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Abstract. This paper develops two first-order system least-squares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are two-stage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired). One approach, which uses L 2 norms to define the FOSLS functional, is shown under certain H 2 regularity assumptions to admit optimal H 1 -like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the Poisson ratio for all variables. The second approach, which is based on H −1 norms, is shown under general assumptions to admit optimal uniform performance for displacement flux in an L 2 norm and for displacement in an H 1 norm. These methods do not degrade as other methods generally do when the material properties approach the incompressible limit.

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