四阶奇异摄动问题的弱Galerkin有限元法

Weak Galerkin finite element methods for fourth order singular perturbation problems

摘要:

本文研究二维和三维情形下四阶奇异摄动问题弱 Galerkin 有限元法的构造与分析.我们引入了弱二阶偏导数算子，对单元内部的位移变量采用连续分片多项式逼近，对单元边界上的位移梯度采用间断分片多项式逼近.基于 Scott-Zhang和 L^2 投影算子的性质，该方法能够得到能量范数的最优误差估计,且针对边界层问题能够得到与摄动参数一致无关的收敛阶.数值算例验证了理论结果.

Abstract:

In this paper, we discuss the construction and analysis of the weak Galerkin (WG) finite element methods for the fourth order singular perturbation problems in two and three dimensions. By introducing the weak second order partial derivative operators, the WG method is constructed by adopting continuous piecewise polynomials of degree k>2 for the approximation to the displacement in the interior of elements, and discontinuous piecewise polynomials of degree k-1 for the approximations to the trace of displacement gradient on the inter-element boundaries. Based on the properties of the Scott-Zhang and L^2 projections, optimal error estimates in energy norm are derived. In addition, for the boundary layer case, we show that the methods are convergent uniformly with respect to the perturbation parameter. Numerical examples are provided to verify the theoretical results.