International audienceThis is a short presentation to the capability of the freefem++ software, in section 1, we recall most of the characteristics of the software, In section 2, we recall how to to build a weak form form of an partial differential equation (PDE) from the strong form of the PDE. In three last sections, we present different problem, tools to illustrated the software. First we do mesh adaptation problem in two and three dimension, secondly, we solve numerically a Phase change with Natural Convection, and the finally to show the HPC possibility we show a Schwarz Domain Decomposition problem on parallel computer

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Three new ideas for anisotropic adaption of unstructured triangular grids are presented, with particular emphasis on fluid flow computations. © 1997 John Wiley & Sons, Ltd.

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Summary. The "parareal in time" algorithm introduced in Lions et al. [2001] enables parallel computation using a decomposition of the interval of time integration. In this paper, we adapt this algorithm to solve the challenging Navier-Stokes problem. The coarse solver, based on a larger timestep, may also involve a coarser discretization in space. This helps to preserve stability and provides for more significant savings.

Summary When written in an Eulerian frame, the conservation laws of continuum mechanics are similar for fluids and solids leading to a single set of variables for a monolithic formulation. Such formulations are well adapted to large displacement fluid‐structure configurations, but stability is a challenging problem because of moving geometries. In this article, the method is presented; time implicit discretizations are proposed with iterative algorithms well posed at each step, at least for small displacements; stability is discussed for an implicit in time finite element method in space by showing that energy decreases with time. The key numerical ingredient is the Characterics‐Galerkin method coupled with a powerful mesh generator. A numerical section discusses implementation issues and presents a few simple tests. It is also shown that contacts are easily handled by extending the method to variational inequalities. This paper deals only with incompressible neo‐Hookean Mooney‐Rivlin hyperelastic material in 2 dimensions in a Newtonian fluid, but the method is not limited to these; compressible and 3D cases will be presented later.

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Abstract. In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.Mathematics Subject Classification. 65N30, 65N50, 76D07, 76S05.