Higher order approximation of interface and boundary conditions

有限元方法是数值逼近偏微分方程解的最经典和最流行的方法之一，特别是来自连续介质力学(流体流动、可变形固体、多孔介质流动)的解。)。物理区域(通常为3维)由小单元(四面体或六面体)组成网格，并在每个单元上搜索近似多项式。当区域具有平滑(非多边形)曲线边界时，通常的做法是构造区域的多面体(或2D中的多边形)近似，从而可以将其划分为具有直面的单元。遗憾的是，这种区域的线性逼近对有限元解的收敛阶带来了限制，特别是当使用高阶单元(二次或多项式)时。补救措施之一是使用高阶边界的近似和沿边界的曲面单元的近似。在实践中，这通常是用等参数单元完成的(边界的近似顺序与每个单元中多项式的次数重合)。*这一领域的理论通常被归结为简单的或学术的方程(典型的泊松方程或更一般的二阶标量椭圆型方程)和它们的最简单的公式，其中，通过相应地选择试探空间，特别地，施加了必要的(或狄利克雷特类型的)边界条件。*本程序的目标是将理论和数值结果推广到流体和固体力学的经典方程组(通常是Stokes、Navier Stokes和Lamé系统)，在具有弯曲边界的区域上，并使用导致弱施加Dirichlet型基本边界条件的公式。**

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.**