The Finite Element Method for Hyperbolic Equations

Part of the Texts in Applied Mathematics book series (TAM, volume 45)


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(h3/2) 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.


Finite Element Method Hyperbolic Equation Stability Estimate Interpolation Error Interpolation Operator 
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© Springer-Verlag Berlin Heidelberg 2009

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