We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge to some “crystalline” perimeter/total variation, and provide an almost explicit formula for the limiting functional.

A Mimetic Discretization method for the linear elasticity problem in mixed weakly symmetric form is developed. The scheme is shown to converge linearly in the mesh size, independently of the incompressibility parameter λ, provided the discrete scalar product satisfies two given conditions. Finally, a family of algebraic scalar products which respect the above conditions is detailed.

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative. Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases. To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step. The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model. Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.

We consider an initial and Dirichlet boundary value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. Discretizing the space-time white noise a modelling error is introduced and a regularized fourth-order linear stochastic parabolic problem is obtained. Fully-discrete approximations to the solution of the regularized problem are constructed by using, for discretization in space, a Galerkin finite element method based on C0 or C1 piecewise polynomials, and, for time-stepping, the Backward Euler method. We derive strong a priori estimates for the modelling error and for the approximation error to the solution of the regularized problem.

We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.

We study here the water waves problem for uneven bottoms in a highly nonlinear regime where the small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is known that, for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom is flat. We generalize here this result with a new class of equations taking into account variable bottom topographies. Of course, many variable depth KdV equations existing in the literature are recovered as particular cases. Various regimes for the topography regimes are investigated and we prove consistency of these models, as well as a full justification for some of them. We also study the problem of wave breaking for our new variable depth and highly nonlinear generalizations of the KdV equations.

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction stays within its natural bounds.