Abstract
The objective of the article is to improve the algorithms for the resolution of the spectral discretization of the vorticity–velocity–pressure formulation of the Navier–Stokes problem in two and three domains. Two algorithms are proposed. The first one is based on the Uzawa method. In the second one we consider a modified global resolution. The two algorithms are implemented and their results are compared.
Highlights
The Navier–Stokes system models the flow of a fluid, for example, the movements of air in the atmosphere, ocean currents, the flow of water in a pipeline and many other fluidflow phenomena
5 Conclusion and future work In this work, we show the efficiency of the global resolution compared to the Uzawa algorithm adapted to the resolution of the discrete problem issued from the spectral discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes problem
We achieved a good convergence with the global resolution through the transformation of the matrix to a symmetric and positive defined one
Summary
The Navier–Stokes system models the flow of a fluid, for example, the movements of air in the atmosphere, ocean currents, the flow of water in a pipeline and many other fluidflow phenomena. Using the Galerkin method and the numerical integration based on the Gauss– Lobatto quadrature formula (5), we deduce for a continuous data f on the following discrete variational formulation: Find (μN , φN , pN ) in YN × XN × MN such that aN (μN , φN ; vN ) + TN (μN , φN ; vN ) + bN (vN , pN ) = (f, vN )N , ∀vN ∈ XN , bN (φN , qN ) = 0, ∀qN ∈ MN ,. IV, Cor. 1.1), problem (6) has a unique solution, since the discrete bilinear form bN (·, ·) coincides with the continuous one on XN × MN due to the exactness of the quadrature formula and verifies the inf-sup condition (see [8], Lem. 3.9).
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