Optimization of meshless galerkin method and applications with some new adaptive algoritms

In this study, among the family of Meshless Methods, we consider Element Free Galerkin Method for solving differential equations. In Element Free Galerkin Method, shape functions are obtained using Moving Least Square Method. Unlike the Finite Element Methods, Meshless Methods do not require construction of a mesh, thus enable us to determine approximate solution using less computational effort. In the first Section, some new properties of shape functions including translation invariance property are derived and these properties are used to obtain shape functions on a given computational domain with considerably less number of floating point operations. This approach also has led to construction of associated algebraic system with less computation work as compared to conventional practice. Furthermore, adaptive algorithms are proposed for one dimensional stationary problems to have better accuracy in approximate solution. In Section two, the adaptive algorithms are generalized to time-dependent linear and nonlinear problems. In Section three, we show that the translation invariance property of shape functions also hold in two dimensional domains and that this property leads to considerable computational savings. Furthermore, a Hessian based adaptive strategy is proposed and implemented for some problems on two dimensional domains. The results show that solutions obtained by adaptive algorithms are much more accurate, however, as the number of steps in adaptivity increases, the computational load also increases which then may require appropriate parallel algorithms for large size problems.