Finite differences, finite elements, and their generalizations are widely used for solving partial differential equations, and their high-order variants have respective advantages and disadvantages. Traditionally, these methods are treated as different (strong vs. weak) formulations and are analyzed using different techniques (Fourier analysis or Green’s functions vs. functional analysis), except for some special cases on regular grids. Recently, the authors introduced a hybrid method, called Adaptive Extended Stencil FEM or AES-FEM (Conley et al., 2016), which combines features of generalized finite differences and Lagrange finite elements to achieve second-order accuracy over unstructured meshes. However, its analysis was incomplete due to the lack of existing mathematical theory that unifies the formulations and analysis of these different methods. In this work, we introduce the framework of generalized weighted residuals to unify the formulation of finite differences, finite elements, and AES-FEM. In addition, we propose a unified analysis of the well-posedness, convergence, and mesh-quality dependency of these different methods. We also report numerical results with AES-FEM to verify our analysis. We show that AES-FEM improves the accuracy of generalized finite differences while reducing the mesh-quality dependency and simplifying the implementation of high-order finite elements.

有限差分，有限元及其推广被广泛用于求解偏微分方程，它们的高阶变式各有优缺点。传统上，除了常规网格上的某些特殊情况外，这些方法被视为不同的（强或弱）公式，并使用不同的技术（傅立叶分析或格林函数与函数分析）进行分析。最近，作者介绍了一种混合方法，称为Adaptive Extended Stencil FEM或AES-FEM（Conley等，2016），它结合了广义有限差分和Lagrange有限元的特征在非结构化网格上实现二阶精度。但是，由于缺乏统一这些不同方法的表述和分析的数学理论，其分析是不完整的。在这项工作中，我们介绍了广义加权残差的框架，以统一有限差分，有限元和AES-FEM的表述。此外，我们提出了对适定性，收敛性和网格质量相关性的统一分析这些不同的方法。我们还使用AES-FEM报告数值结果以验证我们的分析。我们表明，AES-FEM在降低网格质量相关性并简化高阶有限元的实现的同时，提高了广义有限差分的准确性。