Some numerical simulations of multi-scale physical phenomena consume a significant amount of computational resources, since their domains are discretized on high resolution meshes. An enormous wastage of these resources occurs in refinement of sections of the domain where computation of the solution does not require high resolutions. This problem is effectively addressed by adaptive mesh refinement (AMR), a technique of local refinement of a mesh only in sections where needed, thus allowing concentration of effort where it is required. Sections of the domain needing high resolution are generally determined by means of a criterion which may vary depending on the nature of the problem. Fairly straightforward criteria could include comparing the solution to a threshold or the gradient of a solution, that is, its local rate of change to a threshold. While the former criterion is not particularly rigorous and hardly ever represents a physical phenomenon of interest, it is simple to implement. However, the gradient criterion is not as simple to implement as a direct comparison of values, but it is still quick and a good indicator of the effectiveness of the AMR technique. The objective of this thesis is to arrive at an adaptive mesh refinement algorithm for a finite difference scheme using a quadtree decomposition approach. In the AMR algorithm developed, a mesh of increasingly fine resolution permits high resolution computation in sub-domains of interest and low resolution in others. In this thesis work, the gradient of the solution has been considered as the criterion determining the regions of the domain needing refinement. Initial tests using the AMR algorithm demonstrate that the paradigm adopted has considerable promise for a variety of research problems. The tests performed thus far depict that the quantity of computational resources consumed is significantly less while maintaining the quality of the solution. Analysis included comparison of results obtained with analytical solutions for four test problems, as well as a thorough study of a contemporary problem in solid mechanics.