  This work extends the concepts of algebraic flux correction and convex
limiting to continuous high-order Bernstein finite element discretizations of
scalar hyperbolic problems. Using an array of adjustable diffusive fluxes, the
standard Galerkin approximation is transformed into a nonlinear high-resolution
scheme which has the compact sparsity pattern of the piecewise-linear or
multilinear subcell discretization. The representation of this scheme in terms
of invariant domain preserving states makes it possible to prove the validity
of local discrete maximum principles under CFL-like conditions. In contrast to
predictor-corrector approaches based on the flux-corrected transport
methodology, the proposed flux limiting strategy is monolithic; i.e., limited
antidiffusive terms are incorporated into the well-defined residual of a
nonlinear (semi-)discrete problem. A stabilized high-order Galerkin
discretization is recovered if no limiting is performed. In the limited
version, the compact stencil property prevents direct mass exchange between
nodes that are not nearest neighbors. A formal proof of sparsity is provided
for simplicial and box elements. The involved element contributions can be
calculated efficiently making use of matrix-free algorithms and precomputed
element matrices of the reference element. Numerical studies for $\mathbb{Q}_2$
discretizations of linear and nonlinear two-dimensional test problems
illustrate the virtues of monolithic convex limiting based on subcell flux
decompositions.
