A multigrid method for a Petrov-Galerkin discretization of the Stokes equations.
A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation
The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure...
A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation*
The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighboring layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic...
A new approach to the -regularity theorems for linear stationary nonhomogeneous Stokes systems.
A new mixed finite element method based on the Crank-Nicolson scheme for Burgers' equation
In this paper, a new mixed finite element method is used to approximate the solution as well as the flux of the 2D Burgers’ equation. Based on this new formulation, we give the corresponding stable conforming finite element approximation for the pair by using the Crank-Nicolson time-discretization scheme. Optimal error estimates are obtained. Finally, numerical experiments show the efficiency of the new mixed method and justify the theoretical results.
A new regularity criterion for the Navier-Stokes equations.
A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation
A Nonstandard Approach to the Uniqueness Problem for the Navier-Stokes Equations.
A note on the generalized energy inequality in the Navier-Stokes equations
We prove that there exists a suitable weak solution of the Navier-Stokes equation, which satisfies the generalized energy inequality for every nonnegative test function. This improves the famous result on existence of a suitable weak solution which satisfies this inequality for smooth nonnegative test functions with compact support in the space-time.
A note on the generic solvability of the Navier-Stokes equations
A null controllability data assimilation methodology applied to a large scale ocean circulation model
Data assimilation refers to any methodology that uses partial observational data and the dynamics of a system for estimating the model state or its parameters. We consider here a non classical approach to data assimilation based in null controllability introduced in [Puel, C. R. Math. Acad. Sci. Paris 335 (2002) 161–166] and [Puel, SIAM J. Control Optim. 48 (2009) 1089–1111] and we apply it to oceanography. More precisely, we are interested in developing this methodology to recover the unknown final...
A null controllability data assimilation methodology applied to a large scale ocean circulation model*
Data assimilation refers to any methodology that uses partial observational data and the dynamics of a system for estimating the model state or its parameters. We consider here a non classical approach to data assimilation based in null controllability introduced in [Puel, C. R. Math. Acad. Sci. Paris335 (2002) 161–166] and [Puel, SIAM J. Control Optim.48 (2009) 1089–1111] and we apply it to oceanography. More precisely, we are interested in developing this methodology to recover the unknown final...
A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations
A Particle Method to Solve the Navier-Stokes System.
A penalty method for the time-dependent Stokes problem with the slip boundary condition and its finite element approximation
We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization...
A piecewise P2-nonconforming quadrilateral finite element
We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...
A posteriori error estimates for nonlinear problems. -estimates for finite element discretizations of elliptic equations
A Posteriori Error Estimators for the Stokes Equations.
A problem in Rayleigh-Taylor instability
A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equation