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

Abstract

In the last few years meshless methods for numerically solving partial differential equations came 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, e.g., 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 non-linear PDE's. In addition, a need to have flexibility in the selection of approximating functions (e.g., the flexibility to use non-polynomial approximating functions), 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 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 reference 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 understanding 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. AMS(MOS 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Conforming polygonal finite elements

“…The latter shortcoming complicates the imposition of essential boundary conditions in a Galerkin implementation. A unified mathematical analysis of meshfree methods was recently proposed by Babuška and co-workers [21].…”

Smooth, second order, non‐negative meshfree approximants selected by maximum entropy

“…In contrast, typical meshfree methods based on the MLS approximants do not have these properties, and special methods need to be implemented to impose the Dirichlet boundary conditions. As reviewed in [4,19], a number of methods have been proposed to impose essential boundary conditions in the numerical approximation of boundary value problems with noninterpolatory approximants. Penalty formulations present obvious drawbacks, while the Lagrange multipliers method requires a careful choice of the functional spaces to avoid over constraining the approximation or having stability problems.…”

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

“…The Partition of Unity Method (PUM) combines a finite partition of unity covering of the object with a priori knowledge about the behavior of the solution at the interface [8]. The Generalized Finite Element Method [53,19,7] is per se a meshless method and was also developed under the name hp clouds [56]. It was combined with classical FE to improve their approximation capabilities [20,72,21].…”