 1972. Nonhomogeneous Boundary Value Problems and Applications I. New York: Springer-Verlag.
13.
Nochetto, R. H. and C. Verdi. 1988. "Approximation of Degenerate Parabolic Problems Using Numerical Integration, SIAM J. Numer. Anal., 25:784-814.
14.
Sprekels, J.1990. "Shape Memory Alloys: Mathematical Models for a Class of First Order Solid-Solid Phase Transitions in Metals", Control and Cybernetics, 19:287-308.
15.
Strang, G.1972. "Approximation in the Finite Element Method", Numer. Math., 19:81-98.
16.
Visintin, A.1987. "Supercooling and Superheating Effects in Heterogeneous Systems", Quart. Appl. Math., 45:239-263.
17.
Xu, J. 1989. "Theory of Multilevel Methods", Ph.D. dissertation, Cornell University.
