通过自适应扩展网格有限元法（AES-FEM）克服有限元元素质量依赖性
Overcoming Element Quality Dependence of Finite Elements with Adaptive Extended Stencil FEM (AES-FEM)
AI总结：
本文提出AES-FEM，通过通用拉格朗日多项式基函数替代传统基函数，提升有限元在低质量网格下的稳定性与效率，验证其在泊松方程和对流-扩散方程中的准确性与收敛性。
摘要：
有限元方法（FEM）是工程中求解偏微分方程的重要技术，但其稳定性与性能严重依赖于元素形状质量。本文引入自适应扩展网格有限元法（AES-FEM）以克服这一依赖。我们的方法用一组通用拉格朗日多项式（GLP）基函数替代传统基函数，通过局部加权最小二乘近似构造。该方法保持FEM的理论框架，允许以与经典FEM相同的方式施加本质边界条件并积分刚度矩阵。此外，AES-FEM可使用比经典FEM更高的多项式基函数，同时几乎保持刚度矩阵的稀疏性模式。我们描述了AES-FEM的公式和实现，并分析了其一致性和稳定性。我们展示了在2D和3D中泊松方程和时间无关对流-扩散方程的数值实验。数值结果表明，AES-FEM比线性FEM更准确，误差与运行时间比上更高效，并在低质量网格上使迭代求解器具有更好的稳定性和更快的收敛速度。
英文摘要：
The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES-FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial (GLP) basis functions, which we construct using local weighted least-squares approximations. The method preserves the theoretical framework of FEM, and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES-FEM can use higher-degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES-FEM, and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poison equation and a time-independent convection-diffusion equation. The numerical results demonstrate that AES-FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor-quality meshes
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