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Hybridized globally divergence-free LDG methods. I: The Stokes problem. (English) Zbl 1087.76061

Summary: We devise and analyze a new local discontinuous Galerkin (LDG) method for the Stokes equations of incompressible fluid flow. This optimally convergent method is obtained by using an LDG method to discretize a vorticity-velocity formulation of Stokes equations and by applying a new hybridization to the resulting discretization. One of the main features of the hybridized method is that it provides a globally divergence-free approximate velocity without having to construct globally divergence-free finite-dimensional spaces; only elementwise divergence-free basis functions are used. Another important feature is that it has significantly less degrees of freedom than all other LDG methods in the current literature; in particular, the approximation to the pressure is only defined on the faces of the elements. On the other hand, we show that, as expected, the condition number of the Schur-complement matrix for this approximate pressure is of order \( h^{-2}\) in the mesh size \( h\). Finally, we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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