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A two-level method with backtracking for the Navier-Stokes equations. (English) Zbl 0913.76050
The finite element discretizations of the stationary incompressible Navier-Stokes equations lead usually to large systems of nonlinear algebraic equations. Here, for resolving the nonlinearity, the authors study a two-level method, relying on a coarse and a fine mesh, that works with arbitrary pairs of finite element spaces for velocity and pressure satisfying the Ladyzhenskaya-Babuška-Brezzi condition. The method presented is shown to be convergent for all Reynolds numbers. Because the linearization by Newton’s method can cause instabilities at higher Reynolds numbers, the authors use an Oseen-type linearization. Namely, after solving the original nonlinear problem on the coarse mesh, the Oseen problem is solved on a fine mesh. Finally, a coarse mesh correction is performed. If the coarse mesh is fine enough and the step of the fine mesh is not too large, the two-level solution is of the same accuracy as the exact fine mesh solution. In comparison to other methods known from the literature, the method presented in the paper seems to be more efficient.
MSC:
76M10Finite element methods (fluid mechanics)
76D05Navier-Stokes equations (fluid dynamics)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65H10Systems of nonlinear equations (numerical methods)