Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes
Lowest-Order Weak Galerkin Finite Element Methods for Linear Elasticity on Rectangular and Brick Meshes
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DOI：
10.1007/s10915-018-0837-0
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发表时间：
2019-03-01
影响因子：
2.5
通讯作者：
Zheng, Bin
中科院分区：
文献类型：
--
作者：
Harper, Graham;Liu, Jiangguo;Zheng, Bin
本文研究了求解矩形网格和砖网格线弹性问题的最低阶弱伽辽金有限元方法。具体来说，在单元内部和单元界面上分别使用常数向量来近似位移。对于这些常数基函数，它们的离散弱梯度是在局部Raviart-Thomas空间RT[0]d （d=2或3）中计算的，而它们的离散弱散度是作为元素常量计算的。据此计算离散弱应变。然后使用这些量在矩形和砖网格上开发应变-div和梯度-div格式的有限元方案。在2-dim和3-dim情况下的数值实验结果表明，当精确解具有完全正则性时，该方法在位移、应力和位移散度方面具有最优的一阶收敛性。该方法还可以很好地捕获低正则性解。提出了包括Schur互补在内的有效实现策略。在理论分析和数值实验中，还研究了四边形和六面体网格的扩展。
This paper investigates lowest-order weak Galerkin finite element methods for solving linear elasticity problems on rectangular and brick meshes. Specifically, constant vectors are used in element interiors and on element interfaces respectively for approximating displacement. For these constant basis functions, their discrete weak gradients are calculated in the local Raviart-Thomas spaces RT[0]d (d=2 or 3), whereas their discrete weak divergences are calculated as elementwise constants. Discrete weak strains are calculated accordingly. Then these quantities are used to develop finite element schemes in both strain-div and grad-div formulations, on both rectangular and brick meshes. A theoretical analysis supported by numerical experiments in both 2-dim and 3-dim reveal that the methods are locking-free and have optimal 1st order convergence in displacement, stress, and dilation (divergence of displacement), when the exact solution has full regularity. The methods can also capture low-regularity solutions very well. Strategies for efficient implementation including Schur complement are presented. Extension to quadrilateral and hexahedral meshes, in both theoretical analysis and numerical experiments, is also examined.