Element-Free Galerkin Method: Convergence of the continuous and discontinuous shape functions

We consider numerical solutions of second-order elliptic partial di erential equations, such as Laplace's equation, or linear elasticity, in two-dimensional, non-convex domains by the element-free Galerkin method (EFG). This is a meshless method, in which the shape functions are constructed by using weight functions of compact support. For non-convex domains, two approaches to the determination of whether a node a ects approximation at a particular point, a contained path criterion, and the visibility criterion. We show that for non-convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions. The resulting approximation is no longer conforming, and its convergence must be established by inspection of the so-called consistency term. We show that the variant of the element-free Galerkin method which uses the discontinuous shape functions is convergent, and that, in the practically important case of linear shape functions, the convergence rate is not a ected by the discontinuities. The convergence of the discontinuous approximation is rst established by the classical and generalized patch test. As these tests do not provide an estimate of the convergence rate, the rate of convergence in the energy norm is examined, for both the continuous and discontinuous EFG shape functions and for smooth and non-smooth solutions by a direct inspection of the error terms. Introduction The element-free Galerkin method (EFG) is one of the so-called meshless methods. Meshless methods have been proposed in several varieties (see, e.g., an overview in Duarte [1]) as Generalized Finite Di erence Method (Liszka and Orkisz [2]), Smoothed Particle Hydrodynamics (Monaghan [3]), Di use Element Method (Nayroles et al. [4]), Multiquadrics (Kansa [5, 6]), the Element-Free Galerkin Method (Belytschko et al. [7]), Wavelet Galerkin Method (e.g. Qian and Weiss [8]), hp-clouds (Duarte and Oden [9]), Reproducing Kernel Particle Methods (Liu et al. [10]), and the Partition of Unity FEM (Babu ska and Melenk [11, 12]). Meshless methods are a rather interesting complement to traditional nite element techniques. It is possible to (i) construct arbitrarily high order approximation even for di cult fourth-order problems such as Kirchho -Love shells (see Krysl and Belytschko [13]), and (ii) the numerical integrations can be performed on an arbitrary covering of the domain so that expensive (re)meshing can be avoided; see, e.g., Belytschko et al. [7] for the use of background cells, and Krysl and Belytschko [14] for a discussion of the background mesh. In meshless Research Associate, Civil Engineering, Northwestern University, Evanston, IL, USA. Walter P. Murphy Professor of Civil and Mechanical Engineering, Northwestern University, Evanston, IL, USA.