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Local Projection Stabilization for Equal-Order Interpolation

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 24)
Local projection stabilization (LPS) was introduced in Part III, Chapter 3 for a scalar convection-diffusion equation. It will now be extended to the Oseen system
$$-\nu \Delta {\bf u} + ({\bf b} \cdot \nabla){\bf u} + \sigma {\bf u} + \nabla p = {\bf f} \; {\rm in} \; \Omega \subset {{\bf R}^d}$$
(4.1a)
$$\nabla \cdot {\bf u} = 0 \; {\rm in} \; \Omega$$
(4.1b)
$${\bf u} = {\bf 0} \; {\rm on} \; \partial\Omega$$
(4.1c)

As we saw in Chapter 3, the streamline diffusion method (SDFEM) can handle two types of instabilities: that caused by a violation of the discrete infsup (Babu?ska-Brezzi) condition (2.1) and that due to dominant convection. The SDFEM combines the Pressure Stabilized Petrov-Galerkin (PSPG) approach (testing the residual against \({\nabla{\rm q}}\)) with the Streamline Upwind Petrov- Galerkin (SUPG) technique (testing the residual against \({({\rm b}\cdot\nabla)}{\bf v}\))

Despite the extensive theoretical and practical development of the SDFEM, a fundamental flaw in the method - in particular for higher-order interpolations - is that various terms must be added to the weak formulation to guarantee its consistency. Moreover, the requirement of consistency leads to undesirable effects when using residual-based stabilization methods like the SDFEM in optimal control problems; see the discussion at the beginning of Section III.3.3. LPS relaxes the consistency requirement while preserving the main features of the SDFEM approach; in particular, one can use equal-order interpolation without worrying about the Babuška-Brezzi condition. Furthermore, LPS allows us to separate velocity and pressure in the stabilization terms, which for systems of equations means that one can avoid non-physical couplings.

Keywords

Projection Space Element Space Approximation Space Interpolation Operator Subgrid Modelling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

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