标题： 一种用于限制高阶有限元函数的方法：网格有效性应用和保界限制器 显示英文标题

标题： A method for bounding high-order finite element functions: Applications to mesh validity and bounds-preserving limiters

摘要： 我们介绍了一种新颖的方法，用于在有限元逼近中约束高阶多维多项式。该方法涉及预先计算多项式基函数的最佳分段线性边界框，然后可以将其用于局部约束这些基函数的任意组合。此方法可应用于任何元素/基类型以及任何逼近阶数，能够以所需的精度提供局部（即亚单元）极值约束，并且可以在模拟中高效地实时评估。此外，我们表明，与基于凸包属性的传统方法（例如伯恩斯坦多项式）相比，这种方法通常能提供更准确的约束。该技术的有效性已在一些应用中得到证明，例如网格有效性检查和高阶曲线网格优化，在这些应用中，可以确保整个单元内元素雅可比行列式的正性，并且可以对双曲系统施加在整个解多项式上都有效的最大值原理约束器。 显示英文摘要

摘要： We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be used to locally bound any combination of these basis functions. This approach can be applied to any element/basis type at any approximation order, can provide local (i.e., subcell) extremum bounds to a desired level of accuracy, and can be evaluated efficiently on-the-fly in simulations. Furthermore, we show that this approach generally yields more accurate bounds in comparison to traditional methods based on convex hull properties (e.g., Bernstein polynomials). The efficacy of this technique is shown in applications such as mesh validity checks and optimization for high-order curved meshes, where positivity of the element Jacobian determinant can be ensured throughout the entire element, and continuously bounds-preserving limiters for hyperbolic systems, which can enforce maximum principle bounds across the entire solution polynomial.