Error analysis of the reproducing kernel particle method

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Abstract

Interest in meshfree (or meshless) methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in mechanics, especially in dealing with difficult problems involving large deformation, moving discontinuities, etc. In this paper, we provide a theoretical analysis of the reproducing kernel particle method (RKPM), which belongs to the family of meshfree methods. One goal of the paper is to set up a framework for error estimates of RKPM. We introduce the concept of a regular family of particle distributions and derive optimal order error estimates for RKP interpolants on a regular family of particle distributions. The interpolation error estimates can be used to yield error estimates for RKP solutions of BVPs.

Introduction

The finite element method has been the dominant numerical method in computational mechanics for several decades. Recently, a new family of numerical methods has attracted much interest in the community of computational mechanics. This new family of numerical methods shares a common feature that no mesh is needed and shape functions are constructed from sets of particles. These methods are designed to handle more effectively problems with large deformations, moving discontinuities and other difficult problems, and are hailed as numerical methods of the next generation (cf. Preface of [15]).

Currently there is no single universal name for the family of methods, with meshless methods or meshfree methods as possible choices. For example, Meshless methods is the title of a special issue of the journal Computer Methods in Applied Mechanics and Engineering [15] in 1996. Recently, however, the name meshfree methods becomes more popular. Various methods belong to this family, including smooth particle hydrodynamics (SPH) methods [19], [21], [22], diffuse element method (DEM) [23], element free Galerkin (EFG) method [2], [3], reproducing kernel particle method (RKPM) [5], [16], [17], moving least-square reproducing kernel method [14], [18], h-p-Clouds [8], [9], partition of unity finite element method [1], [20].

In this paper, we provide a theoretical analysis of the RKPM, which belongs to the family of meshfree methods. One goal of the paper is to set up a framework for error estimates of RKPM. We introduce the concept of a regular family of particle distributions and derive error estimates for RKP interpolants on a regular family of particle distributions. The interpolation error estimates are used to yield optimal order error estimates for RKP solutions of Neumann boundary value problems. Since the RKP shape functions do not have the Kronecker delta property, the treatment of Dirichlet boundary value conditions is more difficult than in the finite element method. We will show how to derive optimal order error estimates for RKP solutions of Dirichlet boundary value problems (BVPs) in one dimension. Several approaches are proposed in the literature to treat Dirichlet boundary value conditions (cf. [6], [7], [11], [24]) numerically. We discuss error estimates in this paper under sufficient smoothness assumption on the functions being approximated. Error analysis of the method for singular problems will be given in a forthcoming paper [12].

The paper is organized as follows. In Section 2, we introduce some notations to be used later. For convenience of mathematics readers, we provide a precise introduction of RKPM in Section 3, emphasizing the ideas behind the development of the method. In Section 4, we derive optimal order error estimates for RKP interpolants. These results are comparable to those in the theory of the finite element method. In Section 5, we discuss error estimates for RKP solutions of boundary value problems. Numerical results presented in the last section demonstrate convergence orders of RKPM, confirming the theoretical error estimates.

Section snippets

Notations

Throughout the paper, we use the following notations. The letter d is a positive integer and is used for the spatial dimension. We denote Ω⊂Rd to be a nonempty, open bounded set with a Lipschitz continuous boundary. In the one-dimensional case, d=1, we choose Ω=(0,L) for some L>0. A generic point in Rd is denoted by x=(x1,…,xd)T, or y=(y1,…,yd)T or z=(z1,…,zd)T. We use Euclidean norm to measure the vector length:x∥=i=1d|xi|21/2.It is convenient to use the multi-index notation for partial

Reproducing kernel particle approximation

We first introduce the concept of reproducing kernel approximation at the continuous level, which helps in understanding derivation of the RKP approximation.

Interpolation error estimates

We will derive error estimates for the case of quasiuniform support sizes, i.e. there exist two constants c1,c2∈(0,∞) such thatc1rirj⩽c2i,j.For such particle distributions, there exists a parameter r>0 such thatc̃1rirc̃2i.The more general case of arbitrary support sizes will be studied in a forthcoming paper.

Reproducing kernel particle method and error analysis

The RKPM is a Galerkin method combined with the use of RKP spaces. To explain the method in a concrete problem setting, we take a linear elliptic boundary value problem as an example. It is equally fine to consider nonlinear elliptic BVPs if we wish. Since nonhomogeneous Dirichlet boundary conditions can be rendered homogeneous in a standard way (see [13, Chapter 6] or one of many texts on modern PDE), we will assume Dirichlet boundary conditions, if any, are homogeneous. The weak formulation is

Numerical results

In this section, we present some numerical results on convergence orders of RKPM. The numerical results confirm the theoretical prediction. We thank J.S. Chen and C.T. Wu of the University of Iowa for providing us a preliminary code for solving a one-dimensional model problem by RKPM.

We choose the differential equation−u″+ku=0in(0,1)for our calculations. The general solution of Eq. (6.1) can be expressed in terms of exponential functions and is therefore smooth. The differential equation (6.1)

Acknowledgements

We thank one of the referees for his constructive suggestions.

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    The work was supported by NSF under Grant DMS-9874015.

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