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

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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.