We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.

We state a superconvergence result for the lowest order Raviart-Thomas approximation of eigenvalue problems. It is known that a similar superconvergence result holds for the mixed approximation of Laplace problem; here we introduce a new proof, since the one given for the source problem cannot be generalized in a straightforward way to the eigenvalue problem. Numerical experiments confirm thesuperconvergence property and suggest that it also holds for the lowest orderBrezzi-Douglas-Marini approximation.

We derive a posteriori error estimates for singularlyperturbed reaction–diffusion problems which yield a guaranteedupper bound on the discretization error and are fully and easilycomputable. Moreover, they are also locally efficient and robust inthe sense that they represent local lower bounds for the actualerror, up to a generic constant independent in particular of thereaction coefficient. We present our results in the framework ofthe vertex-centered finite volume method but their nature isgeneral for any conforming method, like the piecewise linear finiteelement one. Our estimates are based on a H(div)-conformingreconstruction of the diffusive flux in the lowest-orderRaviart–Thomas–Nédélec space linked with mesh dual to the originalsimplicial one, previously introduced by the last author in thepure diffusion case. They also rely on elaborated Poincaré,Friedrichs, and trace inequalities-based auxiliary estimatesdesigned to cope optimally with the reaction dominance. In order tobring down the ratio of the estimated and actual overall energyerror as close as possible to the optimal value of one,independently of the size of the reaction coefficient, we finallydevelop the ideas of local minimizations of the estimators by localmodifications of the reconstructed diffusive flux. The numericalexperiments presented confirm the guaranteed upper bound,robustness, and excellent efficiency of the derived estimates.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem.An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh.Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms.Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdotf(x,u)=0$ on a closed Riemannian manifold M.For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$

In this paper, we study a postprocessing procedure for improvingaccuracy of the finite volume element approximations of semilinearparabolic problems. The procedure amounts to solve a source problemon a coarser grid and then solve a linear elliptic problem on afiner grid after the time evolution is finished. We derive errorestimates in the L 2 and H 1 norms for the standard finitevolume element scheme and an improved error estimate in the H 1 norm. Numerical results demonstrate the accuracy and efficiency ofthe procedure.

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.

We derive a posteriori estimates for a discretization in space of the standardCahn–Hilliard equation with a double obstacle free energy.The derived estimates are robust and efficient, and in practice are combinedwith a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compareour method with an existing heuristic spatial mesh adaptation algorithm.