Nicholas E. Wilson

On an Efficient Finite Element Method for Navier-Stokes-ω with Strong Mass Conservation

Abstract We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence. Additionally, an improved choice of filtering radius (versus the usual choice of the average mesh width) for the scheme is found, by identifying a connection with a scheme for the velocity-vorticity-helicity NSE formulation. Several numerical experiments are given that demonstrate the performance of the scheme, and the improvement offered by the new choice of the filtering radius.

A subgrid stabilization finite element method for incompressible magnetohydrodynamics

This paper studies a numerical scheme for approximating solutions of incompressible magnetohydrodynamic (MHD) equations that uses eddy viscosity stabilization only on the small scales of the fluid flow. This stabilization scheme for MHD equations uses a Galerkin finite element spatial discretization with Scott-Vogelius mixed finite elements and semi-implicit backward Euler temporal discretization. We prove its unconditional stability and prove how the coarse mesh can be chosen so that optimal convergence can be achieved. We also provide numerical experiments to confirm the theory and demonstrate the effectiveness of the scheme on a test problem for MHD channel flow.

High Accuracy Method for Magnetohydrodynamics System in Elsässer Variables

Abstract A method has been developed recently by the third author, that allows for decoupling of the evolutionary full magnetohydrodynamics (MHD) system in the Elsässer variables. The method entails the implicit discretization of the subproblem terms and the explicit discretization of coupling terms, and was proven to be unconditionally stable. In this paper we build on that result by introducing a high-order accurate deferred correction method, which also decouples the MHD system. We perform the full numerical analysis of the method, proving the unconditional stability and second order accuracy of the two-step method. We also use a test problem to verify numerically the claimed convergence rate.

Enforcing energy, helicity and strong mass conservation in finite element computations for incompressible Navier–Stokes simulations

Abstract We study a finite element scheme for the 3D Navier–Stokes equations (NSE) that globally conserves energy and helicity and, through the use of Scott–Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor–Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme.

On the Leray-deconvolution model for the incompressible magnetohydrodynamics equations

Abstract We extend Leray- α -deconvolution modeling to the incompressible magnetohydrodynamics (MHD). The resulting model is shown to be well-posed, and have attractive limiting behavior both in its filtering radius and order of deconvolution. Additionally, we present and study a numerical scheme for the model, based on an extrapolated Crank–Nicolson finite element method. We show the numerical scheme is unconditionally stable, preserves energy and cross-helicity, and optimally converges to the MHD solution. Numerical experiments are provided that verify convergence rates, and test the scheme on benchmark problems of channel flow over a step and the Orszag–Tang vortex problem.

A connection between coupled and penalty projection timestepping schemes with FE spatial discretization for the Navier–Stokes equations

Abstract We prove that for several inf-sup stable mixed finite elements, the solution of the Chorin/Temam projection methods for Navier–Stokes equations equipped with grad–div stabilization with parameter γ converge to the associated coupled method solution with rate γ−1 as γ → ∞. We prove this result for both backward Euler schemes and BDF2 schemes. Furthermore, we simplify classical numerical analysis of projection methods, allowing us to remove some unnecessary assumptions, such as convexity of the domain. Several numerical experiments are given which verify the convergence rate, and show that projection methods with large grad–div stabilization parameters can dramatically improve accuracy.