We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H-conforming reconstruction of the potential, continuous and piec...

In this paper, we develop and analyze mixed finite element methods for the Stokes and Navier-Stokes equations. Our mixed method is based on the pseudostress-pressure-velocity formulation. The pseudostress is approximated by the Raviart-Thomas, Brezzi-Douglas-Marini, or Brezzi-DouglasFortin-Marini elements, the pressure and the velocity by piecewise discontinuous polynomials of appropriate degre...

We present here the first results of a work on a multiscale resolution method using both Finite Volumes and Elements. Our method relies upon the coupling of grid scales: a coarse one and a fine one. The trick is to build a Finite Element basis on the coarse grid from problems solved on the fine one. In previous works on this subject [1] [2], the Finite Element method was used in both the fine a...

We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the fin...

We study a finite volume element discretization of a nonlinear parabolic equation in a convex polygonal domain. We show existence of the discrete solution and derive error estimates in L2– and H –norms. We also consider a linearized method and provide numerical results to illustrate our theoretical findings.

We study in this paper the finite volume approximation of strong solutions to systems of conservation laws. We derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical h1/2 estimate under a BV assumption on the numerical approximation.

In this paper, a nonlinear control volume finite element (CVFE) scheme for a degenerate Keller– Segel model with anisotropic and heterogeneous diffusion tensors is proposed and analyzed. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in finite element methods. The diffusion term which involves an anisotropic and heterogeneous tensor is discretized on...

A numerical method for the reconstruction of interfaces in finite volume schemes for multiphase flows is presented. The computation of the triple-point at the intersection of three materials in two dimensions of space is addressed. The determination of the normal vectors between pairs of materials is obtained with a finite element approximation. A numerical method for the localization of a trip...

Riemannian metric tensors are used to control the adaptation of meshes for finite element and finite volume computations. To study the numerous metric construction and manipulation techniques, a new method has been developed to visualize two-dimensional metrics without interference from any adaptation algorithm. This method traces a network of orthogonal tensor lines to form a pseudo-mesh visua...

A cell conservative flux recovery technique is developed here for vertexcentered finite volume methods of second order elliptic equations. It is based on solving a local Neumann problem on each control volume using mixed finite element methods. The recovered flux is used to construct a constant free a posteriori error estimator which is proven to be reliable and efficient. Some numerical tests ...