First-Order System Least Squares for Geometrically Nonlinear Elasticity

We present a first-order system least-squares (FOSLS) method to approximate the solution to the equations of geometrically nonlinear elasticity in two dimensions. 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 with a total work bounded by about 25 fine-grid relaxation sweeps.

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