In this thesis, the Galerkin method is used for solving PDEs, with the aid of RKPM meshless method for approximations. Penalty method is used for enforcing Dirichlet boundary conditions. The time and memory usage are tried to be improved in different problems using programming techniques, e.g. vectorization, using kd-tree for nearest neighbour search, using Gauss elimination method for matrix inversion, and choosing the correct kinds of functions to define, to name a few. The results obtained from solving 2D and 3D elliptic problems, e.g. for Laplace and elasticity problems, indicate the high accuracy and speed of the peresented techniques in solving equations. It is also possible to prevent mesh-locking in the use of the penalty method.
