Thierry Gallouët

Non-linear elliptic and parabolic equations involving measure data

In this paper we prove the existence of solutions for equations of the type −div(a(·, Du)) = f in a bounded open set Ω, u = 0 on ∂Ω, where a is a possibly non-linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure. We also consider the equation −div(a(·, Du)) + g(·, u) = f in Ω, u = 0 on ∂Ω (with f ϵ L1(Ω), or f ϵ M(Ω), (·, u) · u ≧ 0) and the parabolic equivalent of the first (elliptic) equation.

Some approximate Godunov schemes to compute shallow-water equations with topography

Abstract We study here the computation of shallow-water equations with topography by Finite Volume methods, in a one-dimensional framework (though all methods introduced may be naturally extended in two dimensions). All methods are based on a discretisation of the topography by a piecewise function constant on each cell of the mesh, from an original idea of Le Roux et al. Whereas the Well-Balanced scheme of Le Roux is based on the exact resolution of each Riemann problem, we consider here approximate Riemann solvers. Several single step methods are derived from this formalism, and numerical results are compared to a fractional step method. Some test cases are presented: convergence towards steady states in subcritical and supercritical configurations, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefaction wave over a step. Numerical schemes, combined with an appropriate high-order extension, provide accurate and convergent approximations.

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Abstract We consider the differential problem (*) { A ( u ) = μ in Ω , u = 0 on ∂ Ω , where Ω is a bounded, open subset of R N , N ≥ 2, A is a monotone operator acting on W 0 1 , p ( Ω ) , p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p -capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L 1 (Ω) and of a measure in W −1, p ′ (Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*) .

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary. One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation

A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

u_t+{\rm div}({\mathbf q} f(u))-\Delta \phi(u)=0

A sequel to a rough Godunov scheme: application to real gases

by a piecewise constant function

NUMERICAL MODELING OF TWO-PHASE FLOWS USING THE TWO-FLUID TWO-PRESSURE APPROACH

u_{{\mathcal D}}

Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations

using a discretization

On an Approximate Godunov Scheme

{\mathcal D}

Resolution of a semilinear equation in L1

in space and time and a finite volume scheme. The convergence of