Mixed FEM and the discontinuous Petrov-Galerkin (DPG) methodology in numerical homogenization

The mixed finite element method uses an approximation of at least two fields (e.g. displacements and stresses). Even though it is difficult to construct stable approximation for such formulations, the mixed methods provide much better convergence of the flux than the standard methods, even for heterogeneous materials with a high contrast of component parameters. Furthermore, the mixed formulations are well-posed for incompressible materials. In this work a novel application of the mixed FEM to compu- tational homogenization is presented and examined. Conformity with the H(div) space is provided by the Piola transformation and the approximation orders are assumed according to the exact sequence of energy spaces. The modified Hellinger-Reissner principle with weakly imposed stress tensor symmetry through the introduction of the Lagrange multiplier is used and the Discontinuous Petrov-Galerkin approach (DPG), with built-in a posteriori error estimation, provides stability of approximation.