We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact ...

In this paper, we study variational inequalities related to ergodic control problems studied by M. Boulbrachène and H. Sissaoui [11, 1996], where the "discount factor" (i.e., the zero order term) is set to 0, we use an overlapping Schwarz method on nomatching grid which consists in decomposing the domain in two subdomains. For α ∈ ]0.1[ we provide the discretization on each subdomain converges ...

The finite element method can be viewed as a machine that automates the discretization of differential equations, taking as input a variational problem, a finite element and a mesh, and producing as output a system of discrete equations. However, the generality of the framework provided by the finite element method is seldom reflected in implementations (realizations), which are often specializ...

A variational data assimilation technique was used to estimate optimal discretization of interpolation operators and derivatives in the nodes adjacent to the rigid boundary. Assimilation of artificially generated observational data in the shallow-water model in a square box and assimilation of real observations in the model of the Black sea are discussed. It is shown in both experiments that co...

An algorithm is derived for a hyperelastic incompressible solid coupled with a Newtonian fluid. It is based on a Eulerian formulation of the full system in which the main variables are the velocities. After a fully implicit discretization in time it is possible to eliminate the displacements and solve a variational equation for the velocities and pressures only. The stability of the method depe...

We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to t...

This contribution is the first part of two papers on the Fully Eulerian formulation for fluid-structure interactions. We derive a monolithic variational formulation for the coupled problem in Eulerian coordinates. Further, we present the Initial Point Set method for capturing the moving interface. For the discretization of this interface problem, we introduce a modified finite element scheme th...

We study anisotropic surface diffusion of curves with a small corner energy regularization. The regularization allows the use of nonconvex free energy densities and turns the evolution law into a 6th order geometric equation. Using a semi-implicit time discretization, we present a variational formulation of this equation for parametric curves, leading to a discretization based on linear finite ...