A hybrid method and unified analysis of generalized finite differences and Lagrange finite elements

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:376

Article

Conley, Rebecca¹  Delaney, Tristan J.^2,3,4  Jiao, Xiangmin^2,3 

[1] St Peters Univ, Dept Math, Jersey City, NJ 07306 USA

[2] SUNY Stony Brook, Dept Appl Math & Stat, Stony Brook, NY 11794 USA

[3] SUNY Stony Brook, Inst Adv Computat Sci, Stony Brook, NY 11794 USA

[4] Synopsys Inc, Mountain View, CA 94043 USA

关键词: Partial differential equations;    Finite element methods;    Generalized finite differences;    Generalized weighted residuals;    Stability;    Convergence;   

DOI  :  10.1016/j.cam.2020.112862

来源: Elsevier

【 摘 要 】

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. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free