Book excerpt: In an effort to improve the error analysis of numerical methods for time-dependent PDEs and obtain reasonable error estimates, Sun developed the concept of numerical smoothness in [29] and [30]. In this dissertation, we prepare the framework for applying numerical smoothness to the error analysis for parabolic equations. The Discontinuous Galerkin (DG) method for solving parabolic equations is considered to be a successful scheme, but the error analysis for the method is limited. To provide the framework, we focus on a class of primal DG methods, namely variations of interior penalty methods. The numerical smoothness technique is used to perform an error analysis for a method in this class known as the Symmetric Interior Penalty Galerkin (SIPG) method. We take our model problem to be the one dimensional heat equation with Dirichlet boundary conditions. Therefore, this work represents a first step in applying Sun's numerical smoothness technique to the error analysis of parabolic equations. Two examples are provided to show how our numerical smoothness indicators can be used. Concluding remarks discuss how this early stage may be expanded to more complex parabolic equations and other numerical schemes.