Higher order approximation of interface and boundary conditions

有限元法是求解偏微分方程，特别是连续介质力学（流体流动、可变形固体、多孔介质流动等）数值逼近解的最经典和最流行的方法之一。。物理域（通常是3维）由小元素（四面体或六面体）网格划分，并在每个元素上以多项式形式搜索近似。******当域具有光滑（非多边形）弯曲边界时，通常的做法是构造域的多面体（或2D中的多边形）近似，从而可以将其划分为具有直面的元素。不幸的是，这种域的线性近似对有限元解的收敛阶有限制，特别是当使用高阶元素（二阶或更多次多项式）时。******补救措施之一是使用高阶边界的近似和沿边界近似的曲面元素。在实践中，这通常是用等参元素来完成的（边界的近似顺序与每个元素中多项式的程度一致）。******这一领域的理论通常被简化为简单的或学术的方程（泊松方程或更一般的二阶标量椭圆方程），并在其最简单的公式中，特别是必要的（或狄利克雷型的）边界条件被强烈地施加，通过选择相应的试验空间。******本程序的目标是将理论和数值结果扩展到流体和固体力学的经典方程组（通常是Stokes， Navier Stokes和lam<s:1>系统），在具有弯曲边界的域上，以及导致Dirichlet型基本边界条件弱强加的公式

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.**