Numerical Methods for Minimizers and Microstructures in Nonlinear Elasticity

A standard finite element method and a finite element trunca-tion method are applied to solve the boundary value problems of nonlinear elasticity with certain nonconvex stored energy func-tions such as those of St. Venant-Kirchhoff materials. Finite element solutions are proved to exist and to be in the form of minimizers in appropriate sets of admissible finite element func-tions for both methods. Convergence of the finite element solu-tions 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 in the problem the finite element truncation method can overcome the difficulty and converges to the absolute minimum while the standard fi-nite element method converges to a poseudominimum which is a minimum in a slightly small set of admissible functions. Subject classification: 65N12, 65N30, 35J60, 49J45, 73C50 Key words: numerical method, minimizer, microstructure, wea