Jean-Luc Guermond

On stability and convergence of projection methods based on pressure Poisson equation

We investigate the proper choices of spatial approximations for velocity and pressure in fractional-step projection methods. Numerical results obtained with classical finite element interpolations are presented. These tests confirm the role of the inf-sup LBB condition in non-incremental and incremental versions of the method for computing viscous incompressible flows

Velocity-Correction Projection Methods for Incompressible Flows

We introduce and study a new class of projection methods---namely, the velocity-correction methods in standard form and in rotational form---for solving the unsteady incompressible Navier--Stokes equations. We show that the rotational form provides improved error estimates in terms of the H1-norm for the velocity and of the L2-norm for the pressure. We also show that the class of fractional-step methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75--111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414--443] can be interpreted as the rotational form of our velocity-correction methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods.

Entropy viscosity method for nonlinear conservation laws

A new class of high-order numerical methods for approximating nonlinear conservation laws is described (entropy viscosity method). The novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand. This new approach does not use any flux or slope limiters, applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation. Various benchmark problems are solved with finite elements, spectral elements and Fourier series to illustrate the capability of the proposed method.

Nonlinear corrections to Darcy's law at low Reynolds numbers

Under fairly general assumptions, for periodic porous media, whose period is of the same order as that of the inclusion, the nonlinear correction to Darcys law is quadratic in terms of the Reynolds number, i.e. cubic with respect to the seepage velocity.

On the approximation of the unsteady Navier-Stokes equations by finite element projection methods

Abstract. This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method.

On the error estimates for the rotational pressure-correction projection methods

In this paper we study the rotational form of the pressure-correction method that was proposed by Timmermans, Minev, and Van De Vosse. We show that the rotational form of the algorithm provides better accuracy in terms of the H1-norm of the velocity and of the L2-norm of the pressure than the standard form.

Convergence Analysis of a Finite Element Projection/Lagrange--Galerkin Method for the Incompressible Navier--Stokes Equations

This paper provides a convergence analysis of a fractional-step method to compute incompressible viscous flows by means of finite element approximations. In the proposed algorithm, the convection, the diffusion, and the incompressibility are treated in three different substeps. The convection is treated first by means of a Lagrange--Galerkin technique, whereas the diffusion and the incompressibility are treated separately in two subsequent substeps by means of a projection method. It is shown that provided the time step,

Discontinuous Galerkin Methods for Friedrichs’ Systems

\delta t,

A splitting method for incompressible flows with variable density based on a pressure Poisson equation

is of

An interior penalty Galerkin method for the MHD equations in heterogeneous domains

{\cal O}(h^{d/4}),