CONSISTENT LOCAL PROJECTION STABILIZED FINITE ELEMENT METHODS

This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. The initial unstable P₁ x P l ,l= 0,1, finite element space is enhanced with solutions of residual-based local problems, and then the static co...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2010-01, Vol.48 (5), p.1801-1825
Main Authors: BARRENECHEA, GABRIEL R., VALENTIN, FRÉDÉRIC
Format: Article
Language:English
Subjects:
Approximation
Cauchy Schwarz inequality
Conservation of mass
Convergence
Degrees of freedom
Error rates
Finite element analysis
Finite element method
Interpolation
Mathematical analysis
Mathematical functions
Mathematical models
Numerical analysis
Polynomials
Projection
Strategy
Studies
Velocity distribution
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Summary:This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. The initial unstable P₁ x P l ,l= 0,1, finite element space is enhanced with solutions of residual-based local problems, and then the static condensation procedure is applied to derive new methods. The approach keeps degrees of freedom unchanged while giving rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure. The resulting methods do not need the use of a macroelement grid structure and are parameter-free. The numerical analysis is carried out showing optimal convergence in natural norms, and moreover, two ways of rendering the velocity field locally mass conservative are proposed. Some numerics validate the theoretical results.
ISSN:0036-1429
1095-7170
DOI:10.1137/090753334