based on a discrete minus one inner product

Department of Mathematics

Texas A&M University

College Station, TX 77843

Applied Math Department

Brookhaven National Laboratory

Upton, NY 11973

The purpose of this paper is to develop and analyze least-squares
approximations for Stokes and elasticity problems. The major advantage of the
least-square formulation is that it does not require that the classical
Ladyzhenskaya-Babuska-Brezzi (LBB) condition be satisfied. We provide two
methods. The first is posed in terms of the velocity-pressure pair without
the introduction of additional variables. The second adds a vorticity
variable. In both cases, we employ least-squares functionals which involve a
discrete inner product which is related to the inner product in
H^{-1}(\d) (the Sobolev space of order minus one on \d). The use of
such inner products (applied to second order problems) was proposed in an
earlier paper by Bramble, Lazarov, and Pasciak [13].

Contributed November 6, 1995.