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“结构保持离散化：有限元，样条和等几何分析”会议提供参与者支持。结构保持离散化是解决一些科学和工程领域中出现的物理模型的计算范式，如计算流体力学、结构力学和宇宙学。这类方法产生忠实的近似，具有几个理想的特性，包括长时间的稳定性和准确性，守恒定律的精确执行（例如，质量，能量，动量），相对于模型参数增强的稳定性，没有数值伪影，减少计算误差。总的来说，这些算法产生了高保真的计算模拟，仍然忠于底层模型的物理特性。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，并讨论结构保持离散化领域的最新发展。这将使该领域的各种观点相互交融，从而更好地理解这些方法，并开发新的算法和理论结果。会议的一个焦点是有限元外部演算（FEEC）框架，这是一类强大的结构保持离散化，它在微分形式的演算中制定了有限元方法。这种方法的一个关键特点是结合了同调代数和泛函分析的工具来开发正则de Rham复合体的有限维子复合体。虽然FEEC框架已成功地应用于具有最小平滑度的de Rham复合体，但最近的进展已将该方法扩展到高阶Sobolev空间，即具有更大平滑度的空间。在FEEC框架下，高阶Sobolev空间的拟合有限元空间的扩展需要使用具有高正则性的分段多项式空间，即光滑多元样条。这是一个被广泛研究和活跃的研究领域，但光滑多项式样条的理论和构造，甚至语言对于有限元分析的研究者来说都是相对陌生的。会议将汇集在有限元分析、多变量样条、等几何分析和代数几何方面的研究人员，以协作和交流当前的趋势，并就共同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，并向研究生和早期职业研究人员展示有限元分析，多元样条理论，等几何分析和应用代数几何的交集。更多的细节可在https://sites.google.com/view/spd2019/home.This上获得，该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.