Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier-Stokes equations
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We propose and study numerically the implicit approximation in time of the Navier–Stokes equations by a Galerkin–collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin–collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is also ensured. As a further ingredient, we employ the Nitsche method to impose all boundary conditions in weak form. As an outlook, we show that the presented approach offers high potential for the simulation of flow problems on dynamic geometries with moving boundaries. For this, the fictitious domain approach, based on Nitsche's method, and cut finite elements are applied with a regular and fixed background mesh. The convergence behavior of the Galerkin–collocation approach combined with Nitsche's method is studied by a numerical experiment. It's accuracy is also illustrated for the problem of flow past an obstacle in a channnel.
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