Modeling of phenomena of physics and engineering will lead to partial differential equations. One of these mathematical models are elliptic interface problems. Many important physical and industrial applications involve mathematical models with very complicated structure that is characterized by free moving surfaces inside the computational domain and by discontinuous or even singular material properties. Examples include composite materials, multi-phase flows, crystal growth, solidification and many others. This type of problems, which we refer to as interface problems, has attracted a lot of attention over the past years. The general form of the equation is as follows: ? ( ?u x ) x + ru = f c + ?? ? , x ? ? = (0 , 1 ) , u (0) = u (1) = 0 With the following jump condition in on interface point ?, [ u ] ? = 0 , [ ?u x ] ? = ? ?. Where ? and r are a smooth functions on both ? - = (0 , ? ) and ? + = ( ?,1 ), . f c ? L 2 (?) , ? ? R and ? ? is a delta functional with the support on the interface ? . We will also assume that ? ± and r ± are sufficiently smooth, i.e. ? ± , r ± ? C 4 (? ± ). Due to presence of these interfaces the problem will contain discontinuities in the coefficients and singularities in the right hand side that are represented by delta functional with the support on the interfaces. As a result, the solution to the interface problem and its derivatives may have jump discontinuities. The proposed method is specifically designed to handle this features of the solution using non-body fitted grids, i.e. the grids are not aligned with the interfaces. In this thesis, two family of method are used for solving the problem. In section 2, the well-known second order immersed interface method is implement as a finite difference method. Based on the [8] the method is designed acording to the modification of the truncation error around the interfaces. Convergence analysis of the method is presented by using a maximum principle method. In section 3, a second order finite element method with linear basis is considerd [10]. To overcome the jump condition, linear basis are modified. Numerical results are presented and efficiency of the method is shown. In [11] , a fourth order finite element is introduction. The method is based on the use of the hermitian basis function. This method is not efficient for solving the problem since value of the solution and its derivative are needed for implementing the method.