scholarly journals Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems

2011 ◽  
Vol 9 (1) ◽  
pp. 373-406 ◽  
Author(s):  
Ivo Babuska ◽  
Robert Lipton
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Local Approximation ◽  
Approximation Spaces ◽  
Multiscale Problems ◽  
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.

GENERALIZED FINITE ELEMENT METHODS — MAIN IDEAS, RESULTS AND PERSPECTIVE

2004 ◽  
Vol 01 (01) ◽  
pp. 67-103 ◽  
Author(s):  
IVO BABUŠKA ◽  
UDAY BANERJEE ◽  
JOHN E. OSBORN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Element Methods ◽  
Meshless Methods ◽  
Generalized Finite Element Method ◽  
Main Ideas ◽  
Generalized Finite Element ◽  
Element Method

This paper is an overview of the main ideas of the Generalized Finite Element Method (GFEM). We present the basic results, experiences with, and potentials of this method. GFEM is a generalization of the classical Finite Element Method — in its h, p, and h-p versions — as well as of the various forms of meshless methods used in engineering.

Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Matthias Maier ◽  
Rolf Rannacher
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Elliptic Problems ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Multiscale Problems ◽  
Multiscale Finite Element ◽  
Heterogeneous Multiscale ◽  
Multiscale Finite Element Methods

AbstractThis paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for

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