Higher order approximation of interface and boundary conditions

界面和边界条件的高阶近似

• 批准号: RGPIN-2015-04610
• 负责人: Urquiza, José
• 金额: $ 0.8万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=674742
• 关键词: Higher; order; approximation; interface; boundary

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.**

有限元方法是求解偏微分方程的最经典和最流行的方法之一，特别是那些来自连续介质力学（流体流动，可变形固体，多孔介质流动，等）的偏微分方程。).物理域（通常是三维的）用小元素（四面体或六面体）进行网格化，并在每个元素上以多项式形式搜索近似值。当域具有平滑（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单元。不幸的是，这种区域的线性近似对有限元解的收敛阶带来了限制，特别是当使用高阶元素（二次或二次以上的多项式）时。补救措施之一是使用高阶边界近似和沿边界近似沿着具有曲面的单元。在实践中，这通常是用等参元素（边界的近似阶数与每个元素中多项式的阶数一致）来完成的。** 这个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），并且在它们的最简单的公式中，特别是通过相应地选择试验空间来施加基本（或Dirichlet类型）边界条件。** 本计划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Lamé系统），具有弯曲边界的区域，以及导致Dirichlet型本质边界条件弱强加的公式。