四川大学学报（自然科学版）2026，Vol.63Issue(1)：83-90,8.DOI:10.19907/j.0490-6756.240353
线弹性问题的一类无稳定子弱有限元法
Stabilizer-free weak Galerkin finite element for linear elastic problems
摘要
Abstract
Linear elastic problem originates from engineering applications such as machinery and aviation.Due to the complexity of differential equation in the linear elastic problem,the analytic solutions of the prob-lem are still unknown.Although the classical finite element methods can be used to numerically solve the lin-ear elastic problem,these methods cannot overcome the"locking"phenomenon.To address this problem,weak Galerkin methods are proposed,which introduces the weak differential operators(weak gradient,weak divergence,etc.)and uses the discontinuous polynomial functions to discrete the differential equation in the problem.In weak Galerkin method,the continuity of the approximation functions usually needs to be guaran-teed by the boundary function and stabilizers.That is to say,the stability of the numerical scheme is guaran-teed by adding the stabilizers,which makes the scheme become very complex and brings the problems of large amount of calculation and slow approximation speed.To keep the stability,the stabilizer-free weak Galerkin methods are proposed,in which the approximation polynomial degree of the weak gradient operator is increased.In this paper,a stabilizer-free weak Galerkin method is proposed for the linear elastic problem.Firstly,the weak gradient and weak divergence operators are introduced,and the piecewise linear and piece-wise quadratic polynomials are used to approximate the internal displacement and boundary displacement of the element,respectively.Then,by introducing the auxiliary variables,a mixed scheme equivalent to the problem is given,and the uniform convergence of the scheme with respect to the Lamé constant is proven,that is,the method can overcome the"locking"phenomenon.In comparison with the weak Galerkin method with stabilizers,the proposed method is very simple and the approximation speed is faster.Numerical ex-amples verify the theoretical results.关键词
线弹性问题/弱有限元方法/稳定子Key words
linear elastic problem/weak Galerkin method/stabilizer分类
数理科学引用本文复制引用
许岳,陈豫眉,谢小平..线弹性问题的一类无稳定子弱有限元法[J].四川大学学报（自然科学版）,2026,63(1):83-90,8.基金项目
国家自然科学基金(12171340) （12171340）
