In this chapter, we first introduce the differences between the finite element method (FEM) and the previously introduced FDM and FVM. Then, the focus will be placed on the Galerkin method for obtaining the weak form of the governing equation. Using thermomechanics as an example, the discretization of the weak form of the governing equations with Neumann boundary conditions into a local algebraic equation will be explained. The discretization of the thermal field and mechanical field, including their couplings, will be introduced separately. Two common types of 2D elements, more specifically, their shape functions, are given to help readers understand the discretization process. Other significant contents of the FEM, such as different element types, stiffness matrix ensembling for obtaining the global algebraic equation, and isoparametric elements, will not be detailed here as they can be easily found in many reference books for the FEM. Finally, MATLAB FEM code is given for solving the problem that has been solved in the FDM and FVM chapters.