Preserving positivity for hyperbolic PDEs using variable-order finite elements with bounded polynomials

The positivity preserving approach of Berzins is generalized by using a derivation based on bounded polynomial approximations and order selection. The approach is extended from the B-spline based methods used previously to the use of more conventional continuous Galerkin elements. The conditions relating to positivity preservation are considered and a numerical example used to demonstrate the performance of the method on a model advection equation problem.