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A stabilized finite element method for the Stokes problem based on polynomial pressure projections. (English) Zbl 1060.76569

Summary: A new stabilized finite element method for Stokes problem is presented. The method is obtained by modification of the mixed variational equation by using local ${L}^{2}$ polynomial pressure projections. Our stabilization approach is motivated by the inherent inconsistency of equal-order approximations for Stokes equations, which leads to an unstable mixed finite element method. Application of pressure projections in conjunction with minimization of the pressure-velocity mismatch eliminates this inconsistency and leads to a stable variational formulation.

Unlike other stabilization methods, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The new method can be implemented at the element level, and for affine families of finite elements on simplicial grids it reduces to a simple modification of weak continuity equation. Numerical results are presented for a variety of equal-order continuous velocity and pressure elements in two and three dimensions.

##### MSC:
 76M10 Finite element methods (fluid mechanics) 76D07 Stokes and related (Oseen, etc.) flows
##### Keywords:
equal-order interpolation; inf-sup condition