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 Lagragian--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.

翻译：对于输运现象的有限元逼近，通常需要应用某种形式的限制手段来确保离散解保持良好性态并满足物理约束。然而，这些限制过程通常在离散节点位置执行，当需要在任意位置（例如自适应网格加密、任意拉格朗日-欧拉求解器中的重映射、重叠网格等）处评估解时，不足以保证格式的鲁棒性。本文提出了一种针对间断伽辽金方法的新型限制方法，该方法能够确保对于任意基函数、逼近阶数和网格单元类型，解始终是连续保界的（即在整个解多项式上成立）。通过对约束泛函进行修正形式化表述，所提出的方法每个单元仅需求解单个空间标量极小化问题，并针对该问题提出了一种高效数值优化算法。通过在高阶非结构间断伽辽金离散格式中强制执行连续约束（涵盖带有最大模原理保持约束的标量输运到带有正值保持约束的可压缩气体动力学），数值实验验证了该方法的有效性。