Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Over the past two decades, meshfree methods have undergone significant development as a numerical tool to solve partial differential equations (PDEs). In contrast to finite elements, the basis functions in meshfree methods are smooth (nonpolynomial functions), and they do not rely on an underlying mesh structure for their construction. These features render meshfree methods to be particularly appealing for higher-order PDEs and for large deformation simulations of solid continua. However, a deficiency that still persists in meshfree Galerkin methods is the inaccuracies in numerical integration, which affects the consistency and stability of the method. Several previous contributions have tackled the issue of integration errors with an eye on consistency, but without explicitly ensuring stability. In this paper, we draw on the recently proposed virtual element method, to present a formulation that guarantees both the consistency and stability of the approximate bilinear form. We adopt maximum-entropy meshfree basis functions, but other meshfree basis functions can also be used within this framework. Numerical results for several two-dimensional and three-dimensional elliptic (Poisson and linear elastostatic) boundary-value problems that demonstrate the effectiveness of the proposed formulation are presented.