Benjamin Müller, University of Duisburg-Essen, Essen, Germany Joint work with Gerhard Starke, Jörg Schröder and Alexander Schwarz The finite element method is an important tool for the simulation of elasticity problems in solid mechanics. It is well known that the linear elastic theory does not cover arising real life problems. Physically more realistic models lead to nonlinear partial differential equations. In this talk we present least squares finite element methods based on the momentum balance and nonlinear constitutive equations for hyperelastic materials. Our approach is motivated by a well-studied least squares formulation for linear elasticity. The idea in this approach is to invert the given stress-strain relation such that it is possible to consider fully incompressible materials. Our aim is to generalize this idea to an approach which takes nonlinear kinematics and nonlinear stress-strain relations into account. General least squares formulations for isotropic homogeneous frame-indifferent hyperelastic materials based on inverse stress-strain relations will be derived. For the special case of a Neo-Hooke material we can consider, similar to linear elasticity, the full incompressible case. A detailed analysis for the nonlinear and the linearized problem will be provided. It will be shown under strong regularity assumptions that the (nonlinear) least squares functional is an efficient and reliable a-posteriori error estimator. A further novelty of our approach, in comparison to other discretization methods, is that next to the displacement u the full first Piola- Kirchhoff stress tensor P is considered and both are approximated simultaneously. At the end of the talk we will illustrate the performance of our method in some numerical experiments using second to lowest order Raviart-Thomas elements for the stress tensor, continuous piecewise quadratic elements for the displacement vector and adaptive refinement.

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