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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 $k,$ $\phantom{\rule{4pt}{0ex}}k=1,\phantom{\rule{4pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}3$, 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.
##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76M10 Finite element methods (fluid mechanics) 76D07 Stokes and related (Oseen, etc.) flows 76D05 Navier-Stokes equations (fluid dynamics) 76M25 Other numerical methods (fluid mechanics)