In simulation sciences, it is desirable to capture the real-world problem features as accurately as possible. Methods popular for scientific simulations such as the finite element method (FEM) and finite volume method (FVM) use piecewise polynomials to approximate various characteristics of a problem, such as the concentration profile and the temperature distribution across the domain. Polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies in order to obtain accurate simulations. This often entails dealing with negative values for the concentration of chemicals, exceeding a percentage value over 100, and other such problems. We consider these inconsistencies in the context of partial differential equations (PDEs). We propose an innovative filter based on convex optimization to deal with the inconsistencies observed in polynomial-based simulations. In two or three spatial dimensions, additional complexities are involved in solving the problems related to structure preservation. We present the construction and application of a structure-preserving filter with a focus on multidimensional PDEs. Methods used such as the Barycentric interpolation for polynomial evaluation at arbitrary points in the domain and an optimized root-finder to identify points of interest improve the filter efficiency, usability, and robustness. Lastly, we present numerical experiments in 2D and 3D using discontinuous Galerkin formulation and demonstrate the filter's efficacy to preserve the desired structure. As a real-world application, implementation of the mathematical biology model involving platelet aggregation and blood coagulation has been reviewed and the issues around FEM implementation of the model are resolved by applying the proposed structure-preserving filter.

翻译：在模拟科学中，尽可能准确地捕捉现实世界问题的特征至关重要。科学模拟中常用的方法，如有限元法（FEM）和有限体积法（FVM），通过分段多项式来逼近问题的各种特性，例如域内的浓度分布和温度分布。然而，在捕捉复杂轮廓时，多项式容易产生伪影，如吉布斯振荡。为获得精确的模拟结果，必须应用高效且准确的方法来处理此类不一致性。这通常涉及处理化学物质浓度的负值、超过100%的百分比值以及其他类似问题。我们将在偏微分方程（PDEs）的背景下考虑这些不一致性。我们提出了一种基于凸优化的创新滤波器，用于处理基于多项式的模拟中出现的不一致性。在二维或三维空间中，解决与结构保持相关的问题涉及额外的复杂性。我们重点围绕多维PDEs，介绍了保持结构滤波器的构建与应用。所用方法，如用于在域内任意点进行多项式评估的重心插值，以及用于识别兴趣点的优化求根器，提高了滤波器的效率、可用性和鲁棒性。最后，我们使用间断伽辽金方法进行了二维和三维数值实验，展示了该滤波器保持所需结构的有效性。作为实际应用，我们审查了涉及血小板聚集和血液凝固的数学生物学模型的实现，并通过应用所提出的保持结构滤波器，解决了该模型在FEM实现中遇到的问题。