Meshfree methods

Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years meshfree approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems. Applications in computational nanotechnology are still somewhat rare, but do exist in the literature. In this chapter we will focus on the mathematical foundation of meshfree methods, and the discussion of various computational techniques presently available for a successful implementation of meshfree methods. At the end of this review we mention some initial applications of meshfree methods to problems in computational nanotechnology, and hope that this introduction will serve as a motivation for others to apply meshfree methods to many other challenging problems in computational nanotechnology