"It is well known that singularities occur when solving elliptic value problems with non-convex domains or when some part of the data or the coefficients of the PDE are not smooth. Such problems and correspondent singularities often arise in practice, for instance, in fracture mechanics, in the material science with heterogeneities, or when dealing with mixed boundary conditions. A great deal is known about the nature of the singularities, which arise in some of these problems. In this thesis, we consider the scalar transmission problems with straight interfaces and with cross points across coefficients and possibly on a non-convex region ($L$-shaped domain). It is known that only $H^{1+au}$ ($0 < au< 1$) regularity on the solution is obtained and therefore the use of finite element method with the piecewise linear continuous function space does not give optimal accuracy. In this thesis, we introduce a new algorithm which are second order accurate on the (weighted) $L_2$, first order accurate on the (weighted) $H_1$ norm and second order accurate for the Stress Intensive Factor (SIF). The new methods take advantage of Mortar techniques. The main feature of the proposed algorithms is that we use primal singular functions {it without} cutting-off functions. The old algorithms use cutting-off functions as a tool of satisfying boundary conditions. In algorithms proposed in this thesis, use instead Mortar finite element technique to match the boundary and interfaces conditions. In this thesis, we also consider non-matching meshes sizes for different coefficients. We note that a new Mortar Lagrange multiplier is required in order to obtain optimal consistence errors for transmission problems. The proposed algorithms are very appealing over other methods because they are very accurate, do not require complicated numerical quadratures or interpolations, it is simple to design PCGs, and it can be generalized to other PDEs and to higher order methods."