Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh \(L^2\) projection.

*(English)*Zbl 1234.65038The authors consider a low equal order finite element method to approximate the stationary Stokes equations in two dimensions. The method does not satisfy the known \(inf\)-\(sup\) condition. This condition is required for instance to get the stability of the finite element methods approximating Stokes equations. A known stabilized form is then introduced and the well posedness for the new discrete problem is proved. The convergence order of the discrete velocity is \(h\) in \(H^1\)-norm whereas the convergence order of the discrete pressure is \(h\) in \(L^2\)-norm. Some postprocessed approximations are derived using the stated discrete velocity and the discrete pressure.

These postprocessed approximations are the \(L^2\) projections into piecewise polynomials of higher degree on some coarse meshes. It is proved that the convergence order of the stated postprocessed approximations is higher than that of the stated discrete velocity and the discrete pressure under some suitable choices for the piecewise polynomials of higher degree which are used to define these postprocessed approximations.

There are then superconvergence results. Two assumptions are assumed to be satisfied in order to get the above stated results. The first one consists of a regularity assumption on the domain on which the problem is posed. The second assumption consists of a regularity assumption on the continuous mixed variational form.

These postprocessed approximations are the \(L^2\) projections into piecewise polynomials of higher degree on some coarse meshes. It is proved that the convergence order of the stated postprocessed approximations is higher than that of the stated discrete velocity and the discrete pressure under some suitable choices for the piecewise polynomials of higher degree which are used to define these postprocessed approximations.

There are then superconvergence results. Two assumptions are assumed to be satisfied in order to get the above stated results. The first one consists of a regularity assumption on the domain on which the problem is posed. The second assumption consists of a regularity assumption on the continuous mixed variational form.

Reviewer: Abdallah Bradji (Annaba)

##### MSC:

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35Q30 | Navier-Stokes equations |

76M10 | Finite element methods applied to problems in fluid mechanics |

76D07 | Stokes and related (Oseen, etc.) flows |

##### Keywords:

inf-sup condition; stabilized finite element approximation; superconvergence; coarse mesh; \(L^2\) projection; piecewise polynomials of higher degree; stationary Stokes equations; stability
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\textit{J. Li} et al., Numer. Methods Partial Differ. Equations 28, No. 1, 115--126 (2012; Zbl 1234.65038)

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