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 problemin mixed weakly symmetric form is developed. The scheme is shown toconverge linearly in the mesh size, independently of theincompressibility parameter λ, provided the discrete scalarproduct satisfies two given conditions. Finally, a family ofalgebraic scalar products which respect the above conditions isdetailed.

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 fora fourth-order linear stochastic parabolic equation, in one spacedimension, forced by an additive space-time white noise.Discretizing the space-time white noise a modelling error isintroduced and a regularized fourth-order linear stochasticparabolic problem is obtained. Fully-discrete approximations to the solution of the regularizedproblem are constructed by using, for discretization in space, aGalerkin finite element method based on C0 or C1piecewise polynomials, and, for time-stepping, the Backward Eulermethod.We derive strong a priori estimates for the modelling error and forthe approximation error to the solution of the regularizedproblem.

We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

We study here the water waves problem for uneven bottoms in a highly nonlinear regime wherethe small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is knownthat, 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 isflat. We generalize here this resultwith 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 justificationfor some of them. We also study the problem of wave breaking for our newvariable depth and highly nonlinear generalizations of the KdV equations.

In this work we describe an efficient model for the simulation of atwo-phase flow made of a gas and a granular solid. The starting point is the two-velocitytwo-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented bya relaxation source term in orderto take into account the pressure equilibrium between the two phases andthe granular stress in the solid phase. We show that the relaxationprocess 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 marchingalgorithm 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 fractionis 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 volumefraction stays within its natural bounds.