A fourth-order unfitted characteristic finite element method for free-boundary problems
A fourth-order unfitted characteristic finite element method for free-boundary problems
复制标题
求解自由边界问题的四阶不拟合特征有限元方法
DOI：
10.1016/j.jcp.2022.111552
复制
发表时间：
2022-08
影响因子：
4.1
通讯作者：
Zheng Weiying
中科院分区：
文献类型：
--
作者：
Ma Chuwen;Zheng Weiying
提出了求解时变偏微分方程自由边界问题的四阶非拟合特征有限元法。域的边界由PDE的解隐式驱动，而PDE是在时变的未知域上提出的。这样，它们就形成了一个耦合的非线性系统。UCFEM的关键是用四阶前向流图显式跟踪域，用四阶后向流图对PDE进行时间离散。采用固定网格上的四阶非拟合有限元法求解微分方程。由于区域边界用三次样条函数显式表示，因此即使区域发生严重变形，也能准确有效地对切割单元进行正交。UCFEM为自由边界问题的高阶数值方法设计提供了一个框架。我们将UCFEM应用于二维对流扩散方程和Navier-Stokes方程，得到了整体的四阶精度。通过大量的数值实验，我们证明了该方法即使在严重变形的区域上也能实现四阶收敛。提出了求解自由边界问题的四阶非拟合特征有限元方法。•领域的发展是由PDE模型的解决方案驱动的。•四阶方法可以处理严重变形的区域。
A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains. • We propose a fourth-order unfitted characteristic finite element method for solving free-boundary problems. • The evolution of the domain is driven by the solution to the PDE model. • The fourth-order method can deal with severely deformed domains.
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DOI：
10.1016/j.jcp.2007.12.029
发表时间：
2008-05
期刊：
J. Comput. Phys.
影响因子：
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作者：
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通讯作者：
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影响因子：
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期刊：
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影响因子：
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影响因子：
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