In the first part of this thesis a novel method is presented to solve problems with weak singularities in two-dimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other mesh-based or mesh-less approaches as the eXtended Finite Element Method (XFEM). The present research considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual integration approach. Developing the formulation in a mapped polar co-ordination, primary basis functions made of Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the well-known literature or solved by well-established techniques, will demonstrate the capability of the method in problems related to composite materials. In the second part, the Equilibrated Singular Basis Functions (EqSBFs) are used in the framework of the Finite Element Method (FEM) while approximately satisfy the PDE in heterogeneous media. EqSBFs are able to automatically reproduce the terms consistent with the singularity order in the vicinity of the singular point. The newly made bases are used as the complimentary part along with the polynomial bases of the FEM to construct a new set of shape functions in the elements adjacent to the singular point. The combination of the two basis sets is performed through ordinary Gauss integration inside each element. It is shown by the numerical results that the combined bases lead to the quality improvement of the solution function as well as its derivatives, especially in the vicinity of the singularity. Key words: Singularity, Heterogeneous, Harmonic, Elasticity, Equilibrated basis functions, Gauss integration, Finite Element Method.