An overlapping local projection stabilization for Galerkin approximations of Stokes and Darcy flow problems. (English) Zbl 1493.65201
Summary: An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming \(\mathbf{P}_1^c\) finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the \(L^2\)-norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 35B45 | A priori estimates in context of PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
finite element method; Stokes problem; Darcy problems; generalized local projection stabilization; inf-sup condition; error estimatesReferences:
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