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...

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...

The numerical minimization of the functional $\mathcal{F}\left(u\right)={\int }_{\mathrm{\Omega }}\varphi \left(x,{\nu }_u\right)\left|Du\right|+{\int }_{\partial \mathrm{\Omega }}\mu ud{\mathcal{H}}^{n-1}-{\int }_{\mathrm{\Omega }}\kappa udx$, $u\in BV\left(\mathrm{\Omega };\left\{-1,1\right\}\right)$ is addressed. The function $\varphi$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $\mathcal{F}$ can be equivalently minimized on the convex set $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ and then regularized with a sequence ${\left\{{\mathcal{F}}_{ϵ}\left(u\right)\right\}}_{ϵ}$, of stricdy convex functionals defined on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$. Then both $\mathcal{F}$ and ${\mathcal{F}}_{ϵ}$, can be discretized by continuous linear finite elements. The convexity property of the functionals on $BV\left(\mathrm{\Omega };\left[-1,1\right]\right)$ is useful in the numerical minimization...

This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. Several multi-scale numerical algorithms have been shown to correctly capture the homogenized limit of solutions of elliptic equations with coefficients modeled as stationary and ergodic random fields. Because theoretical results are available in the continuum setting for such equations, we consider here the case of a second-order...

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive...

Curved triangular $C^m$-elements which can be pieced together with the generalized Bell’s $C^m$-elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order $2(m+1)$ in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the...