Least-squares methods for Stokes equations based on a discrete minus one inner product

• Published in: Journal of computational and applied mathematics Vol. 74; no. 1; pp. 155 - 173
• Main Authors: Bramble, James H., Pasciak, Joseph E.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 05-11-1996; Elsevier
• Subjects: Elasticity equations; Exact sciences and technology; Least-squares; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations, boundary value problems; Preconditioners; Sciences and techniques of general use; Stokes equations; Preconditioners; 65F10; Elasticity equations; Least-squares; 65N30; Stokes equations; Second order equation; Partial differential equations; Stokes problem; Sobolev space; Least squares method; Numerical methods; Elasticity theory; Preconditioning
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/0377-0427(96)00022-2

The purpose of this paper is to develop and analyze least-squares approximations for Stokes and elasticity problems. The major advantage of the least-squares formulation is that it does not require that the classical Ladyzhenskaya-Babǔska-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 (Ω) (the Sobolev space of order minus one on Ω). The use of such inner products (applied to second order problems) was proposed in an earlier paper by Bramble, Lazarov and Pasciak (1994).