Analysis of a Continuous Finite Element Method for Hyperbolic Equations

A finite element method for hyperbolic equations is analyzed in the context of a first order linear problem in $R^2 $. The method is applicable over a triangulation of the domain, and produces a continuous piecewise polynomial approximation, which can be developed in an explicit fashion from triangle to triangle. In a sense, it extends the basic upwind difference scheme to higher order. The method is shown to be stable, and error estimates are obtained. For nth degree approximation, the errors in the approximate solution and its gradient are shown to be at least of order $h^{{{n + 1} / 4}} $ and $h^{{{n - 1} / 2}} $, respectively, assuming sufficient regularity in the solution.