*(English)*Zbl 1037.65120

The authors study the uniform convergence of finite element methods (FEM) for a singularly perturbed saddle point problem in two dimensions. If the perturbation parameter is positive, the saddle point problem has the form of the Stokes equations with the operator $I-{\epsilon}^{2}{\Delta}$ applied to the velocity instead of $-{\Delta}$.

First, the existence and uniqueness of weak solutions is discussed. The results are based on a Babuška-Brezzi (BB) condition and the coercivity in an appropriate energy norm. The critical point is that the appropriate spaces and the energy norm are different in the cases $\epsilon >0$ and $\epsilon =0$. The uniform stability of a FEM relies on discrete versions of the BB condition and the coercivity.

Numerical tests show that some commonly used FEM which are suited for $\epsilon >0$ fail as $\epsilon \to 0$ and vice versa, FEM which work for $\epsilon =0$ diverge if $\epsilon >0$. Then, a new non-conforming finite element for triangular grids is constructed for which stability uniformly in $\epsilon $ is proved. This finite element has nine degrees of freedom in each mesh cell which can be defined by integrals on the edges.

Error estimates which show at least a linear rate of convergence independent of $\epsilon $ are proved for smooth solutions which do not depend on $\epsilon $. For $\epsilon $-dependent solutions with boundary layers, it is shown that the rate of convergence in the energy norm is 0.5 uniformly in $\epsilon $. Both analytical results are supported with numerical examples. Finally, it is shown by a numerical experiment that the new non-conforming FEM works also for a generalized singularly perturbed problem which possesses two parameters. The uniform convergence in the energy norm is proved for smooth solutions which do not depend on these parameters.

##### MSC:

65N30 | Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) |

65N15 | Error bounds (BVP of PDE) |

65N12 | Stability and convergence of numerical methods (BVP of PDE) |

35B25 | Singular perturbations (PDE) |

35Q30 | Stokes and Navier-Stokes equations |

35J25 | Second order elliptic equations, boundary value problems |

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

76M10 | Finite element methods (fluid mechanics) |