In practical problems , usually solutions have properties like crack , discontinuity , singularity . Also , some of them change rapidly over part of the computational domain and slowly over other parts . In these problems , finding an acceptable solution is usually difficult and time consuming . The suggestion is using adaptive strategies that have become very popular . In this thesis , we focus on developing an adaptive technique for the element free Galerkin (EFG) method . This method is a meshless method and uses the moving least squares (MLS) approximation to construct shape functions. The adaptive techniques are powerful tools to simultaneously increase accuracy and reduce the cost of numerical computations . In fact , the aim is to compute a numerical solution in such a way that the error is of a prescribed accuracy and the number of degrees of freedom is as small as possible .