多面体网格上线性弹性的最小二乘弱伽辽金框架
A Least Squares Weak Galerkin Framework for Linear Elasticity on Polytopal Meshes
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中文总结 AI 辅助
研究线性弹性问题，提出最小二乘弱伽辽金有限元方法，利用弱微分算子处理复杂条件，避免离散inf-sup条件，公式对称正定，有几何灵活性，建立唯一性和误差估计，数值实验验证其性能。
中文摘要 AI 辅助
本文开发并分析了一种用于线性弹性的最小二乘弱伽辽金（LS-WG）有限元方法。通过在弱有限元空间上使用弱微分算子，特别是弱梯度、弱应变张量和弱散度，该框架便于处理复杂边界条件和内部界面，同时避免了严格的离散inf-sup条件。所得公式对称正定，在近不可压缩情况下具有稳健的数值性能。此外，该方法具有出色的几何灵活性，可在一般多面体（多边形和多面体）网格上实现。我们建立了数值解的唯一性，并针对定制的离散能量范数导出了最优阶误差估计。大量数值实验证实了理论收敛率，并证明了该方法对近不可压缩材料的稳定性、效率和无锁定性能。
英文摘要
This paper develops and analyzes a least-squares weak Galerkin (LS-WG) finite element method for linear elasticity. By employing weak differential operators, specifically the weak gradient, weak strain tensor, and weak divergence, defined on weak finite element spaces, the proposed framework facilitates the treatment of complex boundary conditions and internal interfaces while avoiding the restrictive discrete inf-sup condition. The resulting formulation is symmetric and positive definite and exhibits robust numerical performance in the nearly incompressible regime. In addition, the proposed method offers exceptional geometric flexibility, allowing implementation on general polytopal (polygonal and polyhedral) meshes. We establish the uniqueness of the numerical solution and derive optimal-order error estimates with respect to a tailored discrete energy norm. Extensive numerical experiments confirm the theoretical convergence rates and demonstrate the method's stability, efficiency, and locking-free performance for nearly incompressible materials.
