Numerical methods for minimizers and microstructures in nonlinear elasticity

A standard finite element method and a finite element truncation method are applied to solve the boundary value problems of nonlinear elasticity with certain nonconvex stored energy functions such as those of St. Venant-Kirchhoff materials. Finite element solutions are proved to exist and of the form of minimizers in appropriate sets of admissible finite element functions for both methods. Convergence of the finite element solutions to a solution in the form of a minimizer or microstructure for the boundary value problem is established. It is also shown that in the presence of Lavrentiev phenomenon the finite element truncation method can overcome the difficulty and converges to the absolute minimum while the standard finite element method converges to a pseudominimum which is a minimum in a slightly smaller set of admissible functions.