First-Order System Least Squares (FOSLS) for Spatial Linear Elasticity: Pure Traction

This paper develops first-order system least-squares (FOSLS) functionals for solving the pure traction problem in three-dimensional linear elasticity. It is a direct extension of an earlier paper on planar elasticity [Z. Cai, T. A. Manteuffel, S. F. McCormick, and S. V. Parter, SIAM J. Numer. Anal., 35 (1998), pp. 320–335]. Two functionals are developed, one involving L norms of the first-order system and the other involving dual norms. These functionals are shown to be equivalent to appropriate product Sobolev norms, uniformly in the Poisson ratio. These results imply that standard finite element discretization and iterative solver techniques can be applied to obtain performance that is optimal—even as the material nears the incompressible limit.