Numerical solution of the scalar double-well problem allowing microstructure

The direct numerical solution of a non-convex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem. The problem (RP) has a minimiser u and a related stress field σ = DW**(⊇u) which is known to coincide with the stress field obtained by solving (P) in a generalised sense involving Young measures. If u h is a finite element solution, σ h := DW**(⊇u h ) is the related discrete stress field. We prove a priori and a posteriori estimates for σ-σ h in L 4/3 (Ω) and weaker weighted estimates for ⊇u - ⊇u h . The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

[1]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[2]  Roger Temam,et al.  On the anti-plane shear problem in finite elasticity , 1981 .

[3]  S. Nash.  Newton-Type Minimization via the Lanczos Method , 1984 .

[4]  R. Kohn,et al.  Numerical study of a relaxed variational problem from optimal design , 1986 .

[5]  J. Ball,et al.  Fine phase mixtures as minimizers of energy , 1987 .

[6]  J. Ball.  A version of the fundamental theorem for young measures , 1989 .

[7]  Donald A. French,et al.  On the convergence of finite-element approximations of a relaxed variational problem , 1990 .

[8]  Patricia Bauman,et al.  A nonconvex variational problem related to change of phase , 1990 .

[9]  Mabry.  Sets which are Well-Distributed and Invariant Relative to All Isometry Invariant Total Extensions of Lebesgue Measure , 1990 .

[10]  Pablo Pedregal,et al.  Characterizations of young measures generated by gradients , 1991 .

[11]  Michel Chipot,et al.  Numerical analysis of oscillations in nonconvex problems , 1991 .

[12]  Mitchell Luskin,et al.  Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem , 1991 .

[13]  Pablo Pedregal,et al.  Weak convergence of integrands and the young measure representation , 1992 .

[14]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[15]  R. D. James,et al.  Proposed experimental tests of a theory of fine microstructure and the two-well problem , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[16]  Michel Chipot,et al.  Numerical approximations in variational problems with potential wells , 1992 .

[17]  R. A. Nicolaides,et al.  Computation of Microstructure Utilizing Young Measure Representations , 1993 .

[18]  Pablo Pedregal,et al.  Jensen's inequality in the calculus of variations , 1994, Differential and Integral Equations.

[19]  B. Brighi,et al.  Approximated convex envelope of a function , 1994 .

[20]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[21]  R. Verfürth.  A posteriori error estimates for nonlinear problems: finite element discretizations of elliptic equations , 1994 .

[22]  Gero Friesecke,et al.  A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems , 1994, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[23]  R. A. Nicolaides,et al.  Strong convergence of numerical solutions to degenerate variational problems , 1995 .

[24]  Tomáš Roubíček,et al.  Relaxation in Optimization Theory and Variational Calculus , 1997 .