Structure-Preserving Discretizations: Finite Elements, Splines, and Isogeometric Analysis

该奖项为将于2019年5月31日至6月1日在匹兹堡大学举行的“结构保全离散化：有限元、样条和等距分析”会议提供支持。结构保持离散是计算流体力学、结构力学和宇宙学等多个科学和工程领域中产生的物理模型的计算范例。这类方法产生了忠实的近似，具有几个理想的特性，包括长期稳定性和精度，严格执行守恒定律(例如，质量、能量、动量)，关于模型参数的增强的稳定性特性，没有数值伪影，以及减少计算误差。总而言之，这些算法产生了高保真的计算模拟，保持了基本模型的物理特性。会议的目标是将具有不同研究背景的数学家和工程师聚集在一起，进行互动、交流、合作和讨论结构保持离散化领域的最新发展。这将使这一领域的不同观点相互交融，导致对这些方法的更好理解，以及新算法和理论结果的发展。会议的焦点之一是有限元外部演算(FEEC)框架，这是一类强大的结构保持离散，它在微分形式演算中形成了有限元方法。这种方法的一个主要特点是结合了同调代数和泛函分析的工具来开发典型De Rham复形的有限维子复形。虽然FEEC框架已经成功地应用于具有最小光滑性的De Rham复形，但最近的进展已将该方法扩展到高阶Sobolev空间，即具有更高光滑性的空间。高阶Sobolev空间的协调有限元空间在FEEC框架下的扩张需要使用具有高正则性的分段多项式空间，即光滑的多元样条。这是一个被广泛研究和活跃的领域，但在有限元分析中，光滑多项式样条曲线的理论和构造，甚至语言都相对鲜为人知。这次会议将汇聚从事有限元分析、多元样条、等距分析和代数几何的研究人员，以协作和交流当前的趋势，并就常见问题分享不同的观点。会议还将定义和讨论这些不同子领域的关键开放问题，并让研究生和早期职业研究人员接触到有限元分析、多元样条理论、等距分析和应用代数几何的交叉。更多细节请参见https://sites.google.com/view/spd2019/home.This奖项，该奖项反映了国家科学基金会的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award provides participants support to the conference "Structure-Preserving Discretizations: Finite Elements, Splines, and Isogeometric Analysis", to be held at University of Pittsburgh on May 31 - June 1, 2019. Structure preserving discretizations are computational paradigms to solve physical models arising in several scientific and engineering fields such as computational fluid dynamics, structural mechanics, and cosmology. This class of methods produce faithful approximations with several desirable properties including long-time stability and accuracy, the exact enforcement of conservation laws (e.g., mass, energy, momentum), enhanced stability properties with respect to model parameters, absence of numerical artifacts, and reduced computational errors. Altogether, these algorithms produce high-fidelity computational simulations that remain true to the physics of the underlying models. The goal of the conference is to bring together mathematicians and engineers with diverse research backgrounds to interact, communicate, collaborate, and discuss recent developments in the field of structure preserving discretizations. This will allow cross-fertilization of various viewpoints in this field, lead to better understandings of these methods, and the development of novel algorithms and theoretical results.One focus of the conference is the finite element exterior calculus (FEEC) framework, a powerful class of structure preserving discretizations that formulates finite element methods in the calculus of differential forms. A key feature of this approach is to combine tools from homological algebra and functional analysis to develop finite dimensional subcomplexes of the canonical de Rham complex. While the FEEC framework has been successfully applied to the de Rham complex with minimal smoothness, recent progress has extended this methodology to higher order Sobolev spaces, i.e., spaces with greater smoothness. The extension of conforming finite element spaces of high-order Sobolev spaces in the FEEC framework necessitates the use of piecewise polynomial spaces with high regularity, i.e., smooth multivariate splines. This is an extensively studied and active research area, but the theory and construction, and even the language of smooth polynomial splines is relatively unknown to researchers in finite element analysis. The conference will bring together researchers working in finite element analysis, multi-variate splines, isogeometric analysis, and algebraic geometry to collaborate and communicate current trends and to share diverse viewpoints on common problems. The conference will also define and discuss critical open problems in these different sub-fields, and expose graduate students and early career researchers to the intersection of finite element analysis, the theory of multivariate splines, isogeometric analysis, and applied algebraic geometry. More details are available at https://sites.google.com/view/spd2019/home.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.