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.

中文翻译：

---

标量守恒定律的高阶连续伽辽金近似的熵守恒性质和熵镇定

本文讨论了标量守恒定律的高阶连续伽辽金 (CG) 有限元离散化的线性和非线性稳定程序的设计。我们证明标准 CG 方法对于平方熵是熵保守的。通常，熵产生/耗散的速率取决于控制方程的残差和熵变量的有限元逼近精度。包含线性高阶稳定性会在熵预算方程中产生一个额外的源/汇。为了平衡每个单元中的熵产生量，我们使用强制双线性形式和无参数熵粘度系数构建熵耗散元素贡献。熵稳定项是高阶一致的，并且在实践中实现了最佳收敛行为。为了在熵稳定性之外强制保留局部边界，我们使用有限元解的伯恩斯坦基表示和新的子单元通量限制程序。潜在的不等式约束确保了局部熵条件和局部最大值原理的有效性。线性和非线性测试问题的数值结果说明了所提出的修改的好处。