  This paper addresses the design of linear and nonlinear stabilization
procedures for high-order continuous Galerkin (CG) finite element
discretizations of scalar conservation laws. We prove that the standard CG
method is entropy conservative for the square entropy. In general, the rate of
entropy production/dissipation depends on the residual of the governing
equation and on the accuracy of the finite element approximation to the entropy
variable. The inclusion of linear high-order stabilization generates an
additional source/sink in the entropy budget equation. To balance the amount of
entropy production in each cell, we construct entropy-dissipative element
contributions using a coercive bilinear form and a parameter-free entropy
viscosity coefficient. The entropy stabilization term is high-order consistent,
and optimal convergence behavior is achieved in practice. To enforce
preservation of local bounds in addition to entropy stability, we use the
Bernstein basis representation of the finite element solution and a new subcell
flux limiting procedure. The underlying inequality constraints ensure the
validity of localized entropy conditions and local maximum principles. The
benefits of the proposed modifications are illustrated by numerical results for
linear and nonlinear test problems.
