Various realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: they are based on the variation
al formulation of the incompressible Navier–Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used math
ematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

In this paper, we propose and analyze a Control Volume Finite Elements (CVFE) scheme for solving possibly degenerated parabolic equations. This scheme does not require the introduction of the so-called Kirchhoff transform in its definition. We prove that the discrete solution obtained \emph{via} the scheme remains in the physical range, and that the natural entropy of the problem decreases with time. The convergence of the method is proved as the discretization steps tend to $0$. Finally, numerical examples illustrate the efficiency of the method.

https://hal.archives-ouvertes.fr/hal-00955091/document

Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.

Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure

Variational Multiscale Methods in Computational Fluid Dynamics

https://hal.archives-ouvertes.fr/hal-01119735/document

Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure

In this paper we consider, in dimension d≥ 2, the standard $$\mathbb{P}_{1}$$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L ∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L 1(Ω), we prove that the unique solution of the discrete problem converges in $$W^{1,q}_0(\Omega)$$ (for every q with $${1 \leq q  1.

Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

In this paper, we introduce a low-cost, high-order stabilized method for the numerical solution of incompressible flow problems. This is a particular type of projection-stabilized method where each targeted operator, such as the pressure gradient or the convection, is stabilized by least-squares terms added to the Galerkin formulation. The main methodological originality is that we replace the projection-stabilized structure by an interpolation-stabilized structure, with reduced computational cost for some choices of the interpolation operator. This stabilization has one level, in the sense that it is defined on a single mesh. We prove the stability of our formulation by means of a specific inf–sup condition, which is the main technical innovation of our paper. We perform a convergence and error estimates analysis, proving the optimal order of accuracy of our method. We include some numerical tests that confirm our theoretical expectations.

A high order term-by-term stabilization solver for incompressible flow problems

This paper introduces a scheme for the numerical solution of a model for two turbulent flows with coupling at an interface. We consider a variational formulation of the coupled model, where the turbulent kinetic energy equation is formulated by transposition. We prove the convergence of the approximation to this formulation for 2D flows by piecewise affine triangular elements. Our main contribution is to prove that the standard Galerkin - finite element approximation of the Laplace equation approximates in L2 norm its solution by transposition, for data with low smoothness. We include some numerical tests for simple geometries that exhibit the behaviour predicted by our analysis.

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A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements

We perform in this paper the numerical analysis of some penalty stabilized solvers for the unsteady Navier---Stokes equations We consider low-order and high-order methods The low-order method is a pure penalty method, while the high-order one is a projection-stabilized method We perform their numerical analysis (stability and convergence) for solutions that only need to bear the natural regularity In this analysis, the stability is based upon specific inf-sup conditions No local orthogonality properties are needed for the projection-interpolation operator The convergence is based upon the representation of the stabilizing terms by means of bubble finite element spaces We include some numerical tests for realistic flows that confirm the theoretical expectations

M. Gómez Mármol

Papers

Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

A high order term-by-term stabilization solver for incompressible flow problems

A model for two coupled turbulent fluids Part III: Numerical approximation by finite elements

Numerical Analysis of Penalty Stabilized Finite Element Discretizations of Evolution Navier---Stokes Equations

Stability of some turbulent vertical models for the ocean mixing boundary layer