Higher order approximation of interface and boundary conditions

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The finite element method is one of the most classical and popular method for the numerical approximation of solutions to partial differential equations, in particular those from continuum mechanics (fluid flows, deformable solids, porous media flows, . ). The physical domain (generally 3-dimensional) is meshed with small elements (tetrahedra or hexahedra) and approximations are searched as polynomials on each of these elements.******When the domain has a smooth (non-polygonal) curved boundary, the common practice is to construct a polyhedral (or polygonal in 2D) approximation of the domain, which can thus be divided into elements with straight faces. Unfortunately, this linear approximation of the domain brings a limitation on the convergence orders of the finite element solutions, in particular when high order elements (polynomials of degree two or more) are used.******One of the remedies is to use approximations of the boundary of higher orders and elements with curved faces along the boundary approximations. In practice this is often done with isoparametric elements (the order of the approximation of the boundary coincides with the degree of the polynomials in each element). ******The theory in this field is often reduced to simple or academic equations (Poisson's equation or more generally second order scalar elliptic equations, typically) and in their simplest formulations where, in particular, essential (or of Dirichlet type) boundary conditions are imposed strongly, by choosing the trial space accordingly. ******The objective of this program is to extend the theoretical and numerical results to classical systems of equations from fluid and solid mechanics (typically, Stokes, Navier Stokes and Lamé systems), on domains with curved boundaries, and with formulations that result in a weak imposition of essential boundary conditions of Dirichlet type.**