First Order System Least Squares for the Pure Traction Problem in
Planar Linear Elasticity
Z. Cai, T. A. Manteuffel, S. F. McCormick, and S. V. Parter
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, then for the displacement itself. One approach, which uses L2
norms to define the FOSLS functional, is shown under full regularity
assumptions to admit optimal H1 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
inverse norms, is shown under general assumptions to admit optimal uniform
performance for displacement flux in an L2 norm and for displacement in an H1
norm. These methods do not degrade as other methods generally do when the
material properties approach the incompressible limit.