An substantial part of modeling and simulation of microelectronic phenomena involves finding a solution to partial differential equations (PDEs). Solving PDEs is an essential step in the simulation of semiconductor processing, device performance, and reliability. A general solution to PDEs is usually unknown and the few known cases are only available for simple problems under very confined conditions. However, engineering problems are diversified and simplified solutions can spoil the analysis. With the advent of the electronic computer, numerical methods with high quality approximations could be applied enabling physical simulations as a helpful tool in engineering analysis and design. In this chapter the Finite Element Method (FEM) is briefly presented as the technique to solve the PDEs. First, the mathematical background of the method is discussed, then its application is analyzed with a general elastic problem. In the last section, numerical schemes for time-dependent problems are treated, closing the set of techniques used within this work.