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.

这项工作将代数通量校正和凸极限的概念扩展到标量双曲线问题的连续高阶Bernstein有限元离散化。使用可调整的扩散通量阵列，将标准的Galerkin近似转换为非线性高分辨率方案，该方案具有分段线性或多线性子像元离散化的紧凑稀疏性模式。用不变域保留状态表示该方案，使得有可能证明类CFL条件下局部离散最大值原理的有效性。与基于通量校正的输运方法的预测器-校正器方法相比，所提出的通量限制策略是整体的，即 将有限的反扩散项合并到非线性（半）离散问题的明确定义的残差中。如果不执行限制，则将恢复稳定的高阶Galerkin离散化。在受限版本中，紧凑的模板属性可防止不是最近邻居的节点之间直接进行质量交换。为简单和框元素提供了稀疏性的正式证明。所涉及的元素贡献可以利用无矩阵算法和参考元素的预先计算的元素矩阵进行有效计算。数值研究 为简单和框元素提供了稀疏性的正式证明。所涉及的元素贡献可以利用无矩阵算法和参考元素的预先计算的元素矩阵进行有效计算。数值研究 为简单和框元素提供了稀疏性的正式证明。所涉及的元素贡献可以利用无矩阵算法和参考元素的预先计算的元素矩阵进行有效计算。数值研究${\mathbb{问}}_2$ 线性和非线性二维测试问题的离散化说明了基于子单元通量分解的整体凸限制的优点。