A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. (English) Zbl 1057.35029

Stokes and Navier-Stokes equations are considered in this paper especially from the numerical point of view. Numerical analysis of a discontinuous Galerkin method with nonoverlapping domain decomposition for steady incompressible Stokes and Navier-Stokes problems is investigated. As a discretization a conforming triangular finite element mesh is used in each subdomain. In each triangle the discretization velocity is a polynomial of degree \(k,\) \(\;k=1,\;2 \text{ or } 3\), while the discretized pressure is of degree \(k-1\). The variational formulation of the problem contains also a jump term on all triangle interfaces due to the domain decomposition method. An inf-sup condition is proved under some hypothesis for the domain decomposition with non-matching grids. Optimal a priori error estimates in the energy norm for the velocity field and the \(L_2\) norm for the pressure are derived first for Stokes and then also for Navier-Stokes problems.


35Q30 Navier-Stokes equations
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76D05 Navier-Stokes equations for incompressible viscous fluids
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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