揭示并利用结构：高阶有限元方法的最佳代码生成
Exposing and exploiting structure: optimal code generation for high-order finite element methods
AI总结：
本文通过优化代码生成平台，揭示有限元结构，实现高阶有限元组装的最优算法复杂度，提出更强大的接口和抽象库及编译器算法。
摘要：
通过扩展Firedrake中的代码生成基础设施，我们引入了能够利用某些有限元固有结构的优化。这包括对立方体单元进行求和因子分解，适用于连续、不连续、H(div)和H(curl)符合元素。我们的实验确认了高阶有限元组装的最优算法复杂度。这通过几个新的贡献实现：更强大的形式编译器与提供有限元的库之间的接口；一个更抽象、更智能的有限元库FInAT，明确传达元素的结构；以及形式编译器算法，以自动利用暴露的结构。
英文摘要：
Code generation based software platforms, such as Firedrake, have become popular tools for developing complicated finite element discretisations of partial differential equations. We extended the code generation infrastructure in Firedrake with optimisations that can exploit the structure inherent to some finite elements. This includes sum factorisation on cuboid cells for continuous, discontinuous, H(div) and H(curl) conforming elements. Our experiments confirm optimal algorithmic complexity for high-order finite element assembly. This is achieved through several novel contributions: the introduction of a more powerful interface between the form compiler and the library providing the finite elements; a more abstract, smarter library of finite elements called FInAT that explicitly communicates the structure of elements; and form compiler algorithms to automatically exploit this exposed structure.
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