Finite Element Methods for Unsteady Problems

In this chapter we deal with the discretization of second-order initial-boundary value problems of parabolic and hyperbolic types. Unlike Chapter 2, the space discretization is handled by finite element methods. We shall begin with some basic properties that are used later to derive appropriate weak formulations of the problems considered. Then, moving on to numerical methods, various temporal discretizations are discussed for parabolic problems: one-step methods (Runge-Kutta), linear multi-step methods (BDF) and the discontinuous Galerkin method. An examination of similar topics for second-order hyperbolic problems follows. The initial value problems that are generated by semi-discretization methods are very stiff, which demands a careful choice of the time integration codes for such problems. Finally, at the end of the chapter some results on error control are given.