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Zbl 1141.76039
Rebholz, Leo G.
An energy- and helicity-conserving finite element scheme for the Navier-Stokes equations. (English)
[J] SIAM J. Numer. Anal. 45, No. 4, 1622-1638 (2007). ISSN 0036-1429; ISSN 1095-7170/e

Summary: We present a new finite element scheme for solving Navier-Stokes equations that exactly conserves both energy $(\int_{\Omega}u^{2})$ and helicity $(\int_{\Omega} u\cdot(\nabla \times u))$ in the absence of viscosity and external force. We prove stability, exact conservation, and convergence for the scheme. Energy and helicity are exactly conserved by using a combination of the usual (convective) form with the rotational form of nonlinearity and solving for both velocity and a projected vorticity in a trapezoidal time discretization. Numerical results are presented that compare the scheme to the usual trapezoidal schemes.

MSC 2000:: *76M10 Finite element methods
76D05 Navier-Stokes equations (fluid dynamics)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)

Keywords: stability; convergence; trapezoidal time discretization

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