Analysis of a two-stage least-squares finite element method for the planar elasticity problem

A new first-order formulation for the two-dimensional elasticity equations is proposed by introducing additional variables which, called stresses here, are the derivatives of displacements. The resulted stress-displacement system can be further decomposed into two dependent subsystems, the stress system and the displacement system recovered from the stresses. For constructing finite element approximations to these subsystems with appropriate boundary conditions, a two-stage least-squares procedure is introduced. The analysis shows that, under suitable regularity assumptions, the rates of convergence of the least-squares approximations for all the unknowns are optimal both in the H¹-norm and in L²-norm. Also, numerical experiments with various Poisson's ratios are examined to demonstrate the theoretical estimates.