on a Discrete Minus One Inner Product

Department of Mathematics

Cornell University

Ithaca, NY 14853 and

Department of Mathematics

Texas A&M University

College Station, TX 77843

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 a least-squares
approximation to a first order system. The first order system represents a
reformulation of a second order elliptic boundary value problem which may be
indefinite and/or nonsymmetric. The approach taken here is novel in that the
least-squares functional employed involves 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 this inner product results in a method of
approximation which is optimal with respect to the required regularity as well
as the order of approximation even when applied to problems with low
regularity solutions. In addition, the discrete system of equations which
needs to be solved in order to compute the resulting approximation is easily
preconditioned, thus providing an efficient method for solving the algebraic
equations. The preconditioner for this discrete system only requires the
construction of preconditioners for standard second order problems, a task
which is well understood.

Contributed November 6, 1995.