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Polynomial robust stability analysis for \(H\)(div)-conforming finite elements for the Stokes equations. (English) Zbl 1462.65192

Summary: In this article, we consider a discontinuous Galerkin method for the discretization of the Stokes problem. We use \(H(\mathrm{div})\)-conforming finite elements, as they provide major benefits such as exact mass conservation and pressure-independent error estimates. The main aspect of this work lies in the analysis of high-order approximations. We show that the considered method is uniformly stable with respect to the polynomial order \(k\) and provides optimal error estimates \(\| {\mathbf{u} - \mathbf{u}_h} \|_{1_h} + \| {\Pi^{Q_h} p-p_h} \|_{0} \leq c \left(h/k \right)^s \| \mathbf{u} \|_{s+1} \). To derive these estimates, we prove a \(k\)-robust Ladyženskaja-Babuška-Brezzi (LBB) condition. This proof is based on a polynomial \(H^2\)-stable extension operator. This extension operator itself is of interest for the numerical analysis of \(C^0\)-continuous discontinuous Galerkin methods for fourth-order problems.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
76S05 Flows in porous media; filtration; seepage
76M10 Finite element methods applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
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