Abstract
The continuous-time Galerkin method is studied for the equation $u_t + u_x = 0$ with periodic solution. If the space of possible approximate solutions is taken to be $C^1 $ piecewise cubic polynomials on mesh of size h, then the $L^2 $-norm of the error is in general no better than $ch^3 $; if the class of possible approximate solutions is taken to be $C^2 $ piecewise cubic polynomials on this mesh, the error is bounded by $ch^4 $ for sufficiently smooth solutions.
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