Generalization of Finite Element Concepts

In this chapter, the authors advance from parabolic problems to consider examples of the two main classes of partial differential equations (PDEs): elliptic and hyperbolic equations. They also consider a third (parabolic) problem, which is governed by a system of PDEs rather than a single (scalar) equation. Hyperbolic problems differ from both parabolic and elliptic equations in that disturbances propagate with finite time and that the solutions may propagate without decay or they may even be sharpening in time, forming what are known as shock waves. The authors demonstrate how the finite element method (FEM) is applied to solve different types of PDEs (in 2Ds), but also consider some new aspects including incorporation of Neumann (flux) boundary conditions, different element types, and coupling. These phenomena create a number of challenges for numerical methods. Three complete Matlab scripts are listed to demonstrate how these features are incorporated in practice.