会议摘要（Abstract）

在计算科学中，一个核心理念是可靠和高效的数值方法应当保持连续问题的关键数学/物理结构。例如，在电磁学中，在离散层面上保持 de Rham 复形结构对构造稳定且高效的算法发挥了基本的作用；这一原则构成了离散微分形式和有限元外微分（Finite Element Exterior Calculus, FEEC）理论的基础。不同问题涉及不同微分结构。离散微分形式的成功为张量场和其他几何对象的离散化等具有挑战性的问题提供了新的见解. 这些问题包括来自广义相对论的爱因斯坦方程离散化以及具有微观结构的连续体模型和算法等领域的重要的应用和需求。本次研讨会的目标是探讨保结构离散化，重点拓展有限元张量微积分的一般框架，旨在促进数值分析、几何力学、离散拓扑和微分几何、网络理论及离散物理等主题之间的联系，将围绕发掘这些联系，从数学结构、有限元理论与算法，以及数值软件和应用的角度进行探讨。同时，研讨会将汇聚从保结构有限元方法领域的领先专家到早期职业研究人员等在内的广泛参与，通过短课程、综述讲座、一般性报告和自由讨论等形式，凝练研究方向，促进新的合作。

In computational sciences, a philosophy is that reliable and efficient numerical methods should preserve the key mathematical and physical structures of continuous problems. For instance, in electromagnetism, preserving the underlying de Rham complex at the discrete level leads to stable and efficient algorithms. This principle guided the development of discrete differential forms and Finite Element Exterior Calculus (FEEC). Different problems involve different differential structures. The success of discrete differential forms offers new insights into the challenging task of discretizing tensor fields and other geometric objects. Crucial applications and needs emerge from areas such as the discretization of the Einstein equations in general relativity and models and algorithms for continua with microstructures.

The goal of this workshop is to explore structure-preserving discretization, with a focus on developing a more general picture of Finite Element Tensor Calculus. We aim to foster connections between topics such as numerical analysis, geometric mechanics, discrete topology and differential geometry, network theories, and discrete physics. These topics will be addressed from the perspectives of mathematical structures, finite element theory and algorithms, and numerical software and applications. The workshop will feature contributions from both leading experts and early career researchers in the field of structure-preserving finite element methods and beyond. The program will include short courses, survey and research talks, as well as free discussions to foster new collaborations.

举办意义(Description of the aim)

本次会议的主要目标是研讨离散张量场和几何对象的基本问题，并开发可靠的数值方法，推动来自相对论和连续体力学的方程的高效求解。

会议的具体主题包括：

- 回答“如何用有限元方法离散化高阶张量等几何对象”的基本问题。

- 通过希尔伯特复形，建立同调代数、分析、离散微分几何和数值分析之间的新联系。

- 着眼于下一代引力波探测器所需的精度和稳定性，开发求解爱因斯坦方程的新型保结构有限元方法。

- 发展统一的框架和方法，用于建模、分析和计算具有微观结构的连续体。

The primary aim of the conference is to address the fundamental problem of discretising tensor fields and geometric objects and develop reliable and numerical methods for solving equations from relativity and continuum mechanics.

 Particular topics of the conference include

- Answering the fundamental question of “how to discretise geometric objects like high-order tensors” with the finite element methodology.

- Developing novel connections between homological algebra, analysis, discrete differential geometry, and numerical analysis via Hilbert complexes.

- Developing structure-preserving finite element approaches for solving the Einstein equations, aiming at the precision and stability required by the next generation of gravitational-wave detectors.

Developing a unified framework and methods for modelling, analysis, and computation of continua with microstructure.