Abstract
We analyze Local Projection Stabilization (LPS) methods for the solution of Stokes problem using equal order finite elements. We investigate their convergence, stability and accuracy properties. The resulting stabilized method is shown to lead to optimal rates of convergence for both velocity and pressure approximations. We distinguish two classes of LPS methods: one-level and two-level methods. Numerical examples using bilinear interpolations are presented to validate the analysis and assess the accuracy of both approaches.
Highlights
Numerical approximations of incompressible flows require a compatibility condition between the discrete velocity and pressure spaces (Girault and Raviart, 1986; Brezzi and Fortin (1991))
The two-level local projection technique has been introduced by Becker and Braack (2001) and Nafa (2008) to circumvent the inf-sup condition and to allow the use of simple equal order interpolations such as P1 P1 and Q1 Q1 velocity-pressure approximations
A weighted difference of the pressure gradient and its local projection is introduced into the continuity equation
Summary
Numerical approximations of incompressible flows require a compatibility condition between the discrete velocity and pressure spaces (Girault and Raviart, 1986; Brezzi and Fortin (1991)). The two-level local projection technique has been introduced by Becker and Braack (2001) and Nafa (2008) to circumvent the inf-sup condition and to allow the use of simple equal order interpolations such as P1 P1 and Q1 Q1 velocity-pressure approximations. This method consists in introducing the L2 -projection of the pressure gradient as a new unknown of the problem. Numerical results are presented to justify the order of convergence and assess the performance accuracy of both LPS methods using bilinear finite element interpolation
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