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A multilevel finite element method in space-time for the Navier-Stokes problem. (English) Zbl 1081.76044

Summary: A multilevel finite element method in space-time for the two-dimensional nonstationary Navier-Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier-Stokes problem is only solved on a single coarsest space-time mesh; subsequent approximations are generated on a succession of refined space-time meshes by solving a linearized Navier-Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the \(J\)-level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: \(h_{j} \sim, h_{j-1}^{3/2}, k_{j} \sim k_{j-1}^{3/2}, j = 2,\cdots ,J\), the \(J\)-level finite element method in space-time provides the same accuracy as the one-level method in space-time in which the fully nonlinear Navier-Stokes problem is solved on a final finest space-time mesh.

MSC:

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
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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