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Divergence-conforming HDG methods for Stokes flows. (English) Zbl 1427.76125

Summary: In this paper, we show that by sending the normal stabilization function to infinity in the hybridizable discontinuous Galerkin methods previously proposed in [N. C. Nguyen et al., Comput. Methods Appl. Mech. Eng. 199, No. 9–12, 582–597 (2010; Zbl 1227.76036)], for Stokes flows, a new class of divergence-conforming methods is obtained which maintains the convergence properties of the original methods. Thus, all the components of the approximate solution, which use polynomial spaces of degree \(k\), converge with the optimal order of \(k+1\) in \(L^2\) for any \(k\geq 0\). Moreover, the postprocessed velocity approximation is also divergence-conforming, exactly divergence-free and converges with order \(k+2\) for \( k\geq 1\) and with order \(1\) for \(k=0\). The novelty of the analysis is that it proceeds by taking the limit when the normal stabilization goes to infinity in the error estimates recently obtained in [B. Cockburn et al., Math. Comput. 80, No. 274, 723–760 (2011; Zbl 1410.76164)].

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
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
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