scholarly journals Generalized finite element methods for quadratic eigenvalue problems

2016 ◽  
Vol 51 (1) ◽  
pp. 147-163 ◽  
Author(s):  
Axel Målqvist ◽  
Daniel Peterseim
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Eigenvalue Problems ◽  
Quadratic Eigenvalue Problems ◽  
Quadratic Eigenvalue ◽  
Generalized Finite Element

scholarly journals Survey of meshless and generalized finite element methods: A unified approach

Acta Numerica ◽  
2003 ◽  
Vol 12 ◽  
pp. 1-125 ◽  
Author(s):  
Ivo Babuška ◽  
Uday Banerjee ◽  
John E. Osborn
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mesh Generation ◽  
Variational Principles ◽  
Meshless Methods ◽  
Nonlinear Pdes ◽  
Time Step ◽  
Complicated Geometry ◽  
Generalized Finite Element ◽  
Selection Of

In the past few years meshless methods for numerically solving partial differential equations have come into the focus of interest, especially in the engineering community. This class of methods was essentially stimulated by difficulties related to mesh generation. Mesh generation is delicate in many situations, for instance, when the domain has complicated geometry; when the mesh changes with time, as in crack propagation, and remeshing is required at each time step; when a Lagrangian formulation is employed, especially with nonlinear PDEs. In addition, the need for flexibility in the selection of approximating functions (e.g., the flexibility to use non-polynomial approximating functions), has played a significant role in the development of meshless methods. There are many recent papers, and two books, on meshless methods; most of them are of an engineering character, without any mathematical analysis.In this paper we address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations, using variational principles. We give a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references to the current literature.The aim of the paper is to provide a survey of a part of this new field, with emphasis on mathematics. We present proofs of essential theorems because we feel these proofs are essential for the understanding of the mathematical aspects of meshless methods, which has approximation theory as a major ingredient. As always, any new field is stimulated by and related to older ideas. This will be visible in our paper.

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