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

该奖项为与会者提供支持会议“结构保持离散化：有限元，样条和等几何分析”，将于2019年5月31日至6月1日在匹兹堡大学举行。结构保持离散化是解决在几个科学和工程领域，如计算流体动力学，结构力学和宇宙学中出现的物理模型的计算范例。 这类方法产生具有几个期望性质的忠实近似，包括长时间稳定性和准确性，守恒定律的精确实施（例如，质量、能量、动量）、相对于模型参数的增强的稳定性属性、没有数值伪影以及减少的计算误差。 总之，这些算法产生高保真的计算模拟，保持真实的物理基础模型。 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。 这将允许在这个领域的各种观点的交叉施肥，导致这些方法的更好的理解，和新的算法和理论成果的发展。会议的焦点之一是有限元外演算（FEEC）框架，一个强大的一类结构保持离散化，制定有限元方法的微分形式的微积分。这种方法的一个关键特征是联合收割机工具从同调代数和功能分析开发有限维子复形的典型的德拉姆复杂。 虽然FEEC框架已经成功地应用于具有最小光滑度的de Rham复形，但最近的进展已经将该方法扩展到高阶Sobolev空间，即，空间更加平滑。 FEEC框架中高阶Sobolev空间的协调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广泛研究和活跃的研究领域，但理论和建设，甚至光滑多项式样条的语言是相对未知的研究人员在有限元分析。 会议将汇集有限元分析，多元样条，等几何分析和代数几何的研究人员，以合作和交流当前的趋势，并分享对常见问题的不同观点。会议还将定义和讨论这些不同子领域的关键开放问题，并使研究生和早期职业研究人员接触有限元分析，多元样条理论，等几何分析和应用代数几何的交叉。更多细节可在www.example.com上获得https://sites.google.com/view/spd2019/home.This奖项反映了NSF的法定使命，并被认为值得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来支持。

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.