Uday Banerjee

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

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

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.

Stable Generalized Finite Element Method (SGFEM)

Abstract The Generalized Finite Element Method (GFEM) is a Partition of Unity Method (PUM), where the trial space of standard Finite Element Method (FEM) is augmented with non-polynomial shape functions with compact support. These shape functions, which are also known as the enrichments, mimic the local behavior of the unknown solution of the underlying variational problem. GFEM has been successfully used to solve a variety of problems with complicated features and microstructure. However, the stiffness matrix of GFEM is badly conditioned (much worse compared to the standard FEM) and there could be a severe loss of accuracy in the computed solution of the associated linear system. In this paper, we address this issue and propose a modification of the GFEM, referred to as the Stable GFEM (SGFEM). We show that SGFEM retains the excellent convergence properties of GFEM, does not require a ramp-function in the presence of blending elements, and the conditioning of the associated stiffness matrix is not worse than that of the standard FEM. Moreover, SGFEM is very robust with respect to the parameters of the enrichments. We show these features of SGFEM on several examples.

Meshless and Generalized Finite Element Methods: A Survey of Some Major Results

In this lecture, we discuss Meshless and Generalized Finite Element Methods. We survey major results in this area with a unified approach.

On principles for the selection of shape functions for the Generalized Finite Element Method

Effective shape functions for the Generalized Finite Element Method should reflect the available information on the solution. This information is partially fuzzy, because the solution is, of course, unknown, and typically the only available information is the solution’s inclusion in a variety of function spaces. It is desirable to choose shape functions that perform robustly over a family of relevant situations. Quantitative notions of robustness are introduced and discussed. We show, in particular, that in one dimension polynomials are robust when the available information consists in inclusions in Sobolev-type spaces that are x-independent.

Estimation of the effect of numerical integration in finite element eigenvalue approximation

SummaryFinite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.

The effect of numerical quadrature in the p-version of the finite element method

We investigate the use of numerical quadrature in the p-version of the finite element method. We describe a set of minimal conditions that the quadrature rules should satisfy, for various types of elements. Under sufficient assumptions of smoothness, we establish optimality of the asymptotic rate of convergence. Some computational results are presented, which illustrate under what conditions overintegration may be useful.

On the approximability and the selection of particle shape functions

Summary.Particle methods, also known as meshless or meshfree methods, have become popular in approximating solutions of partial differential equations, especially in the engineering community. These methods do not employ a mesh, or use it minimally, in the construction of shape functions. There is a wide variety of classes of shape functions that can be used in particle methods. In this paper, we primarily address the issue of selecting a class of shape functions, among this wide variety, that would yield efficient approximation of the unknown solution. We have also made several comments and observations on the order of convergence of the interpolation error, when these shape functions are used; specifically, we have shown that the interpolation error estimate, for certain classes of shape functions, may not indicate the actual order of convergence of the approximation error.

A note on the effect of numerical quadrature in finite element eigenvalue approximation

SummaryIn a recent work by the author and J.E. Osborn, it was shown that the finite element approximation of the eigenpairs of differential operators, when the elements of the underlying matrices are approximated by numerical quadrature, yield optimal order of convergence when the numerical quadrature satisfies a certain precision requirement. In this note we show that this requirement is indeed sharp for eigenvalue approximation. We also show that the optimal order of convergence for approximate eigenvectors can be obtained, using numerical quadrature with less precision.

Higher order stable generalized finite element method

The generalized finite element method (GFEM) is a Galerkin method, where the trial space is obtained by augmenting the trial space of the standard finite element method (FEM) by non-polynomial functions, called enrichments, that mimic the local behavior of the unknown solution of the underlying variational problem. The GFEM has excellent approximation properties, but its conditioning could be much worse than that of the FEM. However, if the enrichments satisfy certain properties, then the conditioning of the GFEM is not worse than that of the standard FEM, and the GFEM is referred to as the stable GFEM (SGFEM). In this paper, we address the higher order SGFEM that yields higher order convergence and suggest a specific modification of the enrichment function that guarantees the required conditioning, yielding a robust implementation of the higher order SGFEM.