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We consider a model of energy minimization arising in the study of themechanical behavior caused by cell contraction within a fibrous biologicalmedium. The macroscopic model is based on the theory of non rank-one convexnonlinear elasticity for phase transitions. We study appropriate numericalapproximations based on the discontinuous Galerkin treatment of highergradients and used succesfully in numerical simulations of experiments. We showthat the discrete minimizers converge in the limit to minimizers of thecontinuous problem. This is achieved by employing the theory of$\Gamma$-convergence of the approximate energy functionals to the continuousmodel when the discretization parameter tends to zero. The analysis is involveddue to the structure of numerical approximations which are defined in spaceswith lower regularity than the space where the minimizers of the continuousvariational problem are sought. This fact leads to the development of a newapproach to $\Gamma$-convergence, appropriate for discontinuous finite elementdiscretizations, which can be applied to quite general energy minimizationproblems. Furthermore, the adoption of exponential terms penalising theinterpenetration of matter requires a new framework based on Orlicz spaces fordiscontinuous Galerkin methods which is developed in this paper as well.

Approximations of energy minimization in cell-induced phase transitions of fibrous biomaterials: $\Gamma$-convergence analysis