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On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. (English) Zbl 1174.65039

Summary: The stability of the \(P_1\)-\(P_0\) mixed-element is established on general Powell-Sabin triangular grids. The piecewise linear finite element solution approximating the velocity is divergence-free pointwise for the Stokes equations. The finite element solution approximating the pressure in the Stokes equations can be obtained as a byproduct if an iterative method is adopted for solving the discrete linear system of equations. Numerical tests are presented confirming the theory on the stability and the optimal order of convergence for the \(P_1\) Powell-Sabin divergence-free finite element method.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
35Q30 Navier-Stokes equations
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