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A robust finite element method for Darcy-Stokes flow. (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)