混合域有限元法的抽象与自动算法
Abstractions and automated algorithms for mixed domain finite element methods
摘要：
混合维偏微分方程（PDE）是定义在不同拓扑维数的域上的耦合未知场的方程。这样的方程自然出现在广泛的科学领域，包括地质学，生理学，生物学和断裂力学。混合维偏微分方程也经常遇到的非标准条件施加在一个子空间的较低的维度，例如通过拉格朗日乘子。本文给出了混合域和余维不超过1的混合维偏微分方程（即nD-mD，|N-m| <= 1）。我们介绍了高层次的数学软件抽象与较低层次的算法，表达和有效地解决这种耦合系统。这里介绍的概念也已在FEniCS有限元软件的背景下实施。我们说明了新的功能，通过一系列的例子，包括约束泊松问题，一组斯托克斯型流动模型和离子电扩散模型。
英文摘要：
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.
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