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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.

35Q30Stokes and Navier-Stokes equations
76M10Finite element methods (fluid mechanics)
76D07Stokes and related (Oseen, etc.) flows
76D05Navier-Stokes equations (fluid dynamics)
76M25Other numerical methods (fluid mechanics)
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