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Finite differences, finite elements, and their generalizations are widelyused for solving partial differential equations, and their high-order variantshave respective advantages and disadvantages. Traditionally, these methods aretreated as different (strong vs. weak) formulations and are analyzed usingdifferent techniques (Fourier analysis or Green's functions vs. functionalanalysis), except for some special cases on regular grids. Recently, theauthors introduced a hybrid method, called Adaptive Extended Stencil FEM orAES-FEM (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002/nme.5246), which combinesfeatures of generalized finite differences and Lagrange finite elements toachieve second-order accuracy over unstructured meshes. However, its analysiswas incomplete due to the lack of existing mathematical theory that unifies theformulations and analysis of these different methods. In this work, weintroduce the framework of generalized weighted residuals to unify theformulation of finite differences, finite elements, and AES-FEM. In addition,we propose a unified analysis of the well-posedness, convergence, andmesh-quality dependency of these different methods. We also report numericalresults with AES-FEM to verify our analysis. We show that AES-FEM improves theaccuracy of generalized finite differences while reducing the mesh-qualitydependency and simplifying the implementation of high-order finite elements.

A Hybrid Method and Unified Analysis of Generalized Finite Differences and Lagrange Finite Elements