Title:
A Least-Squares Finite Element formulation of the Geometrically-Nonlinear
Elasticity Equations

Abstract:
We present a first-order system least-squares method to approximate the
solution to the equations of geometrically-nonlinear elasticity in two
dimensions. With assumptions of regularity on the problem, we show $H1$
equivalence of the norm induced by the FOSLS functional in the case of
pure displacement boundary conditions as well as local convergence of
Newton's method in a nested iteration setting.  Theoretical results hold
for deformations satisfying a small-strain assumption, a set we show to be
largely coincident with the set of deformations allowed by the model. 
Numerical results confirm optimal multigrid performance and finite element
approximation rates of the discrete functional in the pure displacement as
well as the mixed boundary condition case.