Least-squares Methods for Linear Elasticity Based on a Discrete minus One Inner Product

The purpose of this paper is to develop and analyze least-squares approximations for elasticity problems. The major advantage of the least-square formulation is that it does not require that the classical Ladyzhenskaya-Babǔska-Brezzi (LBB) condition be satisfied. By employing least-squares functionals which involve a discrete inner product which is related to the inner product in H−1(Ω) (the Sobolev space of order minus one on Ω) we develop a finite element method which is unconditionally stable for problems with traction type of boundary conditions and for almost and incompressible elastic media. The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov and Pasciak [7].