First-Order System Least Squares for Geometrically Nonlinear Elasticity

Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. 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. Acknowledgements I'd like to thank Tom Manteuffel and Steve McCormick, whose encouragement and help have contributed greatly to my progress in research and professional development. I have greatly enjoyed working with them over the past several years on this project and count myself fortunate to be able to call them my friends. I will forever associate mathematical research with bad puns. It has been my great priviledge to work with the Grandview Group. Ruge have made work fun, especially when we weren't working. I also would like to acknowledge the Applied Mathematics department and NSF VIGRE for funding my career as a graduate student. I am thankful for having the chance to work in such a great department. Thanks to my family for support and encouragement, I am forever grateful.