Abstract This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems.

中文翻译：

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Bernstein 有限元的无矩阵子单元残差分布：单片限制

摘要 本文重点讨论对流问题高阶有限元逼近的残差分布 (RD) 方案中的限制方面。在这项工作中考虑了连续和不连续伽辽金方法。使用对全局非线性系统的元素贡献的代数操作来执行离散最大值原则。无需计算元素矩阵并组装它们的全局对应项，即可执行所需的修改。与标准 Galerkin 离散化相关联的元素向量的分量使用局部子单元权重直接操作，以实现最佳精度。这种低阶非线性 RD 方案最初是为了计算通量校正输运 (FCT) 算法的局部极值递减预测因子而开发的。在本文中，我们将限制器直接纳入残差分布程序，使其适用于平稳问题并导致适定的非线性离散问题。为了规避二阶精度障碍，使用基于二阶导数的平滑度传感器来调整单片限制方法和 FCT 方案的校正因子。所提出方法的收敛行为通过二维测试问题的数值研究来说明。使用基于二阶导数的平滑度传感器调整整体限制方法和 FCT 方案的校正因子。所提出方法的收敛行为通过二维测试问题的数值研究来说明。使用基于二阶导数的平滑度传感器调整整体限制方法和 FCT 方案的校正因子。所提出方法的收敛行为通过二维测试问题的数值研究来说明。