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Bramble, James H.; Pasciak, Joseph E.

Least-squares methods for Stokes equations based on a discrete minus one inner product. (English) Zbl 0867.65055

J. Comput. Appl. Math. 74, No. 1-2, 155-173 (1996).
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-Babuska-Brezzi (LBB) condition be satisfied. Two methods are provided. 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, least-squares functionals are provided which involve a discrete inner product which is related to the inner product in \(H^1(\Omega)\) (the Sobolev space of order minus one in \(\Omega)\).
The use of such inner products (applied to second problems) was proposed in an earlier paper by J. H. Bramble, R. D. Lazarov and J. E. Pasciak [A least-squares approach based on a direct minus one inner product for first order systems, Brookhaven Nat. Laboratory Rep. BNL-60624 (1994)].
Reviewer: I.N.Katz (St.Louis)

Cited in 35 Documents

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
35J25 Boundary value problems for second-order elliptic equations
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows

Keywords:

Stokes equations; preconditioners; least-squares approximations; elasticity problem
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References:

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