This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and Vázquez (2010).

We present the numerical analysis on the Poisson problem of two mixed Petrov-Galerkin finite volume schemes for equations in divergence form $\mathrm{div}\varphi (u,\nabla u)=f$. The first scheme, which has been introduced in [CITE], is a generalization in two dimensions of Keller's box-scheme. The second scheme is the dual of the first one, and is a cell-centered scheme for u and the flux φ. For the first scheme, the two trial finite element spaces are the nonconforming space of Crouzeix-Raviart for the primal unknown u...

We prove the convergence of a finite volume method for a noncoercive linear elliptic problem, with right-hand side in the dual space of the natural energy space of the problem.

We prove the convergence of a finite volume method for a noncoercive linear elliptic problem, with right-hand side in the dual space of the natural energy space of the problem.

We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order $H^1-$norm and $L^2-$norm error estimates.

We consider a new formulation for finite volume element methods, which is satisfied by known finite volume methods and it can be used to introduce new ones. This framework results by approximating the test function in the formulation of finite element method. We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations and prove optimal order H1-norm and L2-norm error estimates.

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...

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation $u_t(x,t)+\text{div}F(x,t,u(x,t))=0$ with the initial condition $u_0\in L^{\infty }\left({ℝ}^N\right)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in L^{i}nfty({ℝ}^N\times {ℝ}_{+})$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L_{loc}^p({ℝ}^N\times {ℝ}_{+})$, 1 ≤ p ≤ +∞. Furthermore, if $u_0\in L^{\infty }\cap {\text{BV}}_{loc}\left({ℝ}^N\right)$, we show that $u\in {\text{BV}}_{loc}({ℝ}^N\times {ℝ}_{+})$, which leads to an “$h^{\frac{1}{4}}$” error estimate between the approximate and the entropy solutions (where h defines the size of the...

We construct finite volume schemes, on unstructured and irregular grids and in any space dimension, for non-linear elliptic equations of the $p$-laplacian kind: $-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f$ (with $1\<p\<\infty$). We prove the existence and uniqueness of the approximate solutions, as well as their strong convergence towards the solution of the PDE. The outcome of some numerical tests are also provided.

We construct finite volume schemes, on unstructured and irregular grids and in any space dimension, for non-linear elliptic equations of the p-Laplacian kind: -div(|∇u|p-2∇u) = ƒ (with 1 < p < ∞). We prove the existence and uniqueness of the approximate solutions, as well as their strong convergence towards the solution of the PDE. The outcome of some numerical tests are also provided.

In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics...