  For finite element approximations of transport phenomena, it is often
necessary to apply a form of limiting to ensure that the discrete solution
remains well-behaved and satisfies physical constraints. However, these
limiting procedures are typically performed at discrete nodal locations, which
is not sufficient to ensure the robustness of the scheme when the solution must
be evaluated at arbitrary locations (e.g., for adaptive mesh refinement,
remapping in arbitrary Lagrangian--Eulerian solvers, overset meshes, etc.). In
this work, a novel limiting approach for discontinuous Galerkin methods is
presented which ensures that the solution is continuously bounds-preserving
(i.e., across the entire solution polynomial) for any arbitrary choice of
basis, approximation order, and mesh element type. Through a modified
formulation for the constraint functionals, the proposed approach requires only
the solution of a single spatial scalar minimization problem per element for
which a highly efficient numerical optimization procedure is presented. The
efficacy of this approach is shown in numerical experiments by enforcing
continuous constraints in high-order unstructured discontinuous Galerkin
discretizations of hyperbolic conservation laws, ranging from scalar transport
with maximum principle preserving constraints to compressible gas dynamics with
positivity-preserving constraints.
