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An adaptive finite element method for the incompressible Navier-Stokes equations on time-dependent domains. (English) Zbl 0847.76028
Heidelberg: Univ. iii, 115 p. (1996).
We develop an algorithm for the numerical solution of the incompressible Navier-Stokes equations on domains varying in time. We describe and analyze the discretization on variable domains. We define an oblique time derivative which allows us to apply the usual energy techniques in order to derive optimal order error estimators. Then we present an adaptive algorithm together with the multigrid scheme on locally refined meshes, and develop an a posteriori error estimator for the Navier-Stokes equations. We also derive an a posteriori indicator for the iteration error and prove stability of the weighted least squares formulation with low order finite elements applied to the stationary Stokes equations. Finally, we describe the discretization of the incompressible Navier-Stokes equations, present a multigrid algorithm for the resulting discrete equations, and apply the adaptive algorithm to a two-dimensional cylinder flow benchmark problem. The error indicator is based on a straightforward generalization of the a posteriori estimator for the Stokes equations. The computations presented are done with the finite element library DEAL.
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs