The Finite Element Method for Hyperbolic Equations

Abstract

In this chapter we apply the finite element method to hyperbolic equations. In Sect. 13.1 we study an initial-boundary value problem for the wave equation, and discuss semidiscrete and completely discrete schemes based on the standard finite element discretization in the spatial variables. In Sect. 13.2 we consider a scalar partial differential equation of first order in two independent variables. We begin by treating the equation as an evolution equation and show a nonoptimal order O(h) error estimate for the standard Galerkin method. Looking instead of the associated boundary value problem as a two-dimensional problem of the type treated in Sect. 11.3, we introduce the streamline diffusion modification and demonstrate a O(h^3/2) convergence estimate. We finally return to the evolution aspect and combine streamline diffusion with the so-called discontinuous Galerkin method to design a time stepping scheme by using two-dimensional approximating functions which may be discontinuous at the time levels.