| In this thesis, we develop a first-order system least-squares (FOSLS) method to approximate the solution to the equations of geometrically-nonlinear elasticity in two dimensions. We consider a Newton-FOSLS type of algorithm to treat the nonlinear problem by successive linear Newton steps, each solved by casting the problem as a first-order system and employing a least-squares finite element discretization.; With assumptions of regularity on the problem; we show H 1 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.; In the case of singular solutions induced by boundary conditions on polygonal domains, we further investigate a weighted-norm least-squares method for recovering optimal finite element convergence properties in terms of both weighted and nonweighted norms. The theory of this general technique is studied in terms of a simplified div-curl system and shown to be similarly effective as applied to the elasticity system. |
