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Hybrid discontinuous Galerkin methods with relaxed \(H(\operatorname{div})\)-conformity for incompressible flows. I. (English) Zbl 1402.35209

The authors deal with the Stokes problem \[ \begin{cases} -\nu \Delta u + \nabla p = f \;\text{ in } \Omega, &\\ \operatorname{div} u=0 \;\text{ in } \Omega,&\\ u=0 \;\text{ on } \Gamma_D,& \end{cases} \] where \(\Gamma_D\) is a subset of \(\partial \Omega\). They propose a new discretization method which is an improved version of the recent (2016) method reported by two of the authors of the present paper. This new method is based on an \(H(\operatorname{div})\)-conforming finite element space and a hybrid discontinuous Galerkin formulation of the viscous forces. The authors present this method in detail, explain why it is more advantageous, perform a thorough \(h\)-version error analysis, and provide some numerical examples.

MSC:

35Q30 Navier-Stokes equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Software:

NGSolve; Netgen
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References:

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