Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions

Abstract

We consider numerical solutions of second-order elliptic partial differential 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 affects approximation at a particular point are used, 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 affected by the discontinuities. The convergence of the discontinuous approximation is first 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.