标题： 抽象与混合域有限元方法的自动化算法 显示英文标题

标题： Abstractions and automated algorithms for mixed domain finite element methods

摘要： 混合维偏微分方程（PDEs）是耦合在不同拓扑维度域上定义的未知场的方程。 这类方程在地质学、生理学、生物学和断裂力学等多个科学领域中自然出现。 在低维子空间上施加非标准条件时，例如通过拉格朗日乘数，也会经常遇到混合维PDEs。 在本文中，我们提出了用于混合域和混合维PDEs（共维数最多为一，即nD-mD且|n-m| <= 1）的有限元离散化的一般抽象和算法。 我们引入了高层次的数学软件抽象以及用于表达和高效求解此类耦合系统的低层次算法。 此处介绍的概念也已在FEniCS有限元软件的背景下实现。 我们通过一系列示例展示了新功能，包括一个约束的泊松问题、一组斯托克斯型流动模型以及一个离子电扩散模型。 显示英文摘要

摘要： Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology, physiology, biology and fracture mechanics. Mixed dimensional PDEs are also commonly encountered when imposing non-standard conditions over a subspace of lower dimension e.g. through a Lagrange multiplier. In this paper, we present general abstractions and algorithms for finite element discretizations of mixed domain and mixed dimensional PDEs of co-dimension up to one (i.e. nD-mD with |n-m| <= 1). We introduce high level mathematical software abstractions together with lower level algorithms for expressing and efficiently solving such coupled systems. The concepts introduced here have also been implemented in the context of the FEniCS finite element software. We illustrate the new features through a range of examples, including a constrained Poisson problem, a set of Stokes-type flow models and a model for ionic electrodiffusion.