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Sharp lower bounds for finite element approximations of the second order laminate minimiser of a finite-well non-convex functional
Lower bounds for finite element approximations of minimisers of non-convex functionals have been studied in the case where the functional has two wells and the function class has affine boundary condition in the first lamination convex hull of the wells.
In this note we give sharp lower bounds to a non-convex functional with three wells over the class of piecewise affine on triangular grid functions with affine boundary condition in the second lamination hull of the wells. If h is triangulation size, the energy of the functional is shown to be bounded below by h1/3 / 500000 the upper bound of c0 h1/3 is already proved for a similar functional in "The appearance of microstructures in problems with incompatible wells and their numerical approach." Chipot, M; Numer. Math 83 (1999) no.3 325-352.