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

TL;DR

This paper introduces a unified framework for analyzing finite differences, finite elements, and a hybrid method called AES-FEM, demonstrating improved accuracy and robustness in solving PDEs.

Contribution

It develops a generalized weighted residuals framework that unifies the formulations and analysis of these methods, including a new analysis of AES-FEM.

Findings

AES-FEM achieves second-order accuracy on unstructured meshes.

AES-FEM reduces mesh-quality dependency compared to generalized finite differences.

Numerical results verify the theoretical analysis and advantages of AES-FEM.

Abstract

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 (Int. J. Num. Meth. Engrg., 2016, DOI:10.1002/nme.5246), 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.