We consider a degenerate parabolic system which models the evolution of nematic liquid crystal with variable degree of orientation. The system is a slight modification to that proposed in [Calderer et al., SIAM J. Math. Anal.33 (2002) 1033–1047], which is a special case of Ericksen's general continuum model in [Ericksen, Arch. Ration. Mech. Anal.113 (1991) 97–120]. We prove the global existence of weak solutions by passing to the limit in a regularized system. Moreover, we propose a practical...

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a “rough” coefficient function $k\left(x\right)$. We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, $k^{\text{'}}$ is in $BV$, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations...

We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a "rough"coefficient function k(x). We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion...

This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial...

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$, we obtain error estimates in $L^2$ of order $𝒪(\Delta t^2+h^{m+1/2})$ where $m$ is the degree of the local polynomials.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methods for the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$, we obtain error estimates in L2 of order $𝒪(\Delta t^2+h^{m+1/2})$ where m is the degree of the local polynomials.

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size $ϵ$. We prove a convergence theorem on solutions of Navier-Stokes Coriolis equations, as $ϵ$ goes to zero, in the well prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus generalize the results obtained on flat boundaries with the classical...

In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate of order h, assuming only the W2,p (for p>2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in...

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H1 finite volume space. We actually prove the convergence of the scheme in a discrete H1 norm, with an error estimate of order O(h) (on meshes...

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...

This work proves the convergence in L¹(ℝ²) towards an Oseen vortex-like solution to the dissipative quasi-geostrophic equations for several sets of initial data with suitable decay at infinity. The relative entropy method applies in a direct way for solving this question in the case of signed initial data and the difficulty lies in showing the existence of unique global solutions for the class of initial data for which all properties needed in the entropy approach are met. However, the estimates...

We propose and analyse two convergent fully discrete schemes to solve the incompressible Navier-Stokes-Nernst-Planck-Poisson system. The first scheme converges to weak solutions satisfying an energy and an entropy dissipation law. The second scheme uses Chorin's projection method to obtain an efficient approximation that converges to strong solutions at optimal rates.

The incompressible MHD equations couple Navier-Stokes equations with Maxwell's equations to describe the flow of a viscous, incompressible, and electrically conducting fluid in a Lipschitz domain $\Omega \subset {ℝ}^3$. We verify convergence of iterates of different coupling and decoupling fully discrete schemes towards weak solutions for vanishing discretization parameters. Optimal first order of convergence is shown in the presence of strong solutions for a splitting scheme which decouples the computation of velocity...

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.

A semidiscretization in time of a fourth order nonlinear parabolic system in several space dimensions arising in quantum semiconductor modelling is studied. The system is numerically treated by introducing an additional nonlinear potential. Exploiting the stability of the discretization, convergence is shown in the multi-dimensional case. Under some assumptions on the regularity of the solution, the rate of convergence proves to be optimal.